Hypersurfaces with constant anisotropic mean curvature in Riemannian manifolds

Calc. Var.DOI 10.1007/s00526-013-0638-0 Calculus of Variations

Hypersurfaces with constant anisotropic mean curvaturein Riemannian manifolds

Jorge H. S. de Lira · Marcelo Melo

Received: 31 August 2011 / Accepted: 25 April 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract We formulate a variational notion of anisotropic mean curvature for immersedhypersurfaces of arbitrary Riemannian manifolds. Hypersurfaces with constant anisotropicmean curvature are characterized as critical points of an elliptic parametric functional subjectto a volume constraint. We provide examples of such hypersurfaces in the case of rotationallyinvariant functionals defined in product spaces. These examples include rotationally invarianthypersurfaces and graphs.

Mathematics Subject Classification (2000) 53C42 · 53A10

1 Introduction

Let Mn+1 be an oriented complete Riemannian manifold and let T M be its tangent bundle.We fix in what follows a differentiable positive function F : T M − {0} → R satisfying thefollowing homogeneity condition

F(y, tη) = t F(y, η), (1)

for all (y, η) ∈ T M, η �= 0, t > 0. Moreover, we suppose that

〈Dζ DF(y, η), ζ 〉T M > 0, (2)

for all (y, η) ∈ T M − {0} and ζ ∈ T(y,η)T M − {0} such that

〈ζ, η〉T M = 0.

Communicated by Y. Giga.

J. H. S. de Lira (B) · M. MeloDepartment of Mathematics, Federal University of Ceará,Campus do Pici, Ceará, Fortaleza 60455-900, Brasile-mail: [email protected]

123

J. H. S. de Lira, M. Melo

Here 〈·, ·〉T M denotes the Sasaki metric in T M and D stands for the Riemannian connec-tion in T M compatible with this metric. Following [1] and [2], we refer to F as an ellipticparametric Lagrangian.

This elliptic parametric Lagrangian F permits to define a parametric functional F asfollows. Given a compact oriented Riemannian manifold Mn with possibly empty boundary∂M and an isometric immersion ψ : M → M , we define

F[ψ] =∫

M

F(ψ, N ) d M, (3)

where the volume element d M is induced by the immersion ψ and N is a unit normal vectorfield along ψ .

The classical example of such a functional is the volume of the immersion, which cor-responds to the Lagrangian F(y, η) = |η|. Other examples may be found for instance at[1–3,6].

The main purpose of this paper is to formulate a suitable notion of anisotropic mean curva-ture for immersed hypersurfaces of Riemannian manifolds. For general ambient manifolds,it is no longer possible to define the anisotropic geometric invariants directly in terms ofa Wulff shape since this relies heavily on the vector structure of the Euclidean space. Ourapproach is to combine the contributions in the seminal articles by Clarenz and von der Mosel[2] and Koiso and Palmer [3].

Indeed in Sect. 2 we follow the variational setting proposed by Clarenz and von der Moselrephrasing their calculations in covariant terms. This requires to consider the geometric struc-ture in T M provided by the Sasaki’s metric. In Theorem 1 we state a first variation formulaof F in terms of an elliptic PDE of the divergence form. This variational characterization ofanisotropic mean curvature is then interpreted in terms of tensor fields defined in the immersedhypersurface. Indeed, in Sect. 3, under the assumption that the parametric Lagrangian F ishorizontally constant, the Euler–Lagrange equation in Theorem 1 is interpreted as the condi-tion that the trace of an anisotropic analog of the Weingarten map is constant. This is closelyrelated to the definition of anisotropic mean curvature presented in the Euclidean case byKoiso and Palmer [3]. We finally deduce a Codazzi’s equation for that anisotropic Weingartenmap.

The examples provided in Sect. 4 are rotationally invariant critical hypersurfaces for theparticular case of a rotationally invariant Lagrangian in the same spirit as had been done in[3] for the Euclidean case. Finally, we prove in Sect. 5 an existence theorem for graphs withconstant anisotropic mean curvature which could be regarded as an anisotropic version ofthe classical Serrin’s result about solutions for equations of the mean curvature type [7]. It isworth to mention that the results in Sects. 4 and 5 are obtained in ambient product manifoldsof the form M = P × R, where P is a Riemannian n-dimensional manifold.

