Annals of Global Analysis and Geometry 18: 541–554, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.
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Gap Phenomena for p-Harmonic Maps ANA-MARIA MATEI Laboratoire de Mathématiques et Physique Théorique UPRES A 6083 du CNRS, Université de Tours, 37200 Parc de Grandmont, Tours, France. e-mail:
[email protected] (Received: 13 April 1999; accepted: 8 October 1999) Abstract. In this paper, we generalize to p-harmonic maps some gap results known for harmonic maps. In particular, we prove that, under a certain level of energy depending on the curvature of the domain and target manifolds, the only p-harmonic maps are the constant ones. The main tools are Bochner–Weitzenböck and Reilly-type formulas involving the p-Laplace operator. Mathematics Subject Classifications (2000): 53C21, 58C99, 58E20. Key words: p-energy, p-harmonic maps, p-Laplacian.
1. Introduction The of harmonic maps, critical points of the energy functional E(φ) = R theory 2 |dφ| ν defined on the space of smooth maps between two Riemannian manig M folds, is one of the most important actual research themes. A natural generalization of the notion of a harmonic map is that R of a p-harmonic map (p > 1), that is a critical point of the p-energy Ep (φ) = M |dφ|p νg . The p-harmonic maps appear in many contexts in physics: non-Newtonian fluids, nonlinear elasticity, glaceology. The aim of this paper is the study of the influences of the curvatures of the domain and target manifolds on the energy of p-harmonic maps. The main instruments for this study are a Bochner–Weitzenböck-type formula involving the p-Laplacian (Lemma 2.1) and a Reilly-type formula which is an extension of the first one in the case where the domain has a nonempty boundary (Lemma 2.2). These formulas are the subject of Section 2. The results that we obtain when the domain manifold is without boundary are essentially the following: denote by RicM the Ricci curvature of the domain manifold (M, g), that we suppose to be compact, and by σ N the sectional curvature of the target manifold (N, h). (i) Generalization of Eells and Sampson’s result [3]. If RicM is nonnegative and σ N is nonpositive, then any p-harmonic map φ: (M, g) → (N, h) is totally geodesic. Furthermore, if
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– RicM is positive definite at least at one point of M, then φ is constant. – σ N is strictly negative, then φ(M) is a geodesic (may be a point). (ii) Gap level for the energy density. If RicM ≥ k0 and σ N ≤ σ1 , where k0 and σ1 are two positive constants, then for any p-harmonic map φ: (M, g) → (N, h): Ep (φ) ≤
m − 1 σ1 m k0
s/2 Ep+s (φ),
∀s ≥ 2.
Moreover, if equality holds for an s ≥ 2, then φ is a totally geodesic dilatation. As a consequence, we obtain that the energy density e(φ) = |dφ|2 of a nonconstant p-harmonic map φ: (M, g) → (N, h) exceeds at least, at one point of M, the level (m/m − 1)(k0 /σ1 ): ||e(φ)||∞ ≥
m k0 , m − 1 σ1
Furthermore, equality implies that φ is a totally geodesic dilatation. (iii) Gap level for the m-energy. Under the same hypotheses as in (ii), we have that for any nonconstant pharmonic map φ: (M, g) → (N, h), (p − 1) m(m − 2) m k0 m/2 Em (φ) ≥ min Vol(M) , 1 p2 m−1 m − 1 σ1
if m ≥ 3,
where Em (φ) is the m-energy which depends in fact only on the conformal class of the metric g, and where Vol(M) is the Riemannian volume of (M, g). For m = 2 we have the lower bound E4 (φ) ≥
8(p − 1) k0 p2 σ1
2 Vol(M).
When M has nonempty boundary, two boundary conditions are classically studied: the Neumann condition (i.e. dφ vanishes on the normal direction to the boundary) and the Dirichlet condition (i.e. φ is constant on ∂M). For M orientable, we proved that if ∂M is convex, then the results (i), (ii), (iii) are still true for p-harmonic maps which satisfy to the Neumann condition. On the other hand, if the mean curvature of ∂M is nonnegative, then the results (i), (ii) are still true for p-harmonic maps satisfying the Dirichlet condition.
