Math. Ann. https://doi.org/10.1007/s00208-018-1700-1

Mathematische Annalen

Strong maximum principle for mean curvature operators on subRiemannian manifolds Jih-Hsin Cheng1 · Hung-Lin Chiu2 · Jenn-Fang Hwang1 · Paul Yang3

Received: 8 November 2016 / Revised: 8 March 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract We study the strong maximum principle for horizontal ( p-)mean curvature operator and p-(sub)Laplacian operator on subRiemannian manifolds including, in particular, Heisenberg groups and Heisenberg cylinders. Under a certain Hormander type condition on vector fields, we show the strong maximum principle holds in higher dimensions for two cases: (a) the touching point is nonsingular; (b) the touching point is an isolated singular point for one of comparison functions. For a background subRiemannian manifold with local symmetry of isometric translations, we have the strong maximum principle for associated graphs which include, among others, intrinsic graphs with constant horizontal ( p-)mean curvature. As applications, we show a rigidity result of horizontal ( p-)minimal hypersurfaces in any higher dimensional Heisenberg cylinder and a pseudo-halfspace theorem for any Heisenberg group. Mathematics Subject Classification Primary 35J70; Secondary 32V20 · 53A10

Communicated by F. C. Marques.

B

Paul Yang [email protected] Jih-Hsin Cheng [email protected] Hung-Lin Chiu [email protected] Jenn-Fang Hwang [email protected]

1

Institute of Mathematics, Academia Sinica, Taipei, Taiwan, ROC

2

Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan, ROC

3

Department of Mathematics, Princeton University, Princeton, NJ 08544, USA

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1 Introduction and statement of the results Hopf probably is the first to study the strong maximum principle (SMP in short) of elliptic operators in its generality ([14] of 1927 or Theorem 3.5 in [13]). For earlier results, under more restrictive hypotheses, see references in [19]. This principle has been extended to certain quasilinear elliptic operators of second order [13]. In 1969 Bony [4] studied, among others, the SMP for linear operators of Hörmander type including some known subelliptic operators. Bony’s SMP has been applied to study various geometric problems. See, for instance, Andrews’ work on noncollapsing in mean-convex mean curvature flow [2] or Brendle’s solution to the Lawson conjecture [5]. In this paper we first extend Bony’s SMP to the quasilinear case and then apply it to (generalized) mean curvature equations in subRiemannian geometry, including p-subLaplacian and usual horizontal ( p-)mean curvature. We consider quasilinear equations Q of second order: Qφ = a i j (x, Dφ)Di j φ + b(x, φ, Dφ)

(1.1)

where x = (x 1 , . . . , x m+1 ) is contained in a domain  of R m+1 , m ≥ 1. The coefficients a i j (x, p) (b(x, z, p), resp.) of Q are assumed to be defined and C ∞ smooth (for simplicity) for all values of (x, p) ((x, z, p), resp.) in the set × R m+1 ( × R× R m+1 , resp.). Let £(X 1 , . . . , X r ) denote the smallest C ∞ -module which contains C ∞ vector fields X 1 , . . . , X r on  and is closed under the Lie bracket (see [4] for precise definition). The following comparison principle is a straightforward application of Bony’s SMP (Theorem 3.1 in [4]) and an idea in the proof for Theorem 10.1 in the book of Gilbarg and Trudinger [13] (see the proof of Theorem A for more details in Sect. 3). Theorem A Let φ, ψ ∈ C ∞ () satisfy Qφ ≥ Qψ in . Assume (1) (a i j ) is nonnegative and a i j = a ji ; (2) ∂b ∂z ≤ 0; (3) there exist vector fields X 1 , . . . , X r and Y of class C ∞ (depending on Dφ(x)) such that a i j (x, Dφ(x))Di j =

r 

X k2 + Y.

k=1

Let  be an integral curve of a vector field Z ∈ £(X 1 , . . . , X r ). Suppose φ − ψ achieves a nonnegative maximum in  at a point of , then the maximum is attained at all points of . In Sect. 3, we give the moving frame version of Theorem A and condition (3) can be replaced by the constant rank condition (see Theorems A and Ã). In applications, we usually assume φ ≤ ψ and φ = ψ at a point p0 . Then we conclude φ ≡ ψ on a hypersurface  containing p0 if the Lie span £(X 1 , . . . , X r ) = C ∞ (, T ). We are going to apply Theorem à to generalized mean curvature equation or p-Laplacian Hφ, p with p ≥ 0.

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Let φ be a (C ∞ smooth, say) defining function (i.e., dφ = 0) on a subRiemannian manifold (M, ·, · ∗ ) of dimension m +1 (see Sect. 2) with |dφ|∗ := ( dφ, dφ ∗ )1/2 = 0 (note that at a point where ·, · ∗ is degenerate, we may have dφ = 0 while |dφ|∗ = 0). Let dv M be a background volume form on M, i.e., a given (m + 1)-form which is nowhere vanishing. With respect to ·, · ∗ we can then talk about the interior product of a 1-form ω with dv M : ωdv M being an m-form such that η∧(ωdv M ) = η, ω ∗ dv M for any 1-form η. We define a function Hφ, p on M with p ≥ 0 by the following formula:   dφ dv M = Hφ, p dv M . (1.2) d 1− p |dφ|∗ For p = 0, Hφ,0 , often denoted as Hφ , is called (Riemannian, subRiemannian, or horizontal) mean curvature while, for p > 0, Hφ, p is so called p-Laplacian or psubLaplacian. In Sect. 2, we give the variational description for Hφ, p . In this paper we mainly deal with the SMP near nonsingular points (where |dφ|∗ = 0, |dψ|∗ = 0). For the SMP near singular points, we only discuss the situation that the reference singular point is isolated for at least one comparison hypersurface. In general, the problem of the SMP near singular points is still open. Let φ and ψ be defining functions for hypersurfaces 1 and 2 in a subRiemannian manifold (M, ·, · ∗ ) of dimension m + 1, resp. (m ≥ 1). I.e., 1 (2 , resp.) is defined by φ = 0 (ψ = 0, resp.). Suppose 1 and 2 are tangent to each other at a point p0 where |dφ|∗ = 0, |dψ|∗ = 0. Define G : T ∗ M → T M by ω(G(η)) = ω, η ∗ for ω, η ∈ T ∗ M. Let ξ := Range(G). Throughout this paper we assume dim ξ = m + 1 − l

(1.3)

near p0 with l being a nonnegative integer unless stated otherwise. We call l the degree of degeneracy of M. Note that dim ker G = l. The following rank condition: rank(£(X 1 , . . . , X m−l )) = m

(1.4)

for any local sections X 1 , . . . , X m−l of ξ, which are independent wherever defined, is important. It means that any (m − l)-dimensional subspace of local sections of ξ can generate m -dimensional spaces. Let £(ξ ∩ T 1 ) = £(X 1 , . . . , X m−l ) where X 1 , . . . , X m−l form a basis of local sections of ξ ∩ T 1 near p0 . Similarly we can define £(ξ ) = £(X 1 , . . . , X m−l , X m+1−l ) where X 1 , . . . , X m−l , X m+1−l form a basis of local sections of ξ near p0 . Note that both £(ξ ∩ T 1 ) and £(ξ ) are independent of choice of a basis of local sections. Then, using Theorem A , we have

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Theorem B Suppose we are in the situation described above, in particular, |dφ|∗ = 0, |dψ|∗ = 0 at p0 . For p ≥ 0, Assume Hψ, p + b(x, ψ, Dψ) ≤ Hφ, p + b(x, φ, Dφ) where b is a C ∞ smooth function for all values of (x, z, ·) and satisfies Moreover, assume ψ ≥ φ near p0 , ψ = φ = 0 at p0 . We have

∂b ∂z

≤ 0.

(a) if we further assume the rank condition (1.4) holds near p0 , then ψ = φ = 0 on 1 near p0 . I.e., 2 coincides with 1 near p0 . (b) in the case of p > 0, if we further assume rank(£(ξ )) = m + 1, then ψ = φ near p0 . Next we want to show that in a certain situation the assumption in Theorem B can be achieved. Suppose we have a one-parameter family of diffeomorphisms a , a ∈ (−δ, δ) for small δ, say, in a small neighborhood U of p0 in M (where φ and ψ are defined), i.e., a = I d for a = 0 and a+b = a ◦ b wherever defined. Definition 1.1 Let (M, ·, · ∗ , dv M ) be a subRiemannian manifold with a background volume form dv M .(M, ·, · ∗ , dv M ) is said to have isometric translations (a ) near p0 ∈ M if there exists a one-parameter family of local diffeomorphisms a , a ∈ (−δ, δ) for small δ > 0, say, in a small neighborhood U of p0 , such that (a) (preserving ·, · ∗ ) a∗ (ω), a∗ (η) ∗ = ω, η ∗ for a ∈ (−δ, δ) and ω, η ∈ T ∗ U. (b) (preserving dv M )a∗ (dv M ) = dv M for a ∈ (−δ, δ). We say the defining function φ of a (local) hypersurface  passing through p0 is compatible with {a } or {a } is compatible with the defining function φ if φ(a (x)) = φ(x) − a

(1.5)

for x ∈  ∩ U (and hence x ∈ U ) and a ∈ (−δ, δ). {a } is said to be transversal to a hypersurface  of U if da (x)  ∈ / Tx   da a=0 for all x ∈ . We will show the existence of some special coordinates for a subRiemannian manifold having local isometric translations as shown below. ˆ Let (M, ·, · ∗ , dv M ) be an (m + 1)-dimensional subRiemannian manTheorem C ifold with a background volume form dv M . Suppose (M, ·, · ∗ , dv M ) has isometric translations a , a ∈ (−δ, δ) for δ > 0, near p0 ∈ M, transversal to a hypersurface  passing through p0 . Then we can find local coordinates x 1 , x 2 , . . . , x m+1 in a neighborhood V of p0 such that (1)  is described by x m+1 = 0 in V, p0 is the origin, x j ◦ a = x j at q ∈ V for any a such that a (q) ∈ V, 1 ≤ j ≤ m, and x m+1 ◦ a = x m+1 + a;

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(2) A(a (q)) = A(q) for any q ∈ V and any a such that a (q) ∈ V if we write dv M (q) = A(q)d x 1 ∧ · · · ∧ d x m+1 . Call this system of special coordinates above in Theorem Cˆ (a system of) translationisometric coordinates. Definition 1.2 Take a system of translation-isometric coordinates x 1 , x 2 , . . . , x m+1 ˆ A a graph is a graph described by as in Theorem C. (x 1 , x 2 , . . . , x m , u(x 1 , x 2 , . . . , x m )). For a a graph, we take the defining function φ = u(x 1 , x 2 , . . . x m ) − x m+1 which is compatible with {a }. For p ≥ 0, we define H p (u)(x 1 , . . . , x m ) := Hφ, p (x 1 , . . . , x m , u(x 1 , x 2 , . . . x m ))

(1.6)

at (x 1 , . . . , x m ) where |dφ|∗ = 0. Making use of translation-isometric coordinates, we have Theorem C , which is an application of Theorem B. Theorem C Suppose (M, ·, · ∗ , dv M ) of dimension m + 1 has isometric translations a near p0 ∈ M, transversal to a hypersurface  passing through p0 . Take a system of translation-isometric coordinates x 1 , x 2 , . . . , x m+1 in a neighborhood V of p0 such that x m+1 = 0 on . Suppose u(v, resp.):  ∩ V → R defines a graph {(x 1 , x 2 , . . . x m , u(x 1 , x 2 , . . . x m ))} ({(x 1 , x 2 , . . . x m , v(x 1 , x 2 , . . . x m ))}, resp.) ⊂ V such that |d(u − x m+1 )|∗ = 0 (|d(v − x m+1 )|∗ = 0, resp.). Assume (1) v ≥ u on  ∩ V and v(0, . . . , 0) = u(0, . . . , 0) = 0; (2) For some p ≥ 0, H p (v) ≤ H p (u) on  ∩ V. Moreover, assume the rank condition (1.4) holds near p0 . Then v ≡ u in a neighborhood of p0 ∈ . Let M be the Heisenberg group Hn considered as a pseudoHermitian manifold and hence a subRiemannian manifold (see Sect. 2 for detailed explanation). We have the Heisenberg translation in the direction ∂ ∂x 1 : la (x 1 , x 2 , . . . , x 2n−1 , x 2n , z) := (x 1 + a, x 2 , . . . , x 2n−1 , x 2n , z − ax n+1 ). (1.7) Applying our theory of translation-isometric coordinates to the situations of la graphs, as well as intrinsic graphs (recall [1] or see Sect. 4), we respectively obtain the corresponding SMP. See Corollary D and Corollary E below. Corollary D Suppose n ≥ 2. Let x 1 = u(η2 , η3 , . . . , η2n , τ ) and x 1 = v(η2 , η3 , . . . , η2n , τ ) be two la graphs defined on a common domain . Assume v = u at p0 ∈ , v ≥ u in , and H (v) ≤ H (u) in . Then v ≡ u near p0 . Corollary E Suppose n ≥ 2. Let v = v(η2 , η3 , . . . , η2n , τ ), u = u(η2 , η3 , . . . , η2n , τ ) be two (C ∞ smooth) intrinsic graphs defined on a neighborhood U of p0 =

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(η02 , η03 , . . . , η02n , τ0 ). Assume v = u at p0 where vηn+1 = 0 and u ηn+1 = 0. Suppose v ≥ u,  v

W ·

W v (v)

 1 + |W v (v)|2



 ≤W · u



W u (u)