2 First variation formula

In this section, we deduce a first variation formula for the parametric functional F . A variationof the immersionψ is a one parameter family of immersionsψs : M → M , with s ∈ (−ε, ε),for some ε > 0, such thatψ0 = ψ . Denoting ψs(x) = ψ(s, x),we define the lifted variationmap

�(s, x) = (ψ(s, x), N (ψ(s, x))), s ∈ (−ε, ε), x ∈ M, (4)

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Hypersurfaces with constant anisotropic

where N (ψ(s, ·)) is a unit normal vector field along ψs . Hence, we are concerned withcalculating the derivative

d

ds

∣∣∣s=0

F[ψ(s, ·)] = d

ds

∣∣∣s=0

∫

M

F(�(s, ·)) d Ms, (5)

where the volume element d Ms is induced from the volume form in M by the immersionψs .The variational vector field associated to the variation � is the vector field along ψ

defined by

�(x) = ψ∗(0, x) · ∂∂s

∣∣∣(0,x)

, x ∈ M. (6)

We decompose � in tangential and normal components according to the expression

�(x) = φ(x)N (ψ(x))+ ψ∗(x) · V (x), x ∈ M, (7)

for a certain function φ ∈ C∞(M) and a vector field V ∈ (T M).From now on, we denote both the ambient and induced metrics by 〈·, ·〉. We denote by

∇ the Riemannian connection in M with respect to the metric induced by ψ = ψ0. TheWeingarten map A of ψ is defined as

ψ∗ AW = −∇ψ∗W N , W ∈ (T M) (8)

and the second fundamental form of ψ is

I I (U,W ) = 〈ψ∗ AU, ψ∗W 〉, U,W ∈ (T M). (9)

In what follows, we consider local coordinates x1, . . . , xn in M and y1, . . . , yn+1 in M .Thus, we associate local coordinates y1, . . . , yn+1, η1, . . . , ηn+1 to points (y, η) ∈ T M bysetting

η = ηk ∂

∂yk

∣∣∣y.

The local components of the induced metric in M and of the ambient metric in M arerespectively denoted by gi j and gkl . Let hi j be the local components of I I . It is well knownthat

∂

∂s

∣∣∣s=0

gi j = −2hi j φ +⟨∇ ∂

∂xiV,

∂

∂x j

⟩+

⟨∂

∂xi,∇ ∂

∂x jV

⟩(10)

and

∇ψ∗ ∂

∂sN

∣∣s=0 = −ψ∗ AV − ψ∗∇φ, (11)

d

ds

∣∣∣s=0

d Ms = (divM V − nHφ) d M, (12)

where divM is the divergence operator in M and H = 1n trA is the mean curvature of ψ .

In order to deduce covariant expressions for the first derivative of the Lagrangian F , wefix some terminology and basic facts about the geometry of the tangent bundle T M .

The canonical projectionπ : T M → M ,π(y, η) = y, determines the vertical distribution

(y, η) ∈ T M → V(y,η) = ker π∗|(y,η) ⊂ T(y,η)T M, (13)

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which is integrable with integral leaves given by the fibers π−1(y) = Ty M . The Riemannianconnection ∇ in M determines a horizontal distribution

(y, η) ∈ M → H(y,η) = ker K(y,η) ⊂ T(y,η)T M, (14)

where the map K : T T M → T M is defined by

K(y,η)(X∗ζ ) = ∇ζ X, ζ ∈ Ty M . (15)

where X ∈ (T M) is a vector field in M with X (y) = η. It turns out that horizontal andvertical subspaces are orthogonal with respect to the Sasaki metric 〈·, ·〉T M in T M definedby

〈ϒ, ϒ〉T M = 〈π∗ϒ,π∗ϒ〉 + 〈Kϒ, K ϒ〉, ϒ, ϒ ∈ (T T M). (16)

Orthogonal projections onto vertical and horizontal subspaces are respectively denoted byπv∗ and πh∗ . In terms of the local coordinates defined earlier, the vertical and horizontal liftsof tangent vectors in M are given by