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2. Bochner–Weitzenböck and Reilly-Type Formulas Throughout this paper we denote by (M, g) and (N, h) two connected Riemannian manifolds where M is compact, m-dimensional, and may have boundary. If ∂M 6 = ∅, in addition, we assume that M is orientable. Let φ: (M, g) → (N, h) be a smooth map. For any p > 1, the p-energy of φ is the real number Z Ep (φ) = |dφ|p νg , M
where νg is the Riemannian volume element induced by g and |dφ| is the Hilbert– Schmidt norm of dφ: |dφ|2x
=
m X
h(dφ(ei ), dφ(ei )),
i=1
(ei ), i = 1, . . . , m being an orthonormal basis of Tx M. A p-harmonic map is a critical point of the p-energy functional. We denote by h , i the Riemannian metric induced on the bundle ∧p T ∗ M ⊗ φ ∗ T N by g and h, and by hh , ii the associated L2 -norm. Let ∇ be the canonical connection of T ∗ M ⊗ φ ∗ T N induced by those of (M, g) and (N, h). The natural extension of ∇ to the bundle ∧p T ∗ M ⊗ φ ∗ T N allows to define the exterior differential d and its adjoint δ w.r.t. the L2 -norm hh, ii (see [2] for details). The Euler–Lagrange equation for p-harmonic maps is 1p φ = 0, where 1p φ := δ(|dφ|p−2 dφ) is the p-Laplacian of φ. Recall that the classical Bochner–Weitzenböck formula gives [2]: 1 dφ = δ∇ dφ + dφ ◦ RicM − Ricφ , where 1 := dδ + δ d is the Hodge–de Rham Laplacian, RicM the Ricci curvature of (M, g), and where Ricφ is given by Ricφx (X)
=
m X
R N (dφ(ei ), dφ(X))dφ(ei ) ,
i=1
(ei ), i = 1, . . . , m being an orthonormal basis of Tx M and R N the curvature tensor of (N, h). It follows that h1 dφ, dφi = 12 1(|dφ|2 ) + |∇dφ|2 + RM,N (dφ), where, in an orthonormal basis of Tx M RM,N (dφ) = hdφ ◦ RicM − Ricφ , dφi m X = hdφ ◦ RicM (ei ), dφ(ei )i − i=1
−
m X i,j =1
R N (dφ(ei ), dφ(ej ), dφ(ei ), dφ(ej )).
(1)
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Several generalizations of this formula in the case of the p-Laplacian were established in the literature [1, 7, 8]. In this paper, we will need the following version. (As the proof is immediate we will give it here for the sake of self-containedness.) LEMMA 2.1. Assume that ∂M = ∅. For any smooth map φ: (M, g) → (N, h): Z Z h1p φ, 1φi νg = (p − 2)|dφ|p−2 |d(|dφ|)|2 + |dφ|p−2 |∇dφ|2 νg + M
M
Z
+
|dφ|p−2 RM,N (dφ) νg .
(2)
M
Proof. Z Z h1p φ, 1φi νg = hδ(|dφ|p−2 dφ), δdφi νg M
Z
M
=
|dφ|p−2 h1 dφ, dφi νg . M
Applying formula (1), we obtain |dφ|p−2 h1 dφ, dφi =
1 |dφ|p−2 1(|dφ|2 ) 2
+
+ |dφ|p−2 |∇dφ|2 + |dφ|p−2 RM,N (dφ).
(3)
On the other hand, 1 |dφ|p−2 1(|dφ|2 ) 2
= 12 δ(|dφ|p−2 d(|dφ|2 )) + (p − 2)|dφ|p−2 |d(|dφ|)|2 .
Replacing in (3), we get |dφ|p−2 h1 dφ, dφi = (p − 2)|dφ|p−2 |d(|dφ|)|2 + |dφ|p−2 |∇dφ|2 + + |dφ|p−2 RM,N (dφ) + 12 δ(|dφ|p−2 d(|dφ|2 )). 2
Integrating this last equality, we obtain the desired result.