1 + |W u (u)|2

in U, and either v or u has constant horizontal (or p-)mean curvature. Then v ≡ u near p0 . For notations in the above corollaries, see Sect. 4. For the operator W u , see the definition in the first line after (4.19). Notice that we have a more geometric description for this theory (Theorem C, Sect. 4), hence a more geometric version of Corollary D (Corollary D, Sect. 4). We remark that the horizontal mean curvature operator of an intrinsic graph u does not belong to the type (1.1) since the second order coefficients contain u itself. So Corollary E does not follow directly from the previous general theorems. However, an intrinsic graph is congruent with an la graph by a rotation (see the proof of Corollary E). So after the rotation, in the new coordinates, there is no longer the dependence of u in the second order coefficients of horizontal (p-)mean curvature. We can proceed with the application of Theorem C as follows. Let  be a (connected and open) domain of R m . Let u, v be two C ∞ smooth, real valued functions on . Let F be a C ∞ smooth vector field on . We remark that for some situations it is sufficient to have only finite differentiability, but for simplicity we work in C ∞ category (unless specified otherwise). Define the Legendrian (or horizontal) normal N F (u) (N F (v), resp.) of u (v, resp.) by N F (u) :=

∇u + F  |∇u + F|



(N F (v) := ∇v+ F , resp.) at points where ∇u + F = 0 (∇v + F = 0, resp.). Define |∇v+ F| the (generalized) horizontal (or p-)mean curvature H F (u) (H F (v), resp.) by H F (u) := div N F (u)

(1.8)

(H F (v) := div N F (v), resp.). We call a point p0 singular with respect to v if ∇v + F = 0 at p0 . Denote the set of all singular points with respect to v by S F (v). In a neighborhood U of a nonsingular point q0 ∈ \S F (v), let N1⊥ (v), N2⊥ (v), ⊥ (v) be an orthonormal basis of the space perpendicular to N (v). Let . . . , Nm−1 F ⊥ (v)) denote the smallest C ∞ -module which contains £(N1⊥ (v), N2⊥ (v), . . . , Nm−1 ⊥ (v), and is closed under the Lie bracket (see [4] for preN1⊥ (v), N2⊥ (v), . . . , Nm−1 ⊥ (v)) at a point q ∈ U cise definition). The rank of £(N1⊥ (v), N2⊥ (v), . . . , Nm−1 is the dimension of the vector space spanned by the vectors Z (q) for all Z ∈ ⊥ (v)). The following result is a special, but important case £(N1⊥ (v), N2⊥ (v), . . . , Nm−1 of Theorem C or Theorem C for degree of degeneracy l = 1.

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Theorem F Suppose m ≥ 3, H F (v) ≤ H F (u), v ≥ u in U ⊂ R m , which is a nonsingular domain for both v and u, and v = u at p0 ∈ U. Assume in U an orthonormal ⊥ (u) of the space perpendicular to N (u) exists and the basis N1⊥ (u), N2⊥ (u), . . . , Nm−1 F ⊥ ⊥ ⊥ (u)) is constant m (similar condition for N (v), rank of £(N1 (u), N2 (u), . . . , Nm−1 F resp.). Then we have v ≡ u in U. In Sect. 5, we show that the rank condition in Theorem F can be replaced by some easy to check conditions, which are formulated as Corollaries G and H. Corollary I is just the special situation of Theorem F for the standard Heisenberg groups. Finally, for the SMP near singular points, we turn to discuss the situation that the reference singular point is isolated.  = (G 1 , . . . , G m ) by  b for G Define G b

G :=

 m  k=1

a Gk , 1k

m 

a Gk , . . . , 2k

k=1

m 

 a

mk

Gk

k=1

where a jk s are real constants such that a jk + a k j = 0 for 1 ≤ j, k ≤ m. For m = 2n,  ∗ by define G  ∗ = (G 2 , −G 1 , G 4 , −G 3 , . . . , G 2n , −G 2n−1 ). G  b , with a 2 j−1,2 j = −a 2 j,2 j−1 = 1, 1 ≤ j ≤ n, a jk = 0 otherwise. ∗ = G Note that G When p0 is an isolated singular point of v, we still have the SMP. Theorem J Suppose m ≥ 3, v ≥ u in  ⊂ R m , such that  ∩ S F (u) = { p0 } ( ∩ S F (v) = { p0 }, resp.) and v = u at p0 . Suppose Hm−1 (S F (v)) = 0 (Hm−1 (S F (u)) = 0, resp.) and div F b > 0 (or div F b < 0). Assume H F (v) ≤ H F (u) in \{{ p0 } ∪ S F (v)} (\{{ p0 } ∪ S F (u)}, resp.) and for each point p ∈ \{ p0 }, there is a neighborhood U of p in which an orthonormal basis ⊥ (u) of the space perpendicular to N (u) exists and the N1⊥ (u), N2⊥ (u), . . . , Nm−1 F ⊥ ⊥ ⊥ (u)) is constant m (similar condition for N (v), rank of £(N1 (u), N2 (u), . . . , Nm−1 F resp.). Then we have v ≡ u in . In the proof of Theorem J, we need to apply the version (see Theorem C in Sect. 6) of the usual maximum principle (in the case of removable singularity) for H F , which is an extension of Theorem C  in [10]. We remark that the condition div F b > 0 or < 0 was first used to extend uniqueness results from even dimension to arbitrary dimension in [9]. In view of Corollary G (Corollary I, resp.), the condition on N F (v) or N F (u) in Theorem F and Theorem J can be replaced by (5.6) ( F = (−x 2 , x 1 , . . . , −x 2n , x 2n−1 ), resp. for m = 2n). Let Hn denote the Heisenberg group of dimension 2n + 1. As a set, Hn is C n × R or R 2n × R. For a hypersurface  in Hn (which may not be a graph over R 2n ), the horizontal (or p-)mean curvature H of  for a defining function ψ is given by H := divb

∇b ψ |∇b ψ|

(1.9)

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where ∇b and divb denote subgradient and subdivergence in Hn , resp. See Sect. 2 for equivalent definitions of mean curvature in subRiemannian geometry. Note that for a graph  over R 2n defined by u, H may be different from H F (u) ( F = (−x 2 , x 1 , . . . , −x 2n , x 2n−1 ) by sign. In fact, if we replace ψ by −ψ in (1.9) , H becomes −H . For the boundary  of a (C ∞ smooth) bounded domain  in Hn , we choose a defining function ψ for , such that ψ < 0 in . In this way H is a positive constant for a Pansu sphere given by the union of all the geodesics of positive constant curvature joining the two poles (see, e.g., [22] for the m = 2 case and [20] for the higher dimensional case). We have Theorem K Let 1 and 2 be two connected, orientable, closed hypersurfaces of constant horizontal ( p-)mean curvature H1 and H2 , resp. in Hn , n ≥ 2. Suppose 2 is inscribed in 1 , i.e., 2 is contained in the closure of the inside of 1 and 1 ∩ 2 is not empty. Assume H2 ≤ H1 and 1 ∩ 2 contains a nonsingular (with respect to both 1 and 2 ) point or an isolated singular point of 1 (2 , resp.). Moreover, assume either 1 or 2 has only isolated singular points. Then 1 ≡ 2 . For further applications we need to extend Theorem J to hypersurfaces of a subRiemannian manifold having isometric translations, touching at an isolated singular point. See Theorem J in Sect. 6. We can now apply Theorem C (or C ) and Theorem J to prove a rigidity result for minimal hypersurfaces in a Heisenberg cylinder (Hn \{0}, ρ −2 ) with n ≥ 2. Here  denotes the standard Heisenberg contact form:  := dz +

n  (x j d x n+ j − x n+ j d x j ). j=1

The associated Heisenberg distance function ρ reads ⎡ ρ := ⎣

2n 

2 (x K )2

⎤1/4 + 4z 2 ⎦

.

K =1

In Sect. 7 we discuss some basic geometry associated to the contact form ρ −2  (using x j , y j instead of x j , x n+ j and both interchangeably) before proving the following result. Theorem L Let  be a closed (compact with no boundary) immersed hypersurface in a Heisenberg cylinder (Hn \{0}, ρ −2 ) with n ≥ 2. Suppose either (a) H ≤ 0 or (b) H ≥ 0 and the interior region of  contains the origin of Hn holds. Then  must be a Heisenberg sphere defined by ρ 4 = c for some constant c > 0. In particular, H ≡ 0.

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Corollary M There does not exist a closed immersed hypersurface of positive constant horizontal ( p-)mean curvature in a Heisenberg cylinder (Hn \{0}, ρ −2 ) with n ≥ 2, whose interior region contains the origin. Let ϕ be a continuous function of τ ∈ [0, ∞). We have the following nonexistence result (pseudo-halfspace theorem).  2 ) Theorem N Let  be a domain of Hn , n ≥ 2, defined by either z > ϕ( x12 +· · · + x2n  2 + z 2 ) where lim or x1 > ϕ( x22 + · · · + x2n τ →∞ ϕ(τ ) = ∞. Then there does not exist any horizontal ( p-minimal hypersurface properly immersed in . The simplest example for Theorem N is ϕ(τ ) = aτ with a > 0. Call associated domains wedge-shaped. Theorem N tells us nonexistence of horizontal ( p-)minimal hypersurfaces in wedge-shaped domains. But Theorem N does not hold for the case a = 0. That is, halfspace theorem does not hold since there are catenoid type horizontal ( p-)minimal hypersurfaces with finite height ([21]) in Hn for n ≥ 2. On the other hand, we do have halfspace theorem for H1 (see [7]). Hoffman and Meeks [15] first proved such a halfspace theorem for R 3 . It fails for R n with n ≥ 4. But above type of pseudo-halfspace theorem still holds for R n with n ≥ 4 by a similar reasoning. There is another notion of mean curvature, called Levi-mean curvature, in the study of real hypersurfaces in C n . We would like to remark that the SMP for such mean curvature operators (generalized to pseudoconvex fully nonlinear Levi-type curvature operators) has been proved by Montanari and Lanconelli [18]. On the other hand, there are other assumptions under which the SMP holds and examples which are not of the type sum of squares and may have degeneracy of arbitrary order. We refer the reader to [3] for more information about this direction.

2 SubRiemannian geometry from the viewpoint of differential forms A subRiemannian manifold is a (C ∞ ) smooth manifold M equipped with a nonnegative inner product ·, · ∗ on T ∗ M, its cotangent bundle, i.e., v, v ∗ ≥ 0 for any cotangent vector v (see, for instance, [23,24]). When ·, · ∗ is positive definite, the definition is equivalent to the usual definition of Riemannian manifold using a positive definite inner product on the tangent bundle T M. However for a degenerate ·, · ∗ , it is difficult to define on the whole T M instead of T ∗ M. So the above definition using T ∗ M of subRiemannian manifold generalizes the notion of Riemannian manifold in a unified way. For M being the Heisenberg group Hn of dimension m = 2n + 1, we recall that the multiplication ◦ of Hn reads (a1 , . . . , an , b1 , . . . , bn , c) ◦ (x1 , . . . , xn , y1 , . . . , yn , z) ⎛ ⎞ n  = ⎝a1 + x1 , . . . , bn + yn , c + z + (b j x j − a j y j )⎠ . j=1

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Let eˆ j =

∂ ∂ ∂ ∂ + y j , eˆ j  = − xj , ∂x j ∂z ∂yj ∂z

1 ≤ j ≤ n be the left-invariant vector fields on Hn , in which x1 , . . . , xn , y1 , . . . , yn , z 1 n n+1 , . . . , x 2n , z used previously). denote the coordinates of Hn (instead  of x , . . . , x , x The (contact) 1-form  ≡ dz + nj=1 (x j dy j − y j d x j ) annihilates eˆ j s and eˆ j  s. We observe that d x1 , dy1 , d x2 , dy2 , . . . , d xn , dyn ,  are dual to eˆ1 , eˆ1 , eˆ2 , eˆ2 , . . . , eˆn , ∂ . Define a nonnegative inner product ·, · Hn or ·, · ∗ by eˆn  , ∂z d x j , d xk ∗ = δ jk , dy j , dyk ∗ = δ jk , d x j , dyk ∗ = 0, , d x j ∗ = , dyk ∗ = ,  ∗ = 0.

(2.1)

We can extend the definition of the above nonnegative inner product to the situation of a general pseudoHermitian manifold. Take e j , e j  = J e j , j = 1, 2, . . . , n to be an orthonormal basis in the kernel of the contact form  with respect to the Levi metric 21 d(·, J ·). Let T be the Reeb vector field of  (such that (T ) = 1 and  d(T, ·) = 0). Denote the dual coframe of e j , e j  , T by θ j , θ j (and ). Now we can  replace d x j , dy j by θ j , θ j in (2.1) to define a nonnegative inner product on a general pseudoHermitian manifold: 





θ j , θ k ∗ = δ jk , θ j , θ k ∗ = δ jk , θ j , θ k ∗ = 0, 

, θ j ∗ = , θ k ∗ = ,  ∗ = 0.