(∂

∂yk

)v ∣∣∣(y,η)

= ∂

∂ηk

∣∣∣(y,η)

, (17)

(∂

∂yk

)h ∣∣∣(y,η)

= ∂

∂yk

∣∣∣(y,η)

− rkl(y)η

l ∂

∂ηr

∣∣∣(y,η)

, (18)

for 1 ≤ k ≤ n + 1, where rkl are the Christoffel symbols of ∇ calculated with respect to

the local chart y1, . . . , yn+1. In particular, given a curve α : (−ε, ε) → T M of the formα(s) = (β(s), η(s)) with α(0) = (y, η), we compute

α′ = (β ′)h + (∇β ′η)v. (19)

Hence, if s → η(s) is the parallel transport of the tangent vector η ∈ Ty M along the curveβ, then the tangent vector α′(0) ∈ T(y,η)T M is the horizontal lift of the tangent vectorβ ′(0) ∈ Ty M .

Vertical and horizontal lifts preserve the metric in the sense that

〈Xh, Y h〉T M = 〈Xv, Y v〉T M = 〈X, Y 〉, (20)

for any vector fields X, Y ∈ (T M). This implies that the fibers π−1(y), y ∈ M, are totallygeodesic. Thus, the coordinates (ηk)n+1

k=1 defined above consist of a global normal coordinatesystem for each fiber. More precisely, it holds that the Riemannian connection D in T Mdetermined by the Sasaki metric satisfies

DXh Y h∣∣(y,η) = (∇X Y )h

∣∣(y,η) − 1

2 (R(X, Y )η)v∣∣(y,η), (21)

DXh Y v = (∇X Y )v∣∣(y,η) + 1

2 (R(η, Y )X)h∣∣(y,η), (22)

DXv Y h = 12 (R(η, X)Y )h

∣∣(y,η), (23)

DXv Y v = 0. (24)

for any X, Y ∈ (T M). Here R is the Riemann curvature tensor of the connection ∇ in M .Finally, the curvature tensor RT M of D satisfies

RT M (Xh, Y v)Zv|(y,η) = − ( 12 R(Y, Z)X + 1

4 R(η, Y )R(η, Z)X)h

(25)

RT M (Xv, Y v)Zv = 0. (26)

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From Gauss equation we conclude, in particular, that the vertical leaves are intrinsically flat.A unit normal vector field N along ψ may be identified with the cross-section of the

pull-back bundle ψ∗T M given by

ϕ(x) = (ψ(x), N (ψ(x))) = �(0, x). (27)

Hence, there exist unique vector fields ξ and χ along ψ such that

DF(ϕ(x)) = ξv(ψ(x))+ χh(ψ(x)). (28)

In terms of local components one has

ξ(x) = gkl(ψ(x))∂F

∂ηl(ϕ(x))

∂

∂yk

∣∣∣ψ(x)

(29)

and

χ(x) = gkl(ψ(x))

(∂F

∂yl(ϕ(x))− m

lr (ψ(x))ηr (x)

∂F

∂ηm(ϕ(x))

)∂

∂yk

∣∣∣ψ(x)

, (30)

for any x ∈ M . Notice that these coordinate expression indeed define well-defined vectorfields along ψ .

Fixed this notation, we may state the first variation formula for F .

Theorem 1 Let Mn be an oriented Riemannian manifold and let Mn+1 be an orientedRiemannian manifold. An isometric immersion ψ : M → M is a critical point of the ellipticparametric functional F defined in (3) if and only if the vector fields ξ and χ defined in (28)satisfy

divMξ + 〈χ, N 〉 = 0, (31)

where N is a unit normal vector field along ψ .If we consider only volume preserving variations of ψ , then the corresponding Euler–

Lagrange equation becomes

divMξ + 〈χ, N 〉 = −nH0, (32)

where H0 is a Lagrange multiplier.