In the case where M is orientable with nonempty boundary, we will denote by η the exterior unit normal vector field of ∂M. The Weingarten operator Ax : Tx ∂M → Tx ∂M at x ∈ ∂M is then given by Ax v = Dv η, where D is the Levi-Civita connection of (M, g). The mean curvature of ∂M is H = htrace A, ηi. In this case we have the following Reilly-type formula: LEMMA 2.2. For any smooth map φ: (M, g) → (N, h): Z Z h1p φ, 1φi νg = (p − 2)|dφ|p−2 |d|dφ||2 + |dφ|p−2 |∇dφ|2 νg + M
M
Z
+
Z |dφ|
M
p−2
RM,N (dφ) νg +
|dφ|p−2 B(φ) νg 0 , (4) ∂M
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where B(φ) = hdφ ◦ A − ∇· (dφ(η)), dφi∂M − h1∂M φ, dφ(η)i + H |dφ(η)|2 and νg 0 is the Riemannian volume element induced by the canonical metric g 0 of ∂M. For p = 2 a similar formula was obtained by Ilias [4]. The original Reilly formula concerning real valued functions can be found in [5]. Proof of Lemma 2.2. Let x ∈ M and let (ei ), i = 1, . . . , m be an orthonormal basis of Tx M. h1p φ, 1φi(x) = −
m X
h∇ei (|dφ|p−2 dφ)(ei ), 1φi(x)
i=1
= −
m X
∇ei (h|dφ|p−2 dφ(·), 1φi)(ei )(x) +
i=1
+
m X
|dφ|p−2 hdφ(ei ), ∇ei 1φi(x)
i=1
= δ(h|dφ|p−2 dφ(·), 1φi)(x) + +
m X
|dφ|p−2 hdφ(ei ), d1φ(ei )i(x)
i=1
= δ(h|dφ|p−2 dφ(·), 1φi)(x) + |dφ|p−2 hdφ, d1φi(x) = δ(h|dφ|p−2 dφ(·), 1φi)(x) + |dφ|p−2 hdφ, 1 dφi(x). Integration gives Z Z Z h1p φ, 1φi νg = δ(h|dφ|p−2 dφ(·), 1φi) νg + |dφ|p−2 hdφ, 1 dφi νg M
M
M
Z
Z
=−
|dφ|p−2 hdφ(η), 1φi νg 0 + ∂M
|dφ|p−2 hdφ, 1 dφi νg . M
We have seen in the proof of Lemma 2.1 that |dφ|p−2 h1 dφ, dφi = (p − 2)|dφ|p−2 |d(|dφ|)|2 + |dφ|p−2 |∇dφ|2 + + |dφ|p−2 RM,N (dφ) + 12 δ(|dφ|p−2 d(|dφ|2 )) . Hence
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ANA-MARIA MATEI
Z |dφ|p−2 h1 dφ, dφi νg M
Z =
(p − 2)|dφ|p−2 |d(|dφ|)|2 + |dφ|p−2 |∇dφ|2 νg + M
Z
+
|dφ|
p−2
M
Z =
1 RM,N (dφ) νg − 2
Z |dφ|p−2 d(|dφ|2 )(η) νg 0 ∂M
(p − 2)|dφ|p−2 |d(|dφ|)|2 + |dφ|p−2 |∇dφ|2 νg + M
Z
+
Z |dφ|
p−2
RM,N (dφ) νg −
|dφ|p−2 h∇η dφ, dφi νg 0 .
M
∂M
Therefore Z h1p φ, 1φi νg M
Z =
(p − 2)|dφ|p−2 |d(|dφ|)|2 + |dφ|p−2 |∇dφ|2 νg + M
Z
+
Z |dφ|
p−2
RM,N (dφ) νg −
M
Z −
|dφ|p−2 hdφ(η), 1φi νg 0 − ∂M
|dφ|p−2 h∇η dφ, dφi νg 0 .
(5)
∂M
A straightforward calculation gives hdφ(η), 1φi + h∇η dφ, dφi = h∇· (dφ(η)), dφi∂M − hdφ ◦ A, dφi∂M + h1∂M φ, dφ(η)i − H |dφ(η)|2 . Replacing in (5), we obtain the desired formula.