(2.2)

Define the bundle morphism G : T ∗ M → T M by ω(G(η)) = ω, η ∗

(2.3)

for ω, η ∈ T ∗ M. In the Riemannian case, G is in fact an isometry. In the pseudoHermitian case, G(T ∗ M) is the contact subbundle ξ of T M, the kernel of . By letting η =  in (2.3), we get G(T ∗ M) ⊂ ξ. On the other hand, it is easy to see  that G(θ j ) = e j , G(θ j ) = e j  (and G() = 0). Since e j , e j  , j = 1, 2, . . . , n span ξ, we have ξ ⊂ G(T ∗ M). For a smooth function ϕ on M, we define the gradient ∇ϕ := G(dϕ). In the pseudoHermitian case, this ∇ϕ is nothing but the subgradient  ∇b ϕ := nj=1 {e j (ϕ)e j + e j  (ϕ)e j  }. Let  ⊂ M be a (smooth) hypersurface in M with a defining function φ such that  = {φ = 0}. Fix a background volume form dv M , i.e., a nonvanishing m + 1 form on M, where m + 1 = dim M (hence dim  = m). For a point ζ where dφ|2∗ := dφ, dφ = 0, for any p ≥ 0, we define subRiemannian area (or volume) element dvφ, p and (generalized) mean curvature Hφ, p (ζ ) for the hypersurface {φ = φ(ζ )} by

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dφ

dvφ, p :=

1− p

|dφ|∗ 

d(dvφ, p ) = d

 dv M , 

dφ

 dv M

1− p

|dφ|∗

:= Hφ, p dv M ,

(2.4)

respectively. Here the interior product for forms is defined so that η ∧ (ω dv M ) = η, ω ∗ dv M . The above notion of subRiemannian area unifies those in Riemannian and pseudoHermitian geometries. For more details (on the case p = 0), please see the Appendix: Generalized Heisenberg Geometry in [8]. For M being Riemannian, we may take the associated volume form as a background volume form. For degenerate M, a background volume form is a choice independent of the (degenerate) metric ·, · ∗ . For the variational formulation, consider the following energy functional:    1 p F p (φ) := |dφ|∗ + H φ dv M  p (H being the prescribed subRiemannian mean curvature or p-subLaplacian) where  ⊂ M is a smooth bounded domain. Let φε = φ + ερ where ρ ∈ C0∞ (). Compute the first variation of F p : (omitting the volume form dv M ) dF p (φε ) |ε=0± dε  = cp S(φ)

p



|dρ|∗ +

p−2

|dφ|∗

\S(φ)

dφ, dρ ∗ +

 

Hρ

(2.5)

where c p = ±1 for p = 1, c p = 0 for 1 < p < ∞, and S(φ) is the set where |dφ|∗ = 0, called the singular set of φ (cf. (1.4) in [11]). From (2.5) we learn that for p = 1, the first term involving the singular set S(φ) is not negligible. So in the proof of the maximum principle (or comparison theorem), we need to worry about the size of S(φ) (see [10,11] for more details). In Lemma 6.2 of this paper, we extend the maximum principle (comparison theorem) to general subRiemannian manifolds. This is necessary in order to show the SMP near singular points (where |dφ|∗ = 0). Define a unit normal ν p to a hypersurface  := {φ = c} by the formula ν p  dv M = dvφ, p .

(2.6)

Given area element dvφ, p and unit normal ν p , we can also define (generalized) mean curvature H, p through a variational formula:  δ f νp



 dvφ, p =



f H, p dvφ, p

(2.7)

for f ∈ C0∞ ().

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Proposition 2.1 On , we have Hφ, p = H, p . Proof Let ιμ denote the interior product with vector μ. From (2.7) we compute  

  f H, p dvφ, p = δ f ν p dvφ, p = L f ν p (dvφ, p )    = (d ◦ ι f ν p + ι f ν p ◦ d)(dvφ, p )  = d( f ν p  dvφ, p ) + ι f ν p (d(dvφ, p ))   = f ν p  dvφ, p + ι f ν p (Hφ, p dv M ) (by (2.4)) ∂   f Hφ, p dvφ, p = 0+ 

since f ∈ C0∞ () and by (2.6). It follows that H, p = Hφ, p .

 

Let (M, J, ) be a pseudoHermitian manifold of dimension 2n +1, considered as a subRiemannian manifold. We take the background volume form dv M :=  ∧ (d)n . Let  be a (smooth) hypersurface of M. At nonsingular points (where T  is transversal to the contact bundle ξ ), we choose orthonormal (with respect to the Levi metric ·, · Levi := 21 d(·, J ·)) basis e1 , e1 , . . . , en−1 , e(n−1) , en in T  ∩ ξ, where e j  = J e j . We choose the horizontal (or Legendrian) normal ν = en  and a defining dφ function φ satisfying dφ(ν) > 0 such that the p-area element dvφ := |dφ| dv M ∗ 





has the expression  ∧ e1 ∧ e1 ∧ · · · en−1 ∧ e(n−1) ∧ en where , e1 , e1 , . . . , en−1 ,   e(n−1) , en , en are dual to T, e1 , e1 , . . . , en−1 , e(n−1) , en , en  . It follows that the associated mean curvature H , called horizontal mean curvature, is the trace of the second fundamental form: H =



p.h.

∇e K ν, e K Levi

K =1,1 ,...,n

(see the Appendix in [6] for more details; note sign difference in [6]). Let |X | Levi := 1/2 X, X Levi for X ∈ ξ. We can also express H in terms of φ as follows: H = divb

∇b φ |∇b φ|levi

where ∇b and divb denote subgradient and subdivergence in (M, J, ). For a graph  := {(x1 , . . . , xn , y1 , . . . , yn , u(x1 , . . . , xn , y1 , . . . , yn ))} in the Heisenberg group Hn , we take the defining function φ(x1 , . . . , xn , y1 , . . . , yn , z) = u(x1 , . . . , xn , y1 , . . . , yn ) − z. Then the resulting H coincides with the definition given by (1.8) with F := (−y1 , x1 , . . . , −yn , xn ).

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Strong maximum principle for mean curvature operators…

3 Proofs of Theorems A, A , Ã, and B Proof of Theorem A A similar idea has been used to prove Theorem 10.1 in [13] for the elliptic situation. From the definition of Q (see (1.1)), we compute the difference of Qφ and Qψ as follows: Qφ − Qψ = a i j (x, Dφ)Di j (φ − ψ) + (a i j (x, Dφ) − a i j (x, Dψ))Di j ψ + (b(x, φ, Dφ) − b(x, φ, Dψ)) + (b(x, φ, Dψ) − b(x, ψ, Dψ)) ≥ 0 (3.1) by assumption. Writing w = φ − ψ, a˜ i j (x) = a i j (x, Dφ(x)), (a i j (x, Dφ) − a i j (x, Dψ))Di j ψ + (b(x, φ, Dφ) − b(x, φ, Dψ)) = b˜ i (x)Di w, b(x, φ, Dψ) − b(x, ψ, Dψ) = a(x)w, ˜ we get Lw := a˜ i j (x)Di j w + b˜ i (x)Di w + a(x)w ˜ ≥ 0. Here the existence of b˜ i (x) is guaranteed by a i j and b being continuously differentiable (in fact they are C ∞ smooth) with respect to the p variables and the mean-value theorem. Noting that the quadratic form (a˜ i j (x)) is nonnegative by condition (1) and a(x) ˜ ≤ 0 by condition (2), we can then apply Theorem 3.1 in [4] to complete the proof.   Let e1 , . . . , em+1 be independent C ∞ vector fields on . Consider second order quasilinear operators Q  of the form: Q  φ = a i j (x, e1 φ, . . . , em+1 φ)ei e j φ + b(x, φ, e1 φ, . . . , em+1 φ)

(3.2)

where x = (x 1 , . . . , x m+1 ) is contained in a domain  of R m+1 , m ≥ 1. We have the following moving frame version of Theorem A. Theorem A . Let φ, ψ ∈ C ∞ () satisfy Q  φ ≥ Q  ψ in . Assume (1) (a i j ) is nonnegative and a i j = a ji ; (2) ∂b ∂z ≤ 0; (3) there exist vector fields X 1 , . . . , X r and Y of class C ∞ (depending on e1 φ(x), . . . , em+1 φ(x)) such that a i j (x, e1 φ(x), . . . , em+1 φ(x))ei e j =

r 

X k2 + Y.

k=1

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Let  be an integral curve of a vector field Z ∈ £(X 1 , . . . , X r ). Suppose φ − ψ achieves a nonnegative maximum in  at a point of . Then the maximum is attained at all points of . Proof a i j ei e j φ = a i j αil α kj ∂l ∂k φ + a i j αil (∂l α kj )∂k φ. Observe that a˜ lk := a i j αil α kj satisfies a˜ lk = a˜ kl since a i j = a ji , and (a˜ lk ) is nonnegative since (a i j ) is nonnegative. Note also that coefficients of first derivatives ∂k φ do not rely on the variable φ. So Q  φ is of the form (1.1) for a certain Qφ which satisfies the conditions (1)–(3) in Theorem A. Thus the conclusion follows from Theorem A.   Let a˜ i j (x) := a i j (x, Dφ(x)) in Theorem A (:= a i j (x, e1 φ(x), . . . , em+1 φ(x)) in Theorem A , resp.). In practice, condition (3) in Theorem A (Theorem A , resp.) can be replaced by rank(a˜ i j (x)) = constant r˜ .

(3.3)

Theorem à Theorem A (Theorem A , resp.) holds if condition (3) is replaced by constant rank condition (3.3). Proof It suffices to show (3.3) implies condition (3).

 

Given a subRiemannian manifold (M, ·, · ∗ ) of dimension m + 1, we recall that G : T ∗ M → T M is defined by ω(G(η)) = ω, η ∗ for ω, η ∈ T ∗ M. Define v, w ∗ := η, ζ ∗ for v, w ∈ ξ := Range(G) and any choice η ∈ G −1 (v), ζ ∈ G −1 (w). It is easy to see that ·, · ∗ is well defined. Assume dim ξ = m + 1 − l for an integer l, 0 ≤ l ≤ m + 1. So ·, · ∗ is positive definite (for l ≤ m) on the vector bundle ξ . There exist (C ∞ ) smooth local sections v1 , . . . , vm+1−l of ξ, orthonormal with respect to ·, · ∗ . We choose any smooth element η j ∈ G −1 (v j ). It follows that ηi , η j ∗ = δi j for 1 ≤ i, j ≤ m + 1 − l. Now given a background volume form dv M , for l ≥ 1, we then choose smooth independent sections ηm+2−l , . . . , ηm+1 of K er G ⊂ T ∗ M such that η j , · ∗ = ·, η j ∗ = 0 for m + 2 − l ≤ j ≤ m + 1 and

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Strong maximum principle for mean curvature operators…

dv M = η1 ∧ · · · ∧ ηm+1 (note that we have freedom to choose a scalar multiple of ηm+2−l , . . . , ηm+1 ). For l = 0, ·, · ∗ is a Riemannian metric on T ∗ M and η1 ∧ · · · ∧ ηm+1 is the Riemannian volume form (up to a sign). So a given volume form dv M is a nonzero scalar multiple of η1 ∧ · · · ∧ ηm+1 . We have shown Lemma 3.1 Let (M, ·, · ∗ , dv M ) be a subRiemannian manifold of dimension m + 1 with a background volume form dv M . Assume dim ξ = m + 1 − l. (cf. (1.3)) Then locally we can choose a suitable (C ∞ smooth) coframe η1 , . . . , ηm+1 such that ηi , η j ∗ = δi j for 1 ≤ i, j ≤ m + 1 − l, ηi , η j ∗ = 0 otherwise, and dv M = η1 ∧ · · · ∧ ηm+1 for 1 ≤ l ≤ m + 1 while dv M is a nonzero scalar multiple of Riemannian volume form for l = 0. Proof of Theorem B Take a coframe ω1 , ω2 , . . . , ωm+1 in T ∗ M (near p0 ), such that dv M = ω1 ∧ω2 ∧· · ·∧ωm+1 . Let e1 , e2 , . . . , em+1 be the dual frame in T M. Compute dφ dv M = (ei φ)ωi  ω1 ∧ ω2 ∧ · · · ∧ ωm+1 = (ei φ)g i j (−1) j−1 ω1 ∧ · · · ∧ ωˆ j ∧ · · · ∧ ωm+1

(3.4)

where g i j = ωi , ω j ∗ and ωˆ j means deleting ω j . From (3.4) and |dφ|∗ = 0 at p0 , we then compute (near p0 )  d



dφ

 dv M

1− p

|dφ|∗  =d +



ei φ 1− p

|dφ|∗ ei φ

1− p |dφ|∗

g i j (−1) j−1 ∧ ω1 ∧ · · · ∧ ωˆ j ∧ · · · ∧ ωm+1

(−1) j−1 d(g i j ω1 ∧ · · · ∧ ωˆ j ∧ · · · ∧ ωm+1 ).

(3.5)

The first term of the right-hand side in (3.5) is the term of second order in φ. We compute it as follows:  d



ei φ 1− p

|dφ|∗  = ek  = ej

g i j (−1) j−1 ∧ ω1 ∧ · · · ∧ ωˆ j ∧ · · · ∧ ωm+1

ei φ

1− p |dφ|∗

ei φ 1− p

|dφ|∗

 ωk g i j (−1) j−1 ∧ ω1 ∧ · · · ∧ ωˆ j ∧ · · · ∧ ωm+1  g i j dv M .

(3.6)

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So by (1.2), (3.5), and (3.6), we have  Hφ, p = e j

ei φ

 g i j + first order terms in φ.

1− p

|dφ|∗

(3.7)

Note that the first order terms in (3.7) do not depend on the variable φ itself. Write  ej

ei φ 1− p |dφ|∗

 gi j = gi j

ei e j φ 1− p

|dφ|∗

+ ( p − 1) + ( p − 1)

g i j (ei φ)(ek φ)(e j el φ)g kl 3− p

|dφ|∗ g i j (ei φ)(ek φ)(el φ)(e j g kl ) 3− p

2|dφ|∗

.