Proof For a fixed x ∈ M , define α(s) = �(s, x) = (ψ(s, x), N (ψ(s, x))). Hence, using(28) and (19) we compute the variation rate of s → F(�(s, x)) as follows

d F

ds

∣∣∣s=0

=⟨

DF, �∗∂

∂s

⟩T M

=⟨πh∗ DF, (ψ∗

∂

∂s)h

⟩T M

+⟨πv∗ DF, (∇

ψ∗ ∂∂s

N )v⟩T M

= 〈χ,�〉 + 〈ξ, ∇�N 〉. (33)

Thus, using (11) we conclude that

d F

ds

∣∣∣s=0

= 〈χ,�〉 − 〈ξ, ψ∗ AV 〉 − 〈ξ, ψ∗∇φ〉

However, the tangential component ξ T of ξ is ψ-related to a tangent vector field ξ em M .Thus,

〈ξ, ψ∗∇φ〉 = 〈ψ∗ξ , ψ∗∇φ〉 = 〈ξ ,∇φ〉 = divM (φ ξ )− φ divM ξ .

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Hence, we compute

divM ξ = gi j⟨∇ ∂

∂xiξ ,

∂

∂x j

⟩= gi j |ψ(0,x)

⟨∇ψ∗ ∂

∂xiξ T , ψ∗

∂

∂x j

⟩

= divMξ − gi j⟨∇ψ∗ ∂

∂xi〈ξ, N 〉N , ψ∗

∂

∂x j

⟩

= divMξ + nH〈ξ, N 〉and then

〈ξ, ψ∗∇φ〉 = divM (φ ξ )− φ divMξ − nHφ〈ξ, N 〉, (34)

what implies that

d F

ds

∣∣∣s=0

= 〈χ,�〉 − 〈ξ, ψ∗ AV 〉 − divM (φ ξ )+ φ divMξ + nHφ〈ξ, N 〉. (35)

Gathering formulae (12) and (35), we compute

d

ds

∣∣∣s=0

F =∫

M

d F

ds

∣∣∣s=0

d M +∫

M

F(ϕ(x))∂

∂s

∣∣∣s=0

d Ms

=∫

M

(〈χ,�〉 − 〈ξ, ψ∗ AV 〉 − divM (φ ξ )+ φ divMξ + nHφ〈ξ, N 〉

)d M

+∫

M

F(ϕ(x)) (divM V − nHφ) d M.

By the Divergence Theorem, we conclude that

d

ds

∣∣∣s=0

F =∫

M

(〈χ,�〉 − 〈ξ, ψ∗ AV 〉 + φ divMξ + nHφ〈ξ, N 〉

−〈∇(F ◦ ϕ), V 〉 − nHφ(F ◦ ϕ)) d M.

However, Euler’s relation gives 〈ξ, N 〉 = F . Hence, we obtain

d

ds

∣∣∣s=0

F =∫

M

(〈χ,�〉 − 〈ξ, ψ∗ AV 〉 + φ divMξ − 〈∇(F ◦ ϕ), V 〉) d M.

On the other hand

〈∇(F ◦ ϕ), V 〉 = 〈(ψ∗V )h, DF(ϕ(x))〉T M + 〈(∇ψ∗V N )v, DF(ϕ(x))〉T M

= 〈ψ∗V, χ〉 + 〈∇ψ∗V N , ξ 〉= 〈ψ∗V, χ〉 − 〈ψ∗ AV, ξ 〉.

Replacing above, we finally obtain

d

ds

∣∣∣s=0

F =∫

M

(〈χ,�〉 − 〈χ,ψ∗V 〉 + φ divMξ) d M

=∫

M

(〈χ, N 〉 + divMξ) φ d M.

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We conclude that ψ : M → M is a critical isometric immersion for the functional F withrespect to arbitrary variations if and only if the Euler–Lagrange equation

divψ(0,M)ξ + 〈χ, N 〉 = 0, (36)

holds. If we restrict ourselves to volume preserving variations, that is, for variations satisfyingthe zero mean condition ∫

M

φ d M =∫

M

〈�, N 〉d M = 0. (37)

then, the condition for extremizing F becomes

divψ(0,M)ξ + 〈χ, N 〉 = −nH0, (38)

Here, H0 is the Lagrange multiplier in the constrained functional

s → F(ψ(s, ·))+ nH0V(s), (39)

where V(s) represents the volume comprised between the hypersurfacesψ0(M) andψs(M),that is,

V(s) =∫

M×[0,s]ι∗d M,

where ι is the inclusion of the immersed solid cylinder⋃s′∈[0,s]

ψs′(M)

in M . It is well-known that

d

ds

∣∣∣s=0

V =∫

M

〈�, N 〉 d M, (40)

what shows that the variations satisfying (37) preserve volume infinitesimally. ��Theorem (1) motivates the following definition.