2
3. Statement and Proof of the Results Denote by σ N the sectional curvature of (N, h). When M has nonempty boundary, ∂M is said to be convex (respectively, strictly convex) at a point x ∈ ∂M if all the eigenvalues of the Weingarten operator Ax (i.e. the principal curvatures) are nonnegative (respectively, positive). The following two propositions generalize a result established by Eells and Sampson [3] for harmonic maps as well as its extension to harmonic maps defined on manifolds with nonempty boundary obtained by Sealey [6]. PROPOSITION 3.1. Let φ: (M, g) → (N, h) be a p-harmonic map and suppose that RicM ≥ 0, σ N ≤ 0. If ∂M 6 = ∅, we assume in addition that one of the following assumptions hold
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(N): ∂M is convex and dφ(η) = 0 (Neumann type condition); (D): H ≥ 0 and φ|∂M = const (Dirichlet type condition). Then (i) φ is totally geodesic. (ii) If RicM is positively definite at some point, then φ is constant. (iii) If σ N < 0 then the image of φ is a geodesic (may be a point). PROPOSITION 3.2. Let φ: (M, g) → (N, h) be a p-harmonic map, and suppose that RicM ≥ 0, σ N ≤ 0. Then (i) If ∂M is convex and strictly convex at least at one point and if dφ(η) = 0, then φ is constant. (ii) If H ≥ 0 and H > 0 at least at one point of ∂M and if φ|∂M is constant, then φ is constant. Proof of Proposition 3.1. Let us start with the case where ∂M 6 = ∅. As φ is p-harmonic, Lemma 2.1 gives Z 0 = (p − 2)|dφ|p−2 |d(|dφ|)|2 + |dφ|p−2 |∇dφ|2 νg + M
Z
+
|dφ|p−2 RM,N (dφ) νg .
(6)
M
By Kato’s inequality, |∇dφ|2 ≥ |d(|dφ|)|2 . Hence (p − 2)|d(|dφ|)|2 + |∇dφ|2 ≥ (p − 1)|d(|dφ|)|2 ≥ 0.
(7)
Assumptions on RicM and σ N implies that RM,N (dφ) = hdφ ◦ RicM − Ricφ , dφi is nonnegative. Indeed, let (ei ), i = 1, . . . , m be an orthonormal basis of Tx M which diagonalizes RicM x . Then hdφ ◦ Ric , dφix = M
m X
hdφ ◦ RicM (ei ), dφ(ei )i
i=1
≥ kx
m X
|dφ(ei )|2 = kx |dφ|2x ≥ 0,
i=1
where kx is the smallest eigenvalue of RicM x . Moreover hRicφ , dφix =
m X i,j =1
|dφ(ei ) ∧ dφ(ej )|2 σ N (dφ(ei ) ∧ dφ(ej )) ≤ 0.
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ANA-MARIA MATEI
It follows from (6) and (7) that |dφ| is constant and |dφ|p−2 |∇dφ|2 = 0 on M. Therefore, either dφ = 0 and then φ is a constant map or dφ 6 = 0 and then ∇ dφ = 0, that is φ is totally geodesic. Now, Equation (6) implies that |dφ|p−2 RM,N (dφ) = 0. Thus, ∀x ∈ M kx |dφ|px = 0 and
φ |dφ|p−2 x hRic , dφix = 0.
− If RicM is positive definite at a point x, (i.e. kx > 0), then |dφ|x = 0. As |dφ| is constant, this implies that φ is constant on M. − If σ N is negative on N, then |dφ|p−2 hRicφ , dφi vanishes if and only if the rank of φ is at most 1. As φ is totally geodesic, then either φ is constant or φ(M) is a (closed) geodesic. Consider now the case where ∂M is nonempty. From Lemma 2.2 and the discussion above, it suffices to prove that under each of the boundary assumptions, (N) and (D), B(φ) is nonnegative. Indeed − if ∂M is convex and dφ(η) = 0 then B(φ) reduces to hdφ ◦ A, dφi. But in an orthonormal basis {ε1 , . . . , εm−1 } of Tx ∂M, x ∈ ∂M, which diagonalizes Ax we have B(φ)(x) =
m−1 X
hdφ(A(εi )), dφ(εi )i =
i=1
≥ µ1
m−1 X
µi |dφ(εi )|2
i=1
m−1 X
|dφ(εi )|2 = µ1 |dφ|2x ≥ 0,
i=1
where µ1 ≤ . . . ≤ µm−1 are the eigenvalues of A at x. − if φ|∂M is constant, then B(φ) reduces to H |dφ(η)|2 = H |dφ|2 which is nonnegative where H is nonnegative. 2 Proof of Proposition 3.2. As in the proof of Proposition 3.1, the hypothesis of Proposition 3.2 implies that all the terms of the right-hand member of (4) are nonnegative. Therefore, |dφ| is constant on M and |dφ|p−2 B(φ) = 0 on ∂M. Now − if ∂M is convex and strictly convex at a point x and dφ(η) = 0, then using the same notations as in the proof of Proposition 3.1, we obtain p 0 = |dφ|p−2 x B(φ)(x) ≥ µ1 |dφ|x .