(3.8)

Define the second order operator a i j (x, e1 φ, . . . , em+1 φ)ei e j by a i j (x, e1 φ, . . . , em+1 φ) :=

g i j (x) 1− p |dφ|∗

+ ( p − 1)

(ei φ)(e j φ) 3− p

|dφ|∗

(3.9)

where ei := g i j e j . Observe that Hφ, p is independent of the choice of (co)frames. By Lemma 3.1, we can choose a suitable coframe (field), denoted as ω˜ 1 , ω˜ 2 , . . . , ω˜ m+1 , such that g˜ i j = δi j for 1 ≤ i, j ≤ m + 1 − l, g˜ i j = 0 otherwise, and dv M = ω˜ 1 ∧ ω˜ 2 ∧ · · · ∧ ω˜ m+1 for l ≥ 1. If l = 0, dv M is a nonzero scalar multiple of the Riemannian volume form ω˜ 1 ∧ ω˜ 2 ∧ · · · ∧ ω˜ m+1 . So we have the same form of second order term and the later argument still works. Let {e˜1 , . . . , e˜m+1 } be dual to {ω˜ 1 , ω˜ 2 , . . . , ω˜ m+1 }. Let

eˇm+1−l :=

m+1−l  1 e˜ j φ(x)e˜ j ∈ ξ |dφ(x)|∗

(3.10)

j=1

where ξ := Range(G) is spanned by the orthonormal basis e˜1 , . . . , e˜m+1−l . Choose another system of orthonormal vectors eˇ1 , . . . , eˇm−l perpendicular to eˇm+1−l in ξ. Also let eˇm+2−l = e˜m+2−l , . . . , eˇm+1 = e˜m+1 . Consider a˜ i j (x) := a i j (x, e˜1 φ(x), . . . , e˜m+1 φ(x)) e˜i φ(x)e˜ j φ(x) δi j + ( p − 1) = 1− p 3− p |dφ(x)|∗ |dφ(x)|∗

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(viewed as a function of x) for 1 ≤ i, j ≤ m + 1 − l; = ( p − 1) Compute a˜ i j (x)e˜i e˜ j =

=

=

=

⎧ ⎨m+1−l 

1 1− p

|dφ(x)|∗ 1

1− p

|dφ(x)|∗ 1

1− p

|dφ(x)|∗ m−l  j=1

X 2j

+



e˜2j + ( p − 1)

⎩ j=1 ⎧ ⎨m+1−l 

m+1  i, j=1

(e˜i φ)(e˜ j φ) 3− p

|dφ|∗

⎫ ⎬ e˜i φ(x)e˜ j φ(x) e ˜ e ˜ i j ⎭ |dφ(x)|2∗

otherwise.

⎫ ⎬

2 eˇ2j + first order + ( p − 1)eˇm+1−l + first order ⎭ ⎩ j=1 ⎧ ⎫ ⎨m−l ⎬  2 eˇ2j + p eˇm+1−l + first order ⎩ ⎭ j=1

√

2

p

eˇ (1− p)/2 m+1−l

+ first order

|dφ(x)|∗

(3.11)

where X j :=

eˇ j (1− p)/2

|dφ(x)|∗

.

Observe that eˇ j φ = 0, and hence we have X j ∈ ξ ∩ T {φ = 0}.

(3.12)

i.e., X j lies in the tangent space of hypersurface defined by φ = 0. It is not hard to see that (a i j ) is symmetric and nonnegative by Cauchy–Schwarz inequality. In view of (3.6)–(3.8), we learn that Hφ, p + b(x, φ, Dφ) is an operator of type Q  φ in (3.2). Observe that condition (3) in Theorem A holds by (3.11). By assumption we have rank(£(X 1 , . . . , X m−l )) = m.

(3.13)

Since dim{φ = 0} = m, the integral curves of all Z ∈ £(X 1 , . . . , X m−l ) will cover a neighborhood of {φ = 0} by (3.12) and (3.13). (a) follows from Theorem A . √ p In case p > 0, X 1 , X 2 , . . . , X m−l , ˇm+1−l form a basis of ξ. By the (1− p)/2 e |dφ(x)|∗

assumption rank(£(ξ )) = m + 1, the integral curves of all Z ∈ £(ξ ) will cover a neighborhood of p0 in M. (b) follows from (3.11) and Theorem A .  

ˆ C , E and 4 Graphs under symmetry and proofs of Theorems C, C, Corollary D Next we want to study when the conditions in Theorem B are satisfied. Let us start with a general subRiemannian manifold (M, ·, · ∗ ) where ·, · ∗ is a nonnegative

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definite inner product on T ∗ M. Take a background volume form dv M . Let 0 be a (local) hypersurface in M. Consider a one-parameter family of diffeomorphisms a : M → M, i.e., 0 = I dentit y, a+b = a ◦ b . We ask when the hypersurface a := a (0 ) has the same mean curvature (function) as 0 . Proposition 4.1 Suppose (M, ·, · ∗ , dv M ) has isometric translations a near p0 ∈ M, compatible with a defining function φ (see Section 1 for the definition). Then for any p ≥ 0, we have Hφ, p (a (x)) = Hφ, p (x). Proof Recall that Hφ, p is defined by  d



dφ 1− p

|dφ|∗

 dv M

:= Hφ, p dv M .

Pulling back the above identity by a , we get  d



d(φ ◦ a ) 1− p

|d(φ ◦ a )|∗

 dv M

= (Hφ, p ◦ a )dv M

(4.1)

by (a) and (b) in Definition 1.1. By the compatibility of a with φ (see (1.5)), we compute the left hand side of (4.1):  d

d(φ ◦ a )



 dv M 1− p |d(φ ◦ a )|∗   dφ  dv M = Hφ, p dv M . =d 1− p |dφ|∗

(4.2)

Comparing (4.1) with (4.2), we obtain Hφ, p ◦ a = Hφ, p .

 

Theorem C Suppose (M, ·, · ∗ , dv M ) of dimension m + 1 has isometric translations a near p0 ∈ M. Let U be the neighborhood of p0 in Definition 1.1. Suppose {a } is transversal to 1 and 2 in U. Choose φ and ψ to be defining functions for hypersurfaces 1 and 2 in U, resp. (I.e., 1 (2 , resp.) is defined by φ = 0 (ψ = 0, resp.)), compatible with {a }. Suppose 1 and 2 are tangent to each other at p0 where |dφ|∗ = 0, |dψ|∗ = 0. Assume (1) For any q ∈ 1 ∩ U, there exists δ > a(q) ≥ 0 such that a(q) (q) ∈ 2 , and (2) For some p ≥ 0, H2 , p (a(q) (q)) ≤ H1 , p (q) for any q ∈ 1 ∩ U. Moreover, assume the rank condition (1.4) holds near p0 . Then 2 coincides with 1 near p0 . Proof For any x near p0 there exist a number b and q ∈ 1 (∩U ) such that x = b (q).

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Strong maximum principle for mean curvature operators…

Let q˜ := a(q) (q) ∈ 2 with a(q) ≥ 0 by condition (1). Choose defining functions ψ and φ compatible with a : first define ψ(a (x)) = ψ(x)−a (φ(a (x)) = φ(x)−a, resp.) for x ∈ 2 (1 , resp.). The same formula then holds for any x near p0 . We compute ˜ ψ(x) = ψ(b (q)) = ψ(b−a(q) (q)) = ψ(q) ˜ − (b − a(q)) = 0 − b + a(q) (since q˜ ∈ 2 ) ≥ φ(q) − b (since q ∈ 1 and a(q) ≥ 0) = φ(−b (x)) − b = φ(x) − (−b) − b = φ(x).

(4.3)

On the other hand, we compute Hψ, p (x) = Hψ, p (b (q)) = Hψ, p (b−a(q) (q)) ˜ = Hψ, p (q) ˜ (by Proposition 4.1) = H2 , p (q) ˜ (by Proposition 2.1 in Sect. 2) = H2 , p (a(q) (q)) ≤ H1 , p (q) (by condition (2)) = Hφ, p (q) (by Proposition 2.1) = Hφ, p (b (q)) (by Proposition 4.1) = Hφ, p (x). The result follows from (4.3), (4.4), and Theorem B with b ≡ 0.

(4.4)  

Let M be the Heisenberg group Hn considered as a pseudoHermitian manifold and hence a subRiemannian manifold (see Sect. 2 for detailed explanation). Suppose two hypersurfaces 1 and 2 in Theorem C are (horizontal) graphs over the x 1 x 2 . . . x 2n hyperplane, defined by v and u, resp. We can take a to be the translation in the last coordinate by the amount a : a (x 1 , x 2 , . . . , x 2n−1 , x 2n , z) = (x 1 , x 2 , . . . , x 2n−1 , x 2n , z + a). The defining functions ψ and φ for 2 and 1 , resp. are given by v(x 1 , x 2 , . . . , x 2n−1 , x 2n ) − z and u(x 1 , x 2 , . . . , x 2n−1 , x 2n ) − z. We can then verify the assumptions of Theorem C or Theorem C (m = 2n) and that condition (2) for the case p = 0 is equivalent to H F (v) ≤ H F (u) (see (1.8)) by identifying H F (v) and H F (u) with the mean curvature of ψ and φ, resp., with respect to a certain subRiemannian manifold (M, ·, · ∗ , dv M ) while condition (1) is the same. So Theorem C or Theorem C includes the Heisenberg group case (see Theorem F and its proof in Sect. 5 for more details). Another situation is that two hypersurfaces 1 and 2 are tangent at p0 vertically in Hn , i.e., the common tangent space at p0 is a hyperplane E perpendicular to the x 1 x 2 . . . x 2n hyperplane. We can then find a one-parameter family of Heisenberg translations in the direction of the normal to the hypersurface at p0 . Let la denote the Heisenberg translation in the direction ∂ ∂x 1 :

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la (x 1 , x 2 , . . . , x 2n−1 , x 2n , z) := (x 1 + a, x 2 , . . . , x 2n−1 , x 2n , z − ax n+1 ). (4.5) Corollary D Suppose 1 and 2 are tangent at p0 vertically in Hn with n ≥ 2. Suppose the (Euclidean) unit normal to the tangent space at p0 is − ∂ ∂x 1 without loss of generality. For p ≥ 0, we assume H2 , p (la(q) (q)) ≤ H1 , p (q) for q ∈ 1 near p0 and la(q) (q) ∈ 2 with a(q) ≥ 0. Then 2 coincides with 1 near p0 . Proof Observe that the Heisenberg group Hn can be viewed as a subRiemannian manifold (see Sect. 2). {la }, the Heisenberg translations (see (4.5)), is a one-parameter family of symmetries transversal to 1 and 2 near p0 . Also we can choose φ and ψ to be defining functions for hypersurfaces 1 and 2 , resp. (I.e., 1 (2 , resp.) is defined by φ = 0 (ψ = 0, resp.)), compatible with {la }. We claim that rank(£(ξ ∩ T 1 )) = m near p0 . Observe that dim ξ ∩ T 1 = 2n − 1 ≥ 3 for n ≥ 2. We can find a J -invariant pair of nonzero vectors X, J X in ξ ∩ T 1 . The Lie bracket [X, J X ] generates the direction out of contact distribution ξ. Therefore rank(£(ξ ∩ T 1 )) = 2n = m near p0 . We then conclude the result by Theorem C.   Proof of Theorem Cˆ Take any system of local coordinates xˆ 1 , xˆ 2 , . . . , xˆ m on  ∩ U where U is a neighborhood of p0 , such that xˆ j ( p0 ) = 0. Observe that V := ∪a∈(−δ  ,δ  ) a ( ∩ U ) for smaller δ  < δ is a neighborhood of p0 due to transversality of a to . Define the last coordinate x m+1 and x 1 , x 2 , . . . , x m in V by (1) x m+1 (a (q)) ˆ = a, (2) x j (a (q)) ˆ = xˆ j (q), ˆ 1≤ j ≤m

(4.6)

for any a ∈ (−δ  , δ  ) and any qˆ ∈  ∩U . Write an arbitrary point q ∈ V as q = b (q) ˆ for b ∈ (−δ  , δ  ). Choose a such that a + b ∈ (−δ  , δ  ). Compute x j (a (q)) = x j (a (b (q))) ˆ = x j (a+b (q)) ˆ = xˆ j (q) ˆ (by (2) of (4.6)) j = x (b (q)) ˆ (by (2) of (4.6)) = x j (q). Similarly we have x m+1 (a (q)) = x m+1 (a (b (q))) ˆ = x m+1 (a+b (q)) ˆ = a + b (by (1) of (4.6))

123

(4.7)

Strong maximum principle for mean curvature operators…

= a + x m+1 (b (q)) ˆ (by (1) of (4.6)) = a + x m+1 (q).

(4.8)

We have proved (1). Moreover, we conclude (2) by (b) of Definition 1.1 and (1). Proof of Theorem C

u(x 1 , x 2 , . . . x m ) − x m+1

Let φ := x m+1 . Observe that v ≥ u implies

and ψ :=

 

v(x 1 , x 2 , . . . x m ) −

ψ ≥ φ.

(4.9)

For any q near p0 , there are a = x m+1 (q) − v(x01 , . . . , x0m ), b = x m+1 (q) − u(x01 , . . . , x0m ) (∈ (−δ, δ)) in which x01 = x 1 (q), . . . , x0m = x m (q), such that q = a (x01 , . . . , x0m , v(x01 , . . . , x0m )) = b (x01 , . . . , x0m , u(x01 , . . . , x0m )).

(4.10)

by transversality of isometric translations. Note that a is compatible with φ and ψ. From (4.10) we compute Hψ, p (q) = Hψ, p (a (x01 , . . . , x0m , v(x01 , . . . , x0m )) = Hψ, p (x01 , . . . , x0m , v(x01 , . . . , x0m )) (by Proposition 4.1) = H p (v)(x01 , . . . , x0m ) (by (1.6)) ≤ H p (u)(x01 , . . . , x0m ) (by assumption (2)) = Hφ, p (x01 , . . . , x0m , u(x01 , . . . , x0m )) (by (1.6)) = Hφ, p (b (x01 , . . . , x0m , u(x01 , . . . , x0m )) (by Proposition 4.1) = Hφ, p (q) (by (4.10)).

(4.11)

In view of (4.9) and (4.11), we have ψ = φ on 1 := {φ = 0} by Theorem B (a). It   follows that v ≡ u in a neighborhood of p0 ∈ . The argument in (4.11) shows the following fact, which will be used to prove Lemma 6.2. Corollary 4.2 Suppose (M, ·, · ∗ , dv M ) of dimension m + 1 with m ≥ 3 has isometric translations a near p ∈ M, transversal to a hypersurface  passing through p. Take a system of translation-isometric coordinates x 1 , x 2 , . . . , x m+1 in a neighborhood V of p such that x m+1 = 0 on . Suppose u :  ∩ V → R defines a graph {(x 1 , x 2 , . . . x m , u(x 1 , x 2 , . . . x m ))} ⊂ V . Let φ(x 1 , x 2 , . . . , x m , x m+1 ) := u(x 1 , x 2 , . . . x m )−x m+1 . Then Hφ (q) = H (u)(π (q)) wherever defined, in which π is the projection to  (i.e., (x 1 , x 2 , . . . , x m , x m+1 ) → (x 1 , x 2 , . . . x m ) in translationisometric coordinates). We now want to apply our theory of translation-isometric coordinates to the situation of la graphs. Observe that η2 = x 2 , η3 = x 3 , . . . , η2n = x 2n , and τ := z + x 1 x n+1 are invariant under la , Heisenberg translations in the direction ∂ ∂x 1 . So an la graph

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u is a hypersurface (u(η2 , η3 , . . . , η2n , τ ), 0, . . . , 0) ◦ (0, η2 , . . . , η2n , τ ) of Hn , parametrized by η2 , η3 , . . . , η2n , τ so that x 1 = u(η2 , η3 , . . . , η2n , τ ), x 2 = η2 , . . . , x 2n = η2n , z = τ − ηn+1 u(η2 , η3 , . . . , η2n , τ ).