Definition 1 The anisotropic mean curvature of the immersion ψ : M → M is defined by

nHF = −divMξ (41)

Hence, if χ = 0, the Euler–Lagrange equation (32) becomes the constant anisotropic meancurvature equation

HF = H0. (42)

Now we consider the anisotropic analogs of capillary hypersurfaces which are solutions tofree boundary problems for isometric immersions in M . These variational problems wereformerly studied in the case of rotationally symmetric Lagrangians in R

3 by Koiso and Palmer[4,5].

Definition 2 Let B ⊂ M be a domain with regular boundary ∂B oriented by a unit normalvector field N0. An isometric immersion ψ : M → M is said to be admissible if ψ(M) ⊂ Band ψ(∂M) ⊂ ∂B. Admissible immersions are denoted by ψ : (M, ∂M) → (B, ∂B). Avariation � of ψ is admissible if each immersion ψ(s, ·), s ∈ (−ε, ε), is admissible. In thiscase, we have 〈�, N0〉 = 0 on ∂M .

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Theorem 2 An admissible isometric immersion ψ : (M, ∂M) → (B, ∂B) is a criticalpoint of F with respect to volume preserving admissible variations if and only if nHF −〈χ, N 〉 is constant and 〈ξ, N0〉 = 0. These critical points are said to be anisotropic capillaryhypersurfaces.

Proof It follows from the proof of Theorem 1 that in the case when ∂M �= ∅ the first variationformula for F reads off as

d

ds

∣∣∣s=0

F =∫

M

(−nHF + 〈χ, N 〉) φ d M +∫

∂M

〈FV − φ ξ, ν〉d∂M, (43)

where d∂M is the volume element in ∂M induced by the immersion ψ and ν is the outwardsunit co-normal vector field along ∂M .

Suppose thatψ is a critical point of F for admissible preserving volume variations. Hence,if we consider only admissible variations with � = 0 on ∂M it holds that

d

ds

∣∣∣s=0

F =∫

M

(−nHF + 〈χ, N 〉) φ d M = 0. (44)

On the other hand since the variations are volume preserving we have∫

M

φ d M = 0

from what we conclude that nHF − 〈χ, N 〉 is constant.Now, suppose by contradiction that there exists y ∈ ∂M such that 〈ξ, N0〉|y �= 0. Hence,

consider a volume preserving admissible variation with variational vector field � satisfying

�|∂M = uν0,

where u : ∂M → R is a function to be chosen later and ν0 is the unit co-normal vector fieldalong ψ(∂M) with respect to ∂B, that is,

ν0 = e1 × . . .× en−1 × N0,

where {ei }n−1i=1 is an orthonormal frame for ψ(∂M) defined near y.

Recall that Euler’s relation implies that 〈ξ, N 〉 = F . Hence

F〈V, ν〉 − φ〈ξ, ν〉 = 〈ξ, N 〉〈�, ν〉 − 〈�, N 〉〈ξ, ν〉= u〈ξ, N 〉〈ν0, ν〉 − u〈ν0, N 〉〈ξ, ν〉= −u〈ξ, N 〉〈N0, N 〉 − u〈ξ, ν〉〈ν, N0〉= −u〈〈ξ, N 〉N + 〈ξ, ν〉ν, N0〉 = −u〈ξ, N0〉.

where we used that 〈�, ν〉 = 〈V, ν〉.Now we fix u = ϕ〈ξ, N0〉 for some functionϕ : ∂M → R with compact support contained

in some neighborhood U of y in ∂M where 〈ξ, N0〉 �= 0. Therefore we have

0 = d

ds

∣∣∣s=0

F = (−nHF + 〈χ, N 〉)∫

M

φ d M +∫

∂M

〈FV − φ ξ, ν〉d∂M

= −∫

∂M

u〈ξ, N0〉d∂M = −∫

U

ϕ〈ξ, N0〉2d∂M < 0,

a contradiction. We conclude from this contradiction that 〈ξ, N0〉 = 0 along ∂M .