As µ1 is positive (∂M is strictly convex at x), dφx = 0 and since |dφ| is constant, φ is constant on M.
GAP PHENOMENA FOR P-HARMONIC MAPS
549
− if H ≥ 0 and H > 0 at a point x and φ|∂M is constant, then p 0 = |dφ|p−2 x B(φ)(x) = Hx |dφ|x .
This implies that dφx = 0 and, since |dφ| is constant, φ is constant on M. 2 The following theorem emphasizes a remarkable relationship between the p-energy and the q-energies, q ≥ p + 2, of a p-harmonic map. THEOREM 3.1. Suppose that there exists two positive constants k0 and σ1 such that RicM ≥ k0 and σ N ≤ σ1 . Let φ: (M, g) → (N, h) be a p-harmonic map. If ∂M 6 = ∅, suppose in addition that one of the two following assumptions is satisfied: − ∂M is convex and dφ(η) = 0; − H ≥ 0 and dφ|∂M = const. Then for any s ≥ 2: m − 1 σ1 s/2 Ep (φ) ≤ Ep+s (φ). m k0 Furthermore, if equality holds for an s ≥ 2, then φ is a totally geodesic dilatation. Denote by e(φ) the 2-energy density of φ: e(φ) = |dφ|2 . A direct consequence of Theorem 3.1 is the following corollary COROLLARY 3.1. Under the same hypotheses as in Theorem 3.1, we have that if φ is not constant, then ||e(φ)||∞ ≥
m k0 . m − 1 σ1
Furthermore, equality implies that φ is a totally geodesic dilatation. Remark. Note that equality holds in Theorem 3.1 and Corollary 3.1 for the canonical embedding i: (S m , can) → (S n , can) as well as for the embedding of the hemisphere i: (S+m , can) → (S n , can) (this latter satifies to the Neumann boundary condition). Indeed, for these maps: e(i) = m =
m k0 m − 1 σ1
(here k0 = m − 1 and σ1 = 1).
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ANA-MARIA MATEI
Proof of Theorem 3.1. Let x ∈ M and let (ei ), i = 1, . . . , m be an orthonormal basis of Tx M which diagonalizes φ ∗ h. Then m X
hRicφ , dφi =
R N (dφ(ei ), dφ(ej ), dφ(ei ), dφ(ej ))
i,j =1
≤ σ1
m X
|dφ(ei ) ∧ dφ(ej )|2
i,j =1
= σ1
m X
(|dφ(ei )|2 |dφ(ej )|2 − hdφ(ei ), dφ(ej )i2 )
i,j =1
= σ1 (|dφ|4 −
m X
|dφ(ei )|4 ).