(4.12)

In coordinates η2 , η3 , . . . , η2n , τ, and x 1 , we write the standard contact form  for Hn as follows:  := dz + x 1 d x n+1 − x n+1 d x 1 +

n  (x j d x n+ j − x n+ j d x j ) j=2

= dτ − 2ηn+1 d x 1 +

n 

(η j dηn+ j − ηn+ j dη j ).

(4.13)

j=2

Observe that ∂ ∂ ∂ ∂ + ηn+2 , . . . , e˚n := n + η2n , ∂η2 ∂τ ∂η ∂τ ∂ ∂ ∂ e˚1 := + 2ηn+1 , e˚n+1 := n+1 , ∂x1 ∂τ ∂η ∂ ∂ ∂ ∂ e˚n+2 := n+2 − η2 , . . . , e˚2n := 2n − ηn . ∂η ∂τ ∂η ∂τ e˚2 :=

form an orthonormal basis of ker  with respect to the Levi metric 21 d(·, J ·) (see Sect. 2 for more explanation). We remark that the above e˚ j , e˚n+ j are the same as the vectors eˆ j , eˆ j  , resp. in Sect. 2, but expressed in different coordinates. Let φ := u(η2 , η3 , . . . , η2n , τ ) − x 1 . We compute e˚1 φ = −1 + 2ηn+1 ∂τ u, e˚n+1 φ = ∂ηn+1 u, e˚2 φ = ∂η2 u + ηn+2 ∂τ u, . . . , e˚2n φ = ∂η2n u − ηn ∂τ u.

(4.14)

 j Hence dφ = (e˚1 φ)d x 1 + 2n j=2 (e˚ j φ)dη + (∂τ φ) has the length with respect to the subRiemannian metric (2.2) as follows: |dφ|2∗ =

2n 

(e˚ j φ)2 = |W (u − x 1 )|2

j=1

= (1 − 2ηn+1 ∂τ u)2 + (∂ηn+1 u)2 +

n 

[(∂η j u + ηn+ j ∂τ u)2 + (∂ηn+ j u − η j ∂τ u)2 ]

j=2

= 1 − 4ηn+1 ∂τ u + |W u|2

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(4.15)

Strong maximum principle for mean curvature operators…

where W is the vector-valued operator (e˚1 , e˚2 , . . . , e˚2n ). Note that the standard volume form d VHn of Hn equals d x 1 ∧ dη2 ∧ · · · ∧ dη2n ∧ . From (2.4) we compute Hφ as follows:  d

 dφ  d VHn |dφ|∗ ⎧⎡ ⎫ ⎤ 2n ⎨ e˚ φ ⎬  e˚ j φ ∂τ φ ⎦ 1 =d ⎣ dx1 + dη j +   d VHn ⎩ |dφ|∗ ⎭ |dφ|∗ |dφ|∗ j=2 ⎡ ⎤      2n e˚ j φ ˚1 φ e ⎦ d VHn . + = ⎣e˚1 e˚ j |dφ|∗ |dφ|∗ j=2

So we have 

      2n e˚ j φ e˚1 φ W (u − x 1 ) e˚ j + =W· |dφ|∗ |dφ|∗ |W (u − x 1 )| j=2   −1 + 2ηn+1 ∂τ u n+1 = 2η ∂τ  1 − 4ηn+1 ∂τ u + |W u|2   ∂ηn+1 u + ∂ηn+1  1 − 4ηn+1 ∂τ u + |W u|2    n  ∂η j u + ηn+ j ∂τ u n+ j + (∂η j + η ∂τ )  1 − 4ηn+1 ∂τ u + |W u|2 j=2   ∂ηn+ j u − η j ∂τ u j + (∂ηn+ j − η ∂τ )  . 1 − 4ηn+1 ∂τ u + |W u|2

Hφ = e˚1

So H (u) has the above expression (see (1.6) with p = 0 for the definition). Note that there is no dependence on u in the second order coefficients of H (u). Proof of Corollary D Let 1 denote the hypersurface defined by u(η2 , η3 , . . . , η2n , τ ) = x 1 . As in the proof of Corollary D, we observe that dim ξ ∩ T 1 = 2n − 1 ≥ 3 for n ≥ 2. We can then find a J -invariant pair of nonzero vectors X, J X in ξ ∩ T 1 . The Lie bracket [X, J X ] generates a direction out of contact distribution ξ. Therefore   rank(£(ξ ∩ T 1 )) = 2n = m near p0 . Corollary D follows from Theorem C . We now want to apply our theory of translation-isometric coordinates to the situation of intrinsic graphs. We are going to show that an intrinsic graph is congruent with an la graph by a rotation. So in the new coordinates, the horizontal (p-)mean curvature no longer depends on u in the second order coefficients. Let us recall [1] that an intrinsic graph u is a hypersurface of Hn , parametrized by η2 , η3 , . . . , η2n , τ so that (x 1 , . . . , x 2n , z) = (0, η2 , η3 , . . . , η2n , τ ) · (u(η2 , η3 , . . . , η2n , τ ), 0, . . . , 0) or

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x 1 = u(η2 , η3 , . . . , η2n , τ ), x 2 = η2 , . . . , x 2n = η2n , z = τ + ηn+1 u(η2 , η3 , . . . , η2n , τ ).

(4.16)

In coordinates η2 , η3 , . . . , η2n , τ, and x 1 , we write the standard contact form  for Hn as follows:  := dz + x 1 d x n+1 − x n+1 d x 1 +

n  (x j d x n+ j − x n+ j d x j ) j=2

n  = dτ + 2x 1 dηn+1 + (η j dηn+ j − ηn+ j dη j ).

(4.17)

j=2

Observe that ∂ ∂ ∂ ∂ ∂ , e˚2 := 2 + ηn+2 , . . . , e˚n := n + η2n , ∂x1 ∂η ∂τ ∂η ∂τ ∂ ∂ := n+1 − 2x 1 , ∂η ∂τ ∂ ∂ ∂ ∂ := n+2 − η2 , . . . , e˚2n := 2n − ηn ∂η ∂τ ∂η ∂τ

e˚1 := e˚n+1 e˚n+2

form an orthonormal basis of ker  with respect to the Levi metric 21 d(·, J ·) (see Sect. 2 for more explanation). Let φ := u(η2 , η3 , . . . , η2n , τ ) − x 1 . We compute e˚1 φ = −1, e˚2 φ = ∂η2 u + ηn+2 ∂τ u, . . . , e˚n+1 φ = ∂ηn+1 u − 2x 1 ∂τ u, . . . , e˚2n φ = ∂η2n u − ηn ∂τ u.

(4.18)

 j Hence dφ = (e˚1 φ)d x 1 + 2n j=2 (e˚ j φ)dη + (∂τ φ) has the length with respect to the subRiemannian metric (2.2) as follows:   2n   |dφ|∗ =  (e˚ j φ)2 = 1 + |W u (u)|2

(4.19)

j=1 u , on the graph described by x 1 = u(η2 , η3 , . . . , η2n , τ ). Here W u := (e˚2 , . . . , e˚n , e˚n+1 ∂ ∂ u e˚n+2 , . . . , e˚2n ) and e˚n+1 := ∂ηn+1 − 2u ∂τ . Note that the standard volume form d VHn

of Hn equals d x 1 ∧ dη2 ∧ · · · ∧ dη2n ∧ . On  := {φ = 0} we have 0 = dφ = 1 j (e˚1 φ)d x + 2n j=2 (e˚ j φ)dη + (∂τ φ). So by (4.18) we get dx1 =

2n  (e˚ j φ)dη j + (∂τ φ). j=2

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(4.20)

Strong maximum principle for mean curvature operators…

Now we compute the area (or volume) element dvφ for the hypersurface {φ = φ( p0 )} : dφ  d VHn |dφ|∗ ⎡ 1 ⎣ (e˚1 φ)dη2 ∧ · · · ∧ dη2n ∧  = |dφ|∗

dvφ :=

+(−1)

⎤ 2n  1 2 j 2n (e˚ j φ)d x ∧ dη ∧ .d ηˆ . ∧ dη ∧ ⎦

j−1

j=2

⎡ =

1 |dφ|∗

⎤ 2n  ⎣−1 − (e˚ j φ)2 ⎦ dη2 ∧ · · · ∧ dη2n ∧  (by (4.20)) j=2

= −|dφ|∗ dη ∧ · · · ∧ dη2n ∧ dτ  = − 1 + |W u (u)|2 dη2 ∧ · · · ∧ dη2n ∧ dτ 2

(4.21)

by (4.17) and (4.19). From (2.4) we compute Hφ as follows:   dφ d  d VHn |dφ|∗ ⎧⎡ ⎫ ⎤ 2n ⎨ e˚ φ ⎬  e˚ j φ ∂τ φ ⎦ 1 =d ⎣ dx1 + dη j +   d VHn ⎩ |dφ|∗ ⎭ |dφ|∗ |dφ|∗ j=2 ⎡ ⎤      2n e˚ j φ e˚1 φ ⎦ d VHn + = ⎣e˚1 e˚ j |dφ|∗ |dφ|∗ j=2

=

2n  j=2

 e˚ j

e˚ j φ |dφ|∗

 d VHn

e˚1 φ −1 since |dφ|∗ is independent of x 1 by (4.19 ) and hence e˚1 ( |dφ| ) = ∂x 1 ( |dφ| ) = 0. So ∗ ∗ we have

Hφ =

2n 

 e˚ j

j=2

=W · u



e˚ j φ |dφ|∗





W u (u) 1 + |W u (u)|2

 (4.22)

at (η2 , η3 , . . . , η2n , τ, u(η2 , η3 , . . . , η2n , τ )). Proof of Corollary E We observe that an intrinsic graph is congruent with an la graph by a rotation. Suppose we have an intrinsic graph described by (4.16). Define a rotation  : (x 1 , . . . , x 2n , z) ∈ Hn → (x˜ 1 , . . . , x˜ 2n , z˜ ) ∈ Hn by

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x˜ 1 = x n+1 , x˜ n+1 = −x 1 , x˜ j = x j , 1 ≤ j ≤ 2n, j = 1, j = n + 1, z˜ = z.

(4.23)

From u(η2 , η3 , . . . , η2n , τ ) − x 1 = 0 and u ηn+1 = 0 at p0 , we can express ηn+1 = ηn+1 (x 1 , η2 , . . . , ηˆ n+1 , . . . , η2n , τ ) near (u( p0 ), η02 , . . . , ηˆ 0n+1 , . . . , η02n , τ0 ) by implicit function theorem. Here ηˆ n+1 means ηn+1 deleted. In view of (4.23) and (4.16), we get x˜ 1 = ηn+1 (x 1 , η2 , . . . , ηˆ n+1 , . . . , η2n , τ ), x˜ j = η j , 2 ≤ j ≤ 2n, j = n + 1, x˜ n+1 = −x 1 , z˜ = z = τ + x 1 ηn+1 (x 1 , η2 , . . . , ηˆ n+1 , . . . , η2n , τ ).

(4.24)

Let ζ = −x 1 and η˜ n+1 (η2 , . . . , ζ, . . . , η2n , τ ) (ηˆ n+1 is replaced by ζ ) := ηn+1 (x 1 , η2 , . . . , ηˆ n+1 , . . . , η2n , τ ).

(4.25)

It follows from (4.24) and (4.25) that the image of an intrinsic graph under the rotation  can be depicted as (η˜ n+1 (η2 , . . . , ζ, . . . , η2n , τ ), 0, . . . , 0)◦(0, η2 , . . . , ζ, . . . , η2n , τ ), an la graph. Let ξ˜ n+1 denote the la graph corresponding to the intrinsic graph v under the rotation . Near p0 the condition v ≥ u implies ξ˜ n+1 ≥ η˜ n+1 (ξ˜ n+1 ≤ η˜ n+1 , resp.) if at p0 , νηn+1 = u ηn+1 < 0 (> 0, resp.). On the other hand, Hξ˜ n+1 (Hη˜ n+1 , resp.) is the same as Hv (Hu , resp.) at the same point in the graph for the case of νηn+1 = u ηn+1 < 0 at p0 . Therefore the condition Hv ≤ Hu is reduced to Hξ˜ n+1 ≤ Hη˜ n+1 in some corresponding small neighborhood when either one is constant. For the case of νηn+1 = u ηn+1 > 0 at p0 , Hξ˜ n+1 (Hη˜ n+1 , resp.) is the same as −Hv (-Hu , resp.) at the same point in the graph. So Hv ≤ Hu is equivalent to Hξ˜ n+1 ≥ Hη˜ n+1 in some corresponding small neighborhood when either one is constant. Now the theorem   follows from Corollary D . We remark that if both v and u do not have constant horizontal (or p-)mean curvature, then we won’t be able to reduce Hv ≤ Hu to Hξ˜ n+1 ≤ (or ≥)Hη˜ n+1 in general. The reason is that we are comparing horizontal (or p-)mean curvature at different pairs of points on two hypersurfaces.