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Since the converse part of the statement is immediate, this ends the proof of the theorem.��

In Sect. 4, we present some examples of anisotropic capillary hypersurfaces in Riemannianproduct spaces. We point out that capillary problems with wetting energy terms may be alsoformulated in the general context of Riemannian spaces (see [5] for the Euclidean case).

3 Anisotropic Weingarten map

In this section, we interpret the anisotropic mean curvature as the trace of a tensor whichplays the role of the Weingarten map in the anisotropic sense. In order to accomplish this, weshould suppose that F is horizontally constant, i.e., that DF has no horizontal components.

Definition 3 Given a cross-section ϕ(x) = (ψ(x), N (ψ(x))), x ∈ M , of the fiber bundleψ∗T M , we define a covariant tensor AF ∈ (T ∗M ⊗ T ∗M) by

AF (X, Y ) = 〈D(ψ∗ X)v DF, (ψ∗Y )v〉T M ◦ ϕ, (45)

for any vector fields X, Y ∈ (T M). We also define the tensor A∗F ∈ (T M ⊗ T ∗M) by

〈A∗F X, Y 〉 = AF (X, Y ), X, Y ∈ (T M). (46)

We observe that the tensor A∗F is well-defined taking tangent vector fields over tangent

vector fields, due to the degree one homogeneity of F . We also remark that the tensors AF

and A∗F are symmetric.

We have in local coordinates that

AF

(ψ∗

∂

∂xi, ψ∗

∂

∂x j

)= ∂yk

∂xi

∂yl

∂x j

∂2 F

∂ηk∂ηl

∣∣∣ϕ(x)

(47)

Definition 4 The tensor AF ∈ (T M ⊗ T ∗M) defined by

〈AF X, Y 〉 = 〈A∗F AX, Y 〉 = AF (AX, Y ), X, Y ∈ (T M), (48)

is the anisotropic Weingarten map.

Definition 5 A parametric Lagrangian F : T M−{0} → R is said to be horizontally constantif πh∗ DF = 0. In particular, if F is horizontally constant, then χ = 0 along ψ .

In Euclidean space, this amounts to be equivalent to imposing that F = F(η). An inter-esting example of such horizontally constant Lagrangians in the case of product spaces isgiven by (58).

Theorem 3 If the parametric Lagrangian F : T M −{0} → R is horizontally constant, thenthe anisotropic Weingarten map is given by

ψ∗ AF = −∇ξ. (49)

In this case nHF = tr AF .

Proof We notice that⟨∇ψ∗ ∂

∂xiξ, ψ∗

∂

∂x j

⟩= ∂

∂xi

⟨ψ∗

∂

∂x j, ξ

⟩−

⟨∇ψ∗ ∂

∂xiψ∗

∂

∂x j, ξ

⟩

= ∂

∂xi

(∂ψk

∂x j

∂F

∂ηk

) ∣∣∣ϕ(x)

−⟨∇ψ∗ ∂

∂xiψ∗

∂

∂x j, ξ

⟩. (50)

Since ∂F∂ηk = 〈DF, ∂

∂ηk 〉 is a function in T M , it follows from (19) that

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∂

∂xi

∂F

∂ηk

∣∣∣ϕ(x)

=⟨

D(ψ∗ ∂

∂xi

)h DF,∂

∂ηk

⟩

T M

+⟨

D(∇ψ∗ ∂

∂xiN

)v DF,∂

∂ηk

⟩

T M

+⟨

DF, D(ψ∗ ∂

∂xi

)h∂

∂ηk

⟩

T M

+⟨

DF, D(∇ψ∗ ∂

∂xiN

)v∂

∂ηk

⟩

T M

. (51)

Since DF has no horizontal components, (51) becomes

∂

∂xi

∂F

∂ηk

∣∣∣ϕ(x)

=⟨

D(∇ψ∗ ∂

∂xiN

)v DF,∂

∂ηk

⟩

T M

+⟨ξ, ∇

ψ∗ ∂

∂xi

∂

∂yk

⟩.