i=1
By Cauchy–Schwarz inequality, |dφ|4 ≤ m
m X
|dφ(ei )|4
(8)
i=1
and, therefore, |dφ|4 −
m X
|dφ(ei )|4 ≤
i=1
m−1 |dφ|4 . m
It follows that hRicφ , dφi ≤
m−1 σ1 |dφ|4 . m
Moreover, as in the proof of Proposition 3.1, hdφ ◦ RicM , dφi ≥ k0 |dφ|2 . Hence RM,N (dφ) ≥ k0 |dφ|2 −
m−1 σ1 |dφ|4 . m
(9)
When ∂M = ∅, Lemma 2.1 gives Z Z 0 ≥ |dφ|p−2 (p − 2)|d|dφ||2 + |∇dφ|2 νg + k0 |dφ|p νg − M
−
m−1 σ1 m
M
Z |dφ|p+2 νg . M
(10)
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GAP PHENOMENA FOR P-HARMONIC MAPS
This implies Ep (φ) ≤
m − 1 σ1 Ep+2 (φ). m k0
By Hölder inequality, for any s > 2, Z Z p+2 Ep+2 (φ) = |dφ| νg = |dφ|p(s−2)/s |dφ|(2p/s)+2 νg M
M
(s−2)/s Z
Z ≤
2/s
|dφ| νg
|dφ|
p
M
p+s
νg
M
= (Ep (φ))(s−2)/s (Ep+s (φ))2/s ≤
m − 1 σ1 Ep+2 (φ) (Ep (φ))−2/s (Ep+s (φ))2/s . m k0
Hence, for any s ≥ 2, m − 1 σ1 s/2 Ep (φ) ≤ Ep+s (φ). m k0 If equality holds in this last inequality for any s ≥ 2, then (10) and (7) imply that φ is totally geodesic and |dφ| is constant. Furthermore, equality must hold in (8) and therefore φ is a totally geodesic dilatation. Now, in the case where ∂M 6 = ∅, it suffices to check that under the hypothesis of the theorem, B(φ) is nonnegative. Thus, inequality (10) is still true and the rest of the proof follows identically as above. 2 Proof of Corollary 3.1. From Theorem 3.1 we have, for any s ≥ 2, s/2 Z p m − 1 σ1 Ep (φ) ≤ |dφ| e(φ) νg . m k0 M If ||e(φ)||∞ <
m k0 , m − 1 σ1
then the right-hand member of this inequality converges to 0 as s → ∞. Therefore, Ep (φ) = 0 and φ is constant. The equality ||e(φ)||∞ =
m k0 m − 1 σ1
implies that e(φ) is constant. We then have equality in Theorem 3.1. Hence φ is a totally geodesic dilatation. 2
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ANA-MARIA MATEI
It is of particular interest to estimate Em (φ) which is a conformal invariant. THEOREM 3.2. Suppose that there exists two positive constants k0 and σ1 such that RicM ≥ k0 and σ N ≤ σ1 and let φ: (M, g) → (N, h) be a nonconstant p-harmonic map. If ∂M 6 = ∅, we suppose in addition that ∂M is convex and φ satisfies the Neumann condition dφ(η) = 0. Then (i) If m ≥ 3, (p − 1) m(m − 2) m k0 m/2 Em (φ) ≥ min Vol(M). , 1 p2 m−1 m − 1 σ1 (ii) If m = 2, E4 (φ) ≥
8(p − 1) k0 p2 σ1
2 Vol(M),
where Vol(M) denotes the Riemannian volume of (M, g). Remark. Note that (p − 1) m(m − 2) min ,1 = 1 p2 m−1
if m ≥ 6
and
2≤p ≤m−
16 . 5
Hence, in this case, the equality holds in inequality (i) above for the standard embeddings i : (S m , can) → (S n , can) and i: (S+m , can) → (S n , can). Proof of Theorem 3.2. (i) m ≥ 3. Using Kato’s inequality |d(|dφ|)|2 ≤ |∇ dφ|2 and the obvious identity |dφ|p−2 |d(|dφ|)|2 = ||dφ|(p/2)−1 d(|dφ|)|2 =
4 |d(|dφ|p/2 )|2 , p2
we obtain from (10) Z Z Z m−1 4(p − 1) p+2 p/2 2 |dφ| νg ≥ |d(|dφ| )| νg + k0 |dφ|p νg . σ1 2 m p M M M In order to simplify the notations, we set f = |dφ|p/2 . Then Z Z Z m−1 4(p − 1) 2 f 2 f 4/p νg ≥ |df | ν + k f 2 νg σ1 g 0 2 m p M M M =
4(p − 1) ||df ||22 + k0 ||f ||22 . p2
(11)
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GAP PHENOMENA FOR P-HARMONIC MAPS
By Hölder inequality, (m−2)/m Z 2/m Z Z 2 4/p 2m/(m−2) (2m)/p f f νg ≤ f νg f νg M
M
=
||f ||22m/(m−2)
M
||f 2/p ||2m .