5 Proofs of Theorem F, Corollaries G, H, and I Proof of Theorem F We want to show that H F (u) at (x 1 , . . . , x m ) is the same as Hφ for φ = u(x 1 , . . . , x m ) − x m+1 at (x 1 , . . . , x m , u(x 1 , . . . , x m )) with respect to a certain subRiemannian structure on R m+1 . Write F = (F1 , . . . , Fm ). Let

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ωm+1 = d x m+1 +

m 

Fj d x j .

(5.1)

j=1

Define subRiemannian metric ·, · F on T ∗ R m+1 by d x i , d x j F = δi j , 1 ≤ i, j ≤ m, η, ωm+1 F = 0 for any one form η. The vectors dual to d x 1 , . . . , d x m , ωm+1 read ∂ ∂ − F j m+1 , 1 ≤ j ≤ m j ∂x ∂x ∂ = . ∂ x m+1

e Fj = em+1

Take the standard volume form d V R m+1 := d x 1 ∧ · · · ∧ d x m ∧ ωm+1 = d x 1 ∧ · · · ∧ d x m ∧ d x m+1 . We compute Hφ as follows: 

 dφ d  ·,· F d V R m+1 |dφ| ·,· F  F  m  e j (φ) F = ej d V R m+1 |dφ| ·,· F

(5.2)

j=1

where ⎛

|dφ| ·,· F

⎞1/2 m  = ⎝ (e Fj (φ))2 ⎠ . j=1

From (5.2) we have Hφ =

m 

 e Fj

j=1

e Fj (φ) |dφ| ·,· F

 .

(5.3)

On the other hand, we compute, for φ = u(x 1 , . . . , x m ) − x m+1 , e Fj (φ) =

∂u + Fj . ∂x j

Hence we have ⎞ 2 1/2 m   ∂u  Rm . =⎝ + F j ⎠ = |∇u + F| ∂x j ⎛

|dφ| ·,· F

j=1

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Observe that

e Fj (φ) |dφ| ·,· F

is independent of x m+1 . It follows that

m 

 e Fj

j=1

e Fj (φ) |dφ| ·,· F



 = div

∇u + F  Rm |∇u + F|

 .

(5.4)

From (5.3), (5.4), we have shown H F (u) at (x 1 , . . . , x m ) equals Hφ at (x 1 , . . . , x m , u(x 1 , . . . , x m )). It is obvious that the translation along x m+1 preserves ·, · F and d V R m+1 . Moreover, x 1 , . . . , x m , x m+1 are translation-isometric coordinates with ⊥ (u) are respect to the x 1 . . . x m hyperplane. Observe that N1⊥ (u), N2⊥ (u), . . . , Nm−1 1 m the x . . . x hyperplane projection of (a choice of) X 1 , . . . , X m−1 in (3.12), resp. The   conclusion follows from Theorem C for l = 1. Write m 

Nα⊥ (v) =

bαk ∂k .

(5.5)

k=1

Corollary G Suppose m ≥ 3, H F (u) ≤ H F (v), u ≥ v in U ⊂ R m , which is a nonsingular domain for both v and u, and u = v at p0 ∈ U. Assume there exists a pair of (α, β), α = β, such that 

j

(∂k F j − ∂ j Fk )(bαk bβ − bαj bβk ) = 0

(5.6)

k< j

(similar condition for Nα⊥ (u), r esp.) in U. Then we have u ≡ v in U. Proof Let v := dv + F j d x j m  = (v j + F j )d x j . j=1

 It follows Observe that v (Nγ⊥ (v)) = 0 since Nα⊥ (v) is perpendicular to ∇v + F. that − v ([Nα⊥ (v), Nβ⊥ (v)]) = dv (Nα⊥ (v), Nβ⊥ (v)) = =

m 

(∂k F j )d x k ∧ d x j (Nα⊥ (v), Nβ⊥ (v))

k, j=1 m 

j

(∂k F j )(bαk bβ − bαj bβk )

k, j=1

=

 j (∂k F j − ∂ j Fk )(bαk bβ − bαj bβk ) k< j

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(5.7)

Strong maximum principle for mean curvature operators…

by (5.5). By assumption we get v ([Nα⊥ (v), Nβ⊥ (v)]) = 0 in view of (5.7). So [Nα⊥ (v), Nβ⊥ (v)] is nonzero and transversal to the kernel of v , the space spanned by all Nγ⊥ (v), γ = 1, 2, . . . , m − 1. We then have that the rank of ⊥ (v))) is constant m in U . Now the conclusion follows £(N1⊥ (v), N2⊥ (v), . . . , Nm−1 from Theorem F.   We now want to find an intrinsic criterion for the Lie span condition rank(£(N1⊥ (u), ⊥ (u))) = m in Theorem F or more generally rank(£(ξ ∩T  )) = m N2⊥ (u), . . . , Nm−1 1 in Theorem C to hold. Suppose  is a hypersurface in a subRiemannian manifold (M, ·, · ∗ ) of dimension m + 1. Define G : T ∗ M → T M by ω(G(η)) = ω, η ∗ for ω, η ∈ T ∗ M. Let ξ := Range(G) ⊂ T M. We assume (as always) dim ξ = constant m + 1 − l where l = dim ker G. We wonder when £(ξ ∩ T ) = £(X 1 , . . . , X m−l ), where X 1 , . . . , X m−l form a basis of local sections of ξ ∩T  (near p0 where ξ = T ), has rank m. Let us start with the case l = 1. Take a nonzero 1-form θ ∈ ker G. It is easy to see that θ annihilates vectors in ξ. So θ (X j ) = 0

(5.8)

for 1 ≤ j ≤ m − 1. We want Lie bracket of a pair of X j to generate a direction not in ξ (which will imply £(X 1 , . . . , X m−1 ) has rank m). Suppose the converse holds. Then we have dθ (X i , X j ) = X i (θ (X j )) − X j (θ (X i )) − θ ([X i , X j ]) = −θ ([X i , X j ]) = 0

(5.9)

by (5.8) for 1 ≤ j ≤ m − 1. It follows from (5.9) that dθ |ξ ×ξ , the bilinear form dθ restricted to ξ × ξ, has rank ≤ 2. So we have proved the following result. Proposition 5.1 Let  be a hypersurface in a subRiemannian manifold (M, ·, · ∗ ) of dimension m + 1. Suppose dim ξ = constant m (i.e., l = 1). Assume rank(dθ |ξ ×ξ ) ≥ 3.

(5.10)

Then there holds (1.4) and hence we have rank(£(ξ ∩ T )) = m.

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We remark that condition (5.10) is independent of the choice of θ. Since dim ker G = 1, another nonzero choice θ˜ ∈ ker G will be a nonzero multiple of θ. That is, θ˜ = λθ with λ = 0. It follows that d θ˜ |ξ ×ξ = λdθ |ξ ×ξ + (dλ ∧ θ )|ξ ×ξ = λdθ |ξ ×ξ . So d θ˜ |ξ ×ξ has the same rank as dθ |ξ ×ξ . Corollary H Suppose m ≥ 3, H F (u) ≤ H F (v), u ≥ v in U ⊂ R m , which is a nonsingular domain for both u and v, and u = v at p0 ∈ U. Assume rank(∂k F j − ∂ j Fk ) ≥ 3

(5.11)

in U. Then we have u ≡ v in U. Proof From the proof of Theorem F, we learn that H F (u) at (x 1 , . . . , x m ) is the same as Hφ for φ = u(x 1 , . . . , x m ) − x m+1 at (x 1 , . . . , x m , u(x 1 , . . . , x m )) with respect to a certain subRiemannian structure on Hn with m = 2n. This subRiemannian structure has the property that dim ξ = constant m (i.e., l = 1). We take θ = ωm+1 = d x m+1 +

m 

Fj d x j

j=1

(see (5.1)). Compute dθ =



(∂k F j − ∂ j Fk )d x k ∧ d x j .

k< j

Therefore condition (5.11) means rank(dθ |ξ ×ξ ) ≥ 3. Note that N1⊥ (v), N2⊥ (v), . . . , ⊥ (v) are the x 1 . . . x m hyperplane projection of (a choice of) X , . . . , X Nm−1 1 m−1 , m+1 resp. in (3.12) for φ = v − x . The conclusion follows from Proposition 5.1 and Theorem F.   Corollary I Suppose m = 2n ≥ 4, H F (u) ≤ H F (v), u ≥ v in U ⊂ R 2n , which is a nonsingular domain for both v and u, and u = v at p0 ∈ U. Assume F = (−x 2 , x 1 , . . . , −x 2n , x 2n−1 ). Then we have u ≡ v in U. Proof For such an F = (−x 2 , x 1 , . . . , −x 2n , x 2n−1 ), we compute ∂2k−1 F2k − ∂2k F2k−1 = 2 for k = 1, . . . , n; ∂k F j − ∂ j Fk = 0 (k < j) otherwise.

(5.12)

So the rank of the matrix (∂k F j − ∂ j Fk ) equals 2n by (5.12). By assumption we have m = 2n ≥ 4 > 3. The conclusion follows from Corollary H.  

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Corollary I provides the SMP of the so called horizontal (or p-)mean curvature for hypersurfaces given by graphs over a domain of the x 1 x 2 . . . x 2n hyperplane in the Heisenberg group Hn (identified with R 2n+1 as a set). We remark that when m = 2, Corollary I does not hold. That is, the SMP of horizontal ( p-)mean curvature for surfaces in H1 does not hold (although the maximum principle holds; see [10] or Theorem C  in the next section) as shown by the following example. Let u = x 1 x 2 + (x 2 )2 and v = x 1 x 2 . It follows from (1.8) that H F (u) = H F (v) = 0 in a nonsingular domain for F = (−x 2 , x 1 ). Observe that u ≥ v in U = {x 1 > 0, x 1 + x 2 > 0}, a nonsingular domain for both u and v, and u = v at (x 1 , 0) ∈ U. But apparently u = v in U \{(x 1 , 0) : x 1 > 0}. We are going to give a sufficient condition for the rank estimate in the case of general l. Let θ 1 , . . . , θ l be a basis of ker G. Choose an (codimension 1, resp.) orthonormal basis X 1 , . . . , X m+1−l (X 1 , . . . , X m−l , resp.) of ξ and its dual forms ω1 , . . . , ωm+1−l (ω1 , . . . , ωm−l , resp.) (not unique). Then we can find unique vector fields T1 , . . . , Tl such that {X 1 , . . . , X m+1−l , T1 , . . . , Tl } is dual to {ω1 , . . . , ωm+1−l , θ 1 , . . . , θ l }. Write

[X i , X j ] =

l 

Wiαj Tα mod ξ

(5.13)

α=1

for 1 ≤ i. j ≤ m + 1 − l (1 ≤ i. j ≤ m − l, resp.). For fixed α we view (Wiαj ) as an (m + 1 − l) × (m + 1 − l) matrix. Since it is skew-symmetric, it has degrees of freedom.

(m+1−l)(m−l) 2

Proposition 5.2 Suppose (m + 1 − l)(m − l) ≥l 2   (m − l)(m − l − 1) ≥ l, resp. . 2

(5.14)

Then rank(£(ξ )) = m + 1 (rank(£(X 1 , . . . , X m−l )) ≥ m, resp.) if dθ α ω1 ∧ · · · ∧ ωm+1−l (ω1 ∧ · · · ∧ ωm−l , resp.), α = 1, . . . , l, are linearly independent, or equivalently [Wiαj ] as (m + 1 − l)× (m + 1 − l) matrix ((m − l)× (m − l) matrix, resp.), α = 1, . . . , l, are linearly independent. Proof In view of (5.13) and (5.14), the linear independence of [Wiαj ], α = 1, . . . , l, implies T1 , . . . , Tl ∈ £(ξ ) (£(X 1 , . . . , X m−l ), resp.) by basic linear algebra. It follows that rank(£(ξ )) ≥ (m+1−l)+l = m+1 (rank(£(X 1 , . . . , X m−l )) ≥ (m−l)+l = m, resp.). On the other hand, clearly we have rank(£(ξ )) ≤ m + 1. By the formula dθ α (X i , X j ) = X i (θ α (X j )) − X j (θ α (X i )) − θ α ([X i , X j ]),

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we obtain dθ α = −



Wiαj ωi ∧ ω j mod θ 1 , . . . , θ l

i< j

by (5.13). We then learn that two conditions are equivalent.

 

6 Singularities and proofs of Theorems C  , J, K, C˜  , J Theorem C (an extension of Theorem C  in [10]). For a bounded domain  in R m ¯ satisfy with m ≥ 2, let v, u ∈ C 2 () ∩ C 0 () H F (v) ≤ H F (u) in \{S F (u) ∪ S F (v)} v ≥ u on ∂. ¯ satisfies div F b > 0 Suppose Hm−1 (S F (u) ∪ S F (v)) = 0 and F ∈ C 1 () ∩ C 0 () (div F b < 0, resp.) in . Then v ≥ u in . Proof By Theorem 5.2 in [10], we have N F (v) = N F (u) in + \{S F (u) ∪ S F (v)} where + := { p ∈ |u( p) − v( p) > 0}. Suppose + is not empty. Then applying Lemma 5.3 in [10] with  = + and F ∗ replaced by F b in the proof, we obtain   v = u in + , a contradiction. We have proved v ≥ u in . Proof of Theorem J Case 1: Suppose v = u at q = p0 . Since p0 is the only singular point of u (v, resp.), q is nonsingular with respect to u (v, resp.). On the other hand, that v − u ≥ 0 and v − u = 0 at q implies that ∇(v − u) = 0 at q. It follows that at q, ∇v + F = ∇u + F = 0 (∇u + F = ∇v + F = 0, resp.) since q is nonsingular with respect to u (v, resp.). So q is also nonsingular with respect to v (u, resp.). Now by Theorem F we obtain v ≡ u in a connected component W of nonsingular (with respect to both v and u) set, containing q. We claim W = \{ p0 }. Otherwise there is a point q  ∈ (S F (v) ∩ W¯ )\{ p0 } ((S F (u) ∩ W¯ )\{ p0 }, resp.) at which ∇u + F = ∇v + F = 0 (∇v + F = ∇u + F = 0, resp.). Therefore q  ∈ S F (u) (q  ∈ S F (v),resp.), a contradiction to p0 being the only singular point of u (v, resp.). Hence v ≡ u in . Case 2: Suppose v > u in \{ p0 }. So there is a small ball B centered at p0 such that v > u in B\{ p0 }. It follows that v ≥u+c on ∂ B for some constant c > 0. By Theorem C  (a version of the usual maximum principle), we conclude that v ≥ u + c in B. But v( p0 ) = u( p0 ) implies 0 ≥ c, a contradiction. We have shown the impossibility of this case.  

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Strong maximum principle for mean curvature operators…

Proof of Theorem K We first observe that the assumption Hm−1 (S F (u)) = 0 or Hm−1 (S F (v)) = 0 in Theorem J is satisfied for m = 2n and F = (−x 2 , x 1 , . . . , −x 2n , x 2n−1 ) by Lemma 5.4 in [10]. Also div F b = 2n > 0. In view of Corollary D, Corollary I, and Theorem J, we finally reach a situation that 1 and 2 are tangent at a nonisolated singular point for both 1 and 2 if they don’t coincide completely. This contradicts the assumption that either 1 or 2 has only isolated singular points.   Lemma 6.1 Suppose |dψ|∗ = 0, |dφ|∗ = 0. Then the following formula ! dψ dφ ∗ dψ − dφ, − |dψ|∗ |dφ|∗ 1 dψ dφ 2 = (|dψ|∗ + |dφ|∗ )| − | 2 |dψ|∗ |dφ|∗ ∗

(6.1)

holds. Proof We compute ! dφ ∗ dψ − dψ − dφ, |dψ|∗ |dφ|∗ dφ, dψ ∗ dψ, dφ ∗ = |dψ|∗ + |dφ|∗ − − |dψ|∗ |dφ|∗ = (|dψ|∗ + |dφ|∗ )(1 − cos ϑ)

(6.2)

where we write dφ, dψ ∗ = dψ, dφ ∗ = |dψ|∗ |dφ|∗ cos ϑ. On the other hand, we have    dψ dφ 2  −  |dψ| |dφ|∗ ∗ ∗     ∗  dψ 2  dφ 2   − 2 dψ, dφ

  = +  |dφ|  |dψ|  |dψ| |dφ| ∗ ∗

= 2(1 − cos ϑ) Now (6.1) follows from (6.2) and (6.3).

∗ ∗

∗

∗

(6.3)  

We remark that the formula (6.1) in vector form first appeared in [12,16,17] independently. The version for Heisenberg group appeared in Lemma 5.1 of [10]. For the case of bounded variation, the reader is referred to [8]. ˜ a hypersurface of a subRiemannian manifold, is called sinA point q˜ ∈ , ˜ gular if ξ ⊂ T  at q˜ (this is equivalent to |dφ|∗ = 0 at q˜ for a defining ˜ mentioned previously). For  ˜ being a graph described by (x 1 , function φ of  2 m 1 2 m 1 2 m x , . . . x , w(x , x , . . . x )), (x , x , . . . x ) ∈ D, in local coordinates, we call ˜ Denote the set of a point q in D singular for w if (q, w(q)) is a singular point of . all singular points in D for w by S D (w) or S(w) if the domain of w is clear in the context.

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Lemma 6.2 Suppose (M, ·, · ∗ , dv M ) of dimension m + 1 has isometric translations a near p0 ∈ M, transversal to a hypersurface  passing through p0 . Take a system of translation-isometric coordinates x 1 , x 2 , . . . , x m+1 in an open neighborhood  of p0 such that x m+1 = 0 on  ∩ . Let V ⊂ V¯ ⊂  be a smaller open neighborhood of p0 . Let v, u ∈ C 2 ( ∩ V ) ∩ C 0 ( ∩ V ) define graphs in  and satisfy H (v) ≤ H (u) in ( ∩ V )\{S(u) ∪ S(v)}, v ≥ u on ∂( ∩ V ).

(6.4) (6.5)

Assume Hm−1 (S(u) ∪ S(v)) = 0. Let ψ := v − x m+1 , φ := u − x m+1 . Then dφ |dφ|∗

mod ker G in ( ∩ u(q) − v(q) > 0}.

V )+ \S(u) ∪

S(v) where ( ∩

V )+

dψ |dψ|∗

=

:= {q ∈  ∩ V :

Proof First Hm−1 (S(u) ∪ S(v)) = 0 means that given any ε > 0, we can find countably many ball B j,ε , j = 1, 2, . . . such that S(u) ∪ S(v) ⊂ ∪∞ j=1 B j,ε ∞ ∞ B ∞ B and H (∂ B ) < ε and we can arrange ∪ ⊂ ∪ m−1 j,ε j,ε 1 j=1 j=1 j=1 j,ε2 for ε1 < ε2 . Since S(u) ∪ S(v) is compact, we can find finitely many B j,ε ’s, say j = 1, 2, . . . , n(ε), still covering S(u) ∪ S(v). Suppose ( ∩ V )+ is not empty. Then by Sard’s theorem there exists a sequence of positive number δi converging to 0 as i goes to infinity, such that ( ∩ V )i+ := { p ∈  ∩ V : u( p) − v( p) > δi } is not empty and ∂( ∩ V )i+ \(S(u) ∪ S(v)) is C 2 smooth. Note that ∂( ∩ V )i+ ∩ ∂( ∩ V ) ⊂ S(u) ∪ S(v) by (6.5). Choose a > 0 (independent of ε and δi ) such that n(ε) [( ∩ V )i+ \ ∪ j=1 B j,ε ] × [−a, a] ⊂ . Now we consider  Iεi

:=

∂([(∩V )i+ \∪ j=1 B j,ε ]×[−a,a]) n(ε)

tan

−1



dψ dφ (ψ − φ) − |dψ|∗ |dφ|∗

  dv M . (6.6)

By Stokes’ theorem we have "



Iεi

dψ − dφ = ∧ 2 n(ε) + [(∩V )i \∪ j=1 B j,ε ]×[−a,a] 1 + (ψ − φ) % −1 + tan (ψ − φ)(Hψ − Hφ )dv M

#

dψ dφ − |dψ|∗ |dφ|∗



$  dv M (6.7)

Observe that ψ − φ = v − u < −δi < 0 in ( ∩ V )i+ × [−a, a] and Hψ (q) − Hφ (q) = H (v)(π (q)) − H (u)(π (q)) ≤ 0 by Corollary 4.2 and (6.4), where π is the projection to  ((x 1 , . . . , x m , x m+1 ) → (x 1 , . . . , x m )). So the second term in the right hand side of (6.7) is nonnegative. As to the first term, we observe that η ∧ (ωdv M ) = η, ω ∗ dv M for any 1-forms η and ω. It then follows from (6.1) in Lemma 6.1 and (6.7) that  Iεi

≥

123

[(∩V )i+ \∪ j=1 B j,ε ]×[−a,a] n(ε)

  |dψ|∗ + |dφ|∗  dψ dφ 2 − dv M (6.8) 2[1 + (ψ − φ)2 ]  |dψ|∗ |dφ|∗ ∗

Strong maximum principle for mean curvature operators…

On the other hand, look at the boundary integral in (6.6): ψ − φ = v − u = δi on n(ε) ∂( ∩ V )i+ × [−ai , ai ] → 0 as i → ∞, the boundary integral on [( ∩ V )i+ \ ∪ j=1

B j,ε ] × {−ai } cancels with the boundary integral on [( ∩ V )i+ \ ∪n(ε) j=1 B j,ε ] × {ai }

dψ dφ due to translation invariance and orientation, and |dψ| − |dφ| is bounded while ∗ ∗ n(ε) i j=1 Hm−1 (∂ B j,ε ) < ε. So we conclude that Iε ≤ ε for i = i(ε) large enough from (6.6). Together with (6.8) we conclude that

   dψ dφ   − =0  |dψ| |dφ|∗ ∗ ∗ It follows that

dψ |dψ|∗

=

dφ |dφ|∗

mod ker G in ( ∩ V )+ \S(u) ∪ S(v).

 

Lemma 6.3 Suppose (M, ·, · ∗ , dv M ) of dimension m + 1 has isometric translations a near p0 ∈ M, transversal to a hypersurface  passing through p0 . Take a system of translation-isometric coordinates x 1 , x 2 , . . . , x m+1 in an open neighborhood  of p0 such that x m+1 = 0 on  ∩ and p0 is the origin (0, . . . , 0). Let v, u :  ∩ → R be two graphs in . Let ψ := v(x 1 , x 2 , . . . , x m ) − x m+1 , φ := u(x 1 , x 2 , . . . , x m ) − dψ dφ = |dφ| mod ker G. Moreover, we assume the rank condition x m+1 . Suppose |dψ| ∗ ∗ (1.4). Then ∇v = ∇u in  ∩ . dψ dφ = |dφ| mod ker G implies ξ ∩ T p {ψ = Proof Note that ξ = ∩θ∈ker G ker θ. So |dψ| ∗ ∗ c( p)} = ξ ∩ T p {φ = 0}. By a translation (depending on p) in the x m+1 direction, we can translate the hypersurface {ψ = c( p)} to the fixed hypersurface {ψ = 0}. Since translations are isometries, we then have

ξ ∩ T( p,v( ¯ ∗ (ξ ∩ T( p,u( ¯ p)) ¯ {ψ = 0} = ( p) ¯ p)) ¯ {φ = 0})

(6.9)

where p¯ ∈  ∩  and ( p) ¯ is a translation in the x m+1 direction, depending on p. ¯ By {ψ = 0}) = (T {ψ = 0}) ( £(ξ ∩ T {φ = (1.4) we have £(ξ ∩ T( p,v( ¯ p)) ¯ ( p,v( ¯ p)) ¯ ( p,u( ¯ p)) ¯ {φ = 0}), resp.) since rank(£(ξ ∩ T {ψ = 0})) = m 0}) = (T( p,u( ¯ p)) ¯ ( p,v( ¯ p)) ¯ (rank(£(ξ ∩T( p,u( ¯ p)) ¯ {φ = 0})) = m, resp.) and ξ ∩T( p,v( ¯ p)) ¯ {ψ = 0} ⊂ T( p,v( ¯ p)) ¯ {ψ = {φ = 0} ⊂ T {φ = 0}, resp.). Here (E) denotes the space of 0} (ξ ∩ T( p,u( ¯ p)) ¯ ( p,u( ¯ p)) ¯ all (C ∞ smooth) sections of the vector bundle E. It follows from (6.9) that T( p,v( ¯ ∗ (T( p,u( ¯ p)) ¯ {ψ = 0} = ( p) ¯ p)) ¯ {φ = 0}). We then have ∇v = ∇u in  ∩ .

 

˜  . Suppose (M, ·, · ∗ , dv M ) of dimension m +1 has isometric translations Theorem C a near p0 ∈ M, transversal to a hypersurface  passing through p0 . Take a system of translation-isometric coordinates x 1 , x 2 , . . . , x m+1 in an open neighborhood  of p0 such that x m+1 = 0 on  ∩ . Let V ⊂ V¯ ⊂  be a smaller open neighborhood of p0 . Let v, u ∈ C 2 ( ∩ V ) ∩ C 0 ( ∩ V ) define graphs in  and satisfy

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H (v) ≤ H (u) in ( ∩ V )\{S∩V (u) ∪ S∩V (v)}, v ≥ u on ∂( ∩ V ). Assume the rank condition (1.4) and Hm−1 (S∩V (u) ∪ S∩V (v)) = 0. Then v ≥ u in  ∩ V. dψ dφ Proof By Lemma 6.2 we get |dψ| = |dφ| mod ker G in ( ∩ V )+ \S(u) ∪ S(v) ∗ ∗ + where ( ∩ V ) := {q ∈  ∩ V : u(q) − v(q) > 0}. By Lemma 6.3 we get ∇v = ∇u in ( ∩ V )+ and hence u − v = constant > 0 in (  ∩ V )+ . On the other hand, we have v ≥ u on ∂( ∩ V ) by assumption. We get contradiction by continuity of v and   u. So ( ∩ V )+ is an empty set. We then conclude that v ≥ u in  ∩ V.

Theorem J . Suppose (M, ·, · ∗ , dv M ) of dimension m + 1 has isometric translations a near p0 ∈ M, transversal to a hypersurface  passing through p0 . Take a system of translation-isometric coordinates x 1 , x 2 , . . . , x m+1 in a neighborhood  of p0 such that x m+1 = 0 on (connected) ∩. Suppose u (v, resp.): ∩ → R defines a graph {(x 1 , x 2 , . . . x m , u(x 1 , x 2 , . . . x m ))}({(x 1 , x 2 , . . . x m , v(x 1 , x 2 , . . . x m ))}, resp.) ⊂ . Assume (1) v ≥ u on  ∩  such that S∩ (u) = { p0 } (S∩ (v) = { p0 }, resp.), p0 = (0, . . . , 0), and v( p0 ) = u( p0 ) = 0; (2) H (v) ≤ H (u) in ( ∩ )\{{ p0 } ∪ S∩ (v)} ((∩)\{{ p0 } ∪ S∩ (u)}, resp.). Suppose Hm−1 (S∩ (v)) = 0(Hm−1 (S∩ (u)) = 0, resp.). Moreover, we assume the rank condition (1.4) holds. Then we have v ≡ u in  ∩ . Proof The idea is similar as in the proof of Theorem J. Case 1: Suppose v = u at q = p0 . Observe that q is nonsingular with respect to u (v, resp.) since p0 is the only singular point of u (v, resp.). On the other hand, we have ∇(v − u) = 0 at q since v − u ≥ 0 and v − u = 0 at q. It follows that dψ = dv − d x m+1 = du − d x m+1 = dφ at (q, v(q) = (q, u(q)). So ξ  ker dφ at (q, u(q)) implies ξ  ker dψ at (q, v(q)). That is, q is also nonsingular with respect to v (u, resp.). By Theorem C we obtain v ≡ u in a connected component W of nonsingular (with respect to both v and u) set, containing q. We claim W = ( ∩ )\{ p0 }. Otherwise there is a point q  ∈ (S∩ (v) ∩ W¯ )\{ p0 } ((S∩ (u) ∩ W¯ )\{ p0 }, resp.) at which ∇v = ∇u, and hence ξ ⊂ ker dψ = ker dφ at (q  , v(q  ) = (q  , u(q  )). So q  is also a singular point of u (v, resp.), a contradiction to p0 being the only singular point of u (v, resp.). Hence v ≡ u in  ∩ . Case 2: Suppose v > u in ( ∩ )\{ p0 }. So there is a small ball B ⊂  ∩ , centered at p0 such that v > u in B\{ p0 }. It follows that v ≥u+c

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Strong maximum principle for mean curvature operators…

on ∂ B for some constant c > 0. By Theorem C˜  , we conclude that v ≥ u + c in B. But v( p0 ) = u( p0 ) implies 0 ≥ c, a contradiction. We have shown the impossibility of this case.  