(12)
Since f ∈ H12 (M), we can use the following Sobolev type inequality [4] to estimate ||f ||2m/(m−2) in terms of ||df ||22 and ||f ||22 : ||f ||22m/(m−2) ≤
4(m − 1) Vol(M)−2/m ||df ||22 + Vol(M)−2/m ||f ||22 . m(m − 2)k0
(13)
From (11), (12) and (13), we obtain 4(m − 1)2 σ1 Vol(M)−2/m ||df ||22 ||f 2/p ||2m + m2 (m − 2) k0 m−1 σ1 Vol(M)−2/m ||f ||22 ||f 2/p ||2m m 4(p − 1) ≥ ||df ||22 + k0 ||f ||22 . p2 +
Hence
2 4(p − 1) 4(m − 1)2 σ1 −2/m 2 p 0 ≥ − 2 Vol(M) ||f ||m ||df ||22 + 2 p m (m − 2)k0 2 m−1 −2/m 2 + k0 − ||f p ||m ||f ||22 . σ1 Vol(M) m
This ends the proof, since 2/m Z 2/p 2 m ||f ||m = |dφ| νg = (Em (φ))2/m . M
(ii) m = 2. In order to simplify the notations, we set f = |dφ|p/2 . When M is a surface, we obtain from (11): Z 4(p − 1) 1 σ f 2 f 4/p νg ≥ ||df ||22 + k0 ||f ||22 . 2 1 2 p M By Schwarz inequality 1/2 Z 1/2 Z Z 2 4/p 4 8/p f f νg ≤ f νg f νg = ||f ||24 ||f 2/p ||24 . M
M
(14)
(15)
M
Since f ∈ H12 (M), we can use the following Sobolev type inequality [4] to ‘majorate’ ||f ||24 in terms of ||df ||22 and ||f ||22 : ||f ||24 ≤
1 Vol(M)−1/2 ||df ||22 + Vol(M)−1/2 ||f ||22 . k0
(16)
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ANA-MARIA MATEI
From (14), (15) and (16), we obtain 1 2
σ1 Vol(M)−1/2 ||df ||22 ||f 2/p ||24 + 12 σ1 Vol(M)−1/2 ||f ||22 ||f 2/p ||24 k0 ≥
4(p − 1) ||df ||22 + k0 ||f ||22 . p2
Finally,
4(p − 1) σ1 −1/2 2/p 2 0 ≥ − Vol(M) ||f ||4 ||df ||22 + p2 2k0 σ1 + k0 − Vol(M)−1/2 ||f 2/p ||24 ||f ||22 . 2 This ends the proof since 1/2 Z 2/p 2 4 ||f ||4 = |dφ| = (E4 (φ))1/2 . M
2 Acknowledgements The author wishes to thank Professor Ahmad El Soufi for his helpful advice and Professor Said Ilias for his remarks. References 1. 2. 3. 4. 5. 6. 7. 8.
Duzaar, F.: Regularity of p-harmonic maps into a closed hemisphere, Boll. UMI, B(7) 5 (1991), 157–170. Eells, J. and Lemaire, L.: Selected Topics in Harmonic Maps, Regional Conf. Ser. Math. 50, Amer. Math. Soc., Providence, RI, 1983. Eells, J. and Sampson, J.H.: Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160. Ilias, S.: Inégalités de Sobolev et resultats d’isolement pour les applications harmoniques, J. Funct. Anal. 139(1) (1996), 182–195. Reilly, R.: Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J. 26 (1977), 459–472. Sealey, H. C. J.: Harmonic maps of small energy, Bull. London Math. Soc. 13 (1981), 405–408. Takeuchi, H.: Some conformal properties of p-harmonic maps and a regularity for sphere-valued p-harmonic maps, J. Math. Soc. Japan 46(2) (1994), 217–234. Xin, Y. L. and Yang, Y.: Regularity of p-harmonic maps into certain manifolds with positive sectional curvature, J. Reine Angew. Math. 466 (1995), 1–17.