7 Applications: uniqueness and nonexistence Consider the Heisenberg cylinder Hn \{0} with C R structure same as Hn and contact form θ=

1 , ρ2

 denoted as (Hn \{0}, ρ −2 ), where  := dz + nj=1 (x j dy j − y j d x j ) and ρ :=  [( nj=1 (x 2j + y 2j ))2 + 4z 2 ]1/4 . Here x1 , y1 , . . . , xn , yn , z denote the coordinates of Hn . Topologically Hn \{0} is homeomorphic to S 2n × R + through the map  (x1 , y1 , . . . , xn , yn , z) →

x1 yn z ,..., , 2 ρ ρ ρ



 ,ρ

where the Heisenberg sphere S 2n ⊂ Hn is defined by ρ = 1. Next we want to compute horizontal ( p-)mean curvature of a hypersurface of (Hn \{0}, ρ −2 ), described by a defining function φ. Take an orthonormal basis e I := ρ eˆ I , 1 ≤ I ≤ 2n, with respect ∂ where to the Levi metric (see Sect. 2) and T := ρ 2 ∂z ∂ ∂ + y j , for 1 ≤ I = j ≤ n ∂x j ∂z ∂ ∂ eˆ I = eˆ j  = − x j , for n + 1 ≤ I = j  = n + j ≤ 2n. ∂yj ∂z

eˆ I = eˆ j =

So the dual coframe is θ I = ρ −1 d x I (x j = x j , x n+ j = y j for 1 ≤ j ≤ n) and θ = ρ −2  and the associated subRiemannian metric ·, · ∗ on Hn \{0} is given by θ I , θ K ∗ = δ I K , ·, θ ∗ = θ, · ∗ = 0 (cf. (2.2)). We have the volume form d V := θ 1 ∧ θ 2 ∧ · · · ∧ θ 2n ∧ θ = ρ −(2n+2) d x 1 ∧ d x 2 ∧ · · · ∧ d x 2n ∧ . (7.1) Compute dφ = (e I φ)θ I + (T φ)θ ∂φ = (eˆ I φ)d x I + , ∂z

(7.2)

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and hence |dφ|2∗ := dφ, dφ ∗ 2n 2n   2 2 = (e I φ) = ρ (eˆ I φ)2 . I =1

(7.3)

I =1

From (7.1), (7.2), we compute dφ  dV |dφ|∗ = (−1) I −1

eI φ 1 θ ∧ · · · ∧ θˆ I ∧ · · · ∧ θ 2n ∧ θ |dφ|∗

(θˆ I means θ I deleted) eˆ I φ −(2n+1) 1 ρ d x ∧ · · · ∧ d xˆ I ∧ · · · ∧ d x 2n ∧  = (−1) I −1 |dφ| Hn dφ = ρ −(2n+1)  Hn d VHn . |dφ| Hn

(7.4)

where d VHn denotes the standard volume form of Hn , which is d x 1 ∧ · · · ∧ d xˆ I ∧ · · · ∧ d x 2n ∧  and | · | Hn denotes the length with respect to the standard subRiemannian  2 1/2 . Taking exterior metric in Hn (see (2.1)). It follows that |dφ| Hn = ( 2n I =1 (eˆ I φ) ) differentiation of (7.4) gives 

 dφ  dV Hφ d V := d |dφ|∗   dφ −(2n+1) d  Hn d VHn = ρ |dφ| Hn eˆ I φ + eˆ I (ρ −(2n+1) )d VHn |dφ| Hn # $ −(2n+1) ˆ −2n−2 eˆ I φ = ρ eˆ I ρ d VHn Hφ − (2n + 1)ρ |dφ| Hn

(7.5)

where Hˆ φ denotes the horizontal ( p-)mean curvature with respect to the standard eˆ I φ subRiemannian metric in Hn . Observe that |dφ| eˆ I is the horizontal normal to hyperH n

φ

surfaces defined by φ = constant, denoted as e2n . By (7.1) and (7.5), we obtain φ Hφ = ρ Hˆ φ − (2n + 1)e2n ρ

(7.6)

 I 2 1/2 , we want to (cf. Lemma 7.2 in [10]). For φ = u(r ) − z where r = [ 2n I =1 (x ) ] get a formula for Hφ in terms of u and its derivatives. First we compute |dφ| Hn as follows:

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Strong maximum principle for mean curvature operators… 2n  (eˆ I φ)2

|dφ|2Hn =

I =1 2n 



(u  (r )∂ I r − x I )2

=

I =1 2n  

u  (r )

=

I =1 

xI  − xI r

2

= (u  (r ))2 + r 2

(7.7)



where x I := x n+ j for I = j, x I := −x j for I = n + j, 1 ≤ j ≤ n and note that 2n I I  eˆ I φ I =1 x x = 0. Next noting that ∂z ( |dφ| H ) = 0, we compute n

 eˆ I (φ) |dφ| Hn    u  (r )∇r − (x I ) = div R 2n  (u  (r ))2 + r 2

Hˆ φ = eˆ I



(7.8)



by (7.7). Observe that ∂ I x I = 0 and 2n 

  x I ∂r ( (u  (r ))2 + r 2 )−1 ∂ I r

I =1 2n    = r −1 ∂r ( (u  (r ))2 + r 2 )−1 x I x I = 0. I =1

So we can reduce (7.8) to  Hˆ φ = div R 2n 

u  (r )∇r



 (u  (r ))2 + r 2 u  (r )r 2n−1

∇r



 (u  (r ))2 + r 2 r 2n−1       ∇r u  (r )r 2n−1 ∇r since div R 2n =0 = ∇ · 2n−1 r r 2n−1 (u  (r ))2 + r 2   1 u  (r )r 2n−1 d  (since |∇r |2 = 1). = 2n−1 dr (u  (r ))2 + r 2 r

= div R 2n

(7.9)

On the other hand, we compute φ

e2n ρ =

eˆ I φ r 2 (r u  (r ) − 2u(r ))  eˆ I ρ = |dφ| Hn ρ 3 (u  (r ))2 + r 2

(7.10)

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for φ = u(r ) − z. Substituting (7.9) and (7.10) into (7.6), we obtain ρ

d Hφ = 2n−1 r dr



−(2n + 1)



u  (r )r 2n−1



(u  (r ))2 + r 2

r 2 (r u  (r ) − 2u(r ))  . ρ 3 (u  (r ))2 + r 2

(7.11)

For φ = u(r ) − z with u(r ) = cr 2 , c being a constant, we get 2(2n − 1)c ρ Hφ = √ 1 + 4c2 r 2(2n − 1)c = (1 + 4c2 )1/4

(7.12)

at points where φ = cr 2 − z = 0. That is to say, the hypersurface in the Heisenberg cylinder (Hn \{0}, ρ −2 ), defined by z = cr 2 , has constant horizontal ( p-)mean curvature as shown in (7.12) (note that at p0 ∈  := {φ = c}, we have Hφ ( p0 ) = H ( p0 ), the horizontal ( p-)mean curvature of . See Proposition 2.1 in Sect. 2). Next we want to compute the horizontal ( p-)mean curvature of Heisenberg spheres defined by ρ 4 = c > 0. Let φ = ρ 4 − c. We compute 

eˆ I φ = (∂ I + x I ∂z )((r 2 )2 + 4z 2 ) 

= 4(r 2 x I + 2x I z). It follows that |dφ|2Hn =

2n 

(eˆ I φ)2

I =1

= 16(r 6 + 4r 2 z 2 ) = 16r 2 ρ 4 .

(7.13)

Then a straightforward computation shows r Hˆ φ = (2n + 1) 2 . ρ On the other hand, since 4ρ 3 eˆ I ρ = eˆ I φ for φ = ρ 4 − c, we have φ

e2n ρ = =

2n  eˆ I φ eˆ I ρ |dφ| Hn I =1 2n  I =1

123

(eˆ I φ)2 |dφ| Hn (4ρ 3 )

(7.14)

Strong maximum principle for mean curvature operators…

=

|dφ| Hn r = 3 4ρ ρ

(7.15)

by (7.13). Substituting (7.14) and (7.15) into (7.6) gives Hφ = ρ(2n + 1)

r r − (2n + 1) = 0. 2 ρ ρ

So this means that Heisenberg spheres {ρ 4 = c} are horizontally ( p-)minimal hypersurfaces in the Heisenberg cylinder (Hn \{0}, ρ −2 ). We summarize what we obtain so far as follows: Proposition 7.1 Let  be a hypersurface in the Heisenberg cylinder (Hn \{0}, ρ −2 ) with n ≥ 1. We have (a) Suppose  is defined by z = cr 2 for a constant c. Then  is a hypersurface of 2(2n−1)c constant horizontal ( p-)mean curvature with constant (1+4c 2 )1/4 ;

(b) Suppose  is defined by ρ 4 = c for a constant c > 0. Then  is a horizontally ( p-)minimal hypersurface. Observe that the dilation τλ : (x 1 , . . . , x 2n , z) → (λx 1 , . . . , λx 2n , λ2 z) preserves = ρ −2  for any λ ∈ R\{0}. So τλ is a pseudoHermitian isomorphism of the Heisenberg cylinder (Hn \{0}, ρ −2 ). We can now prove a uniqueness result stated in Theorem L in Sect. 1. ρ −2 , i.e., τλ∗ (ρ −2 )

Proof of Theorem L For the case (a), we take a Heisenberg sphere S(c1 ) defined by ρ 4 = c1 for c1 large enough so that the interior region {ρ 4 < c1 } of S(c1 ) contains . Decrease (or take) c1 to reach a constant c2 > 0 so that S(c2 ) is tangent to  at some point p0 while  lies in {ρ 4 ≤ c2 }. Observe that H ≤ 0 = HS(c2 ) near p0 by the assumption and Proposition 7.1(b). It follows from the SMP (Theorem C and Theorem J ) that  must coincide with S(c2 ). Similarly for the case (b), we can first find a Heisenberg sphere S(c3 ) defined by ρ 4 = c3 for c3 small enough so that S(c3 ) is contained in the interior region of . Increase (or take) c3 to reach a constant c4 > 0 so that S(c4 ) is tangent to  at some point q while {ρ 4 ≤ c4 } is contained in the interior region of . Now observe that H ≥ 0 = HS(c4 ) near q by the assumption and Proposition 7.1(b). So it follows that  = S(c4 ) by the SMP.   We can also show a nonexistence result (pseudo-halfspace theorem).  2 )}, we consider Proof of Theorem N For the first case  = {z > ϕ( x12 + · · · + x2n comparison hypersurfaces-horizontal hyperplanes {z = c, a constant}. Starting from

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c = c0 < minτ ∈[0,∞) ϕ(τ ) (existence by the assumption: limτ →∞ ϕ(τ ) = ∞), we increase c to reach c = c1 such that the hyperplane {z = c1 } is tangent to  at some point p1 at the first time. The existence of such p1 ∈ {z = c1 } ∩  is due to the immersion being proper. Note that the hyperplane {z = c1 } is horizontally ( p-)minimal and its singular set consists of one isolated singular point (0, . . . , 0, c1 ). We can then apply the SMP (Corollary I or Theorem J)at p1 to conclude that  ⊂

2 )}. But such the hyperplane {z = c1 } which touches ∂ = {z = ϕ( x12 + · · · + x2n  is not properly immersed in .  2 + z 2 )}, we consider compariFor the second case  = {x1 > ϕ( x22 + · · · + x2n son hypersurfaces -vertical hyperplanes {x1 = c, a constant}. By a similar reasoning as for the first case, we can find c = c1 such that the hyperplane {x1 = c1 } is tangent to  at some point q. Note that the hyperplane {x1 = c1 } is horizontally ( p-)minimal and has no singular points. Apply the SMP (Corollary  D) to this situation to conclude 2 + z 2 )}. But such that  ⊂ {x1 = c1 } which touches ∂ = {x1 = ϕ( x22 + · · · + x2n  is not properly immersed in .  

The simplest example is ϕ(τ ) = aτ with a > 0. Call associated domains wedgeshaped. Theorem N tells us nonexistence of p -minimal hypersurfaces in wedge-shaped domains. But Theorem N does not hold for the case a = 0. That is, halfspace theorem does not hold since there are catenoid type horizontal ( p-)minimal hypersurfaces with finite height ( [21]) in Hn for n ≥ 2. On the other hand, we do have halfspace theorem for H1 (see [7]). Acknowledgements J.-H. Cheng, H.-L. Chiu, and J.-F. Hwang would like to thank the Ministry of Science and Technology of Taiwan for the support of the projects: MOST 104-2115-M-001-011-MY2, 104-2115M-008-003-MY2, and 104-2115-M-001-009-MY2, resp. J.-H. Cheng is also grateful to the National Center for Theoretical Sciences of Taiwan for the constant support. P. Yang would like to thank the NSF of the US for the support of the project: DMS-1509505.

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