Now, the first term in (50) becomes

∂

∂xi

⟨ψ∗

∂

∂x j, ξ

⟩=

⟨D(

∇ψ∗ ∂

∂xiN

)v DF,

(ψ∗

∂

∂x j

)v⟩

T M

+⟨ξ,∂ψk

∂x j∇ψ∗ ∂

∂xi

∂

∂yk

⟩+

⟨ξ,

∂2ψk

∂xi∂x j

∂

∂yk

⟩

=⟨

D(∇ψ∗ ∂

∂xiN

)v DF,

(ψ∗

∂

∂x j

)v⟩

T M

+⟨ξ, ∇

ψ∗ ∂

∂xiψ∗

∂

∂x j

⟩.

We then conclude that⟨∇ψ∗ ∂

∂xiξ, ψ∗

∂

∂x j

⟩=

⟨D(

∇ψ∗ ∂

∂xiN

)v DF,

(ψ∗

∂

∂x j

)v⟩

T M

.

Therefore (∇ψ∗ ∂

∂xiξ

)T

= ψ∗ AF∂

∂xi.

However, Euler’s formula implies that

〈ξ, N 〉 = 〈DF, N v〉T M = ηk ∂F

∂ηk= F.

Thus,⟨∇ψ∗ ∂

∂xiξ, N

⟩= ψ∗

∂

∂xi〈ξ, N 〉 −

⟨ξ, ∇

ψ∗ ∂

∂xiN

⟩= ∂

∂xi(F ◦ ϕ)−

⟨ξ, ∇

ψ∗ ∂

∂xiN

⟩

=⟨

DF, (ψ∗∂

∂xi)h

⟩T M

+⟨

DF, (∇ψ∗ ∂

∂xiN )v

⟩T M

−⟨ξ, ∇

ψ∗ ∂

∂xiN

⟩

=⟨

DF, (∇ψ∗ ∂

∂xiN )v

⟩T M

−⟨ξ, ∇

ψ∗ ∂

∂xiN

⟩= 0.

We conclude that

∇ξ = ψ∗ AF (52)

123


and in particular

divMξ=gi j⟨∇ψ∗ ∂

∂xiξ, ψ∗

∂

∂x j

⟩=gi j

⟨D(∇

ψ∗ ∂

∂xiN )v DF, (ψ∗

∂

∂x j)v

⟩

T M

= trgAF A, (53)

where (aij )

ni, j=1 are the local components of A.

This finishes the proof of the theorem. ��Now, we establish an anisotropic analog of the Codazzi’s equation which we will be very

useful later for proving the existence of graphs with constant anisotropic mean curvature.

Corollary 1 The anisotropic Weingarten AF defined in (48) satisfies the followinganisotropic version of the Codazzi’s equation

ψ∗(∇X AF )Y − ψ∗(∇Y AF )X = −R(ψ∗ X, ψ∗Y )ξ, (54)

where X, Y are vector fields in M and R is the Riemann’s curvature tensor in M.

Proof It follows from (49) that

ψ∗(∇X AF )Y = ψ∗(∇X AF Y − AF∇X Y )

= ∇ψ∗ Xψ∗ AF Y − 〈∇ψ∗ Xψ∗ AF Y, η〉η + ∇ψ∗∇X Y ξ

= −∇ψ∗ X ∇ψ∗Y ξ + 〈∇ψ∗ X ∇ψ∗Y ξ, η〉η + ∇ψ∗∇X Y ξ

= −∇ψ∗ X ∇ψ∗Y ξ + ψ∗ X〈∇ψ∗Y ξ, η〉η − 〈∇ψ∗Y ξ, ∇ψ∗ Xη〉η + ∇ψ∗∇X Y ξ.

However, since ∇ψ∗Y ξ has not a component in the direction of the η and ∇ψ∗ Xη is a tangentvector, we obtain

ψ∗(∇X AF )Y = −∇ψ∗ X ∇ψ∗Y ξ − 〈AF Y, AX〉η + ∇ψ∗∇X Y ξ.

Similarly, one has

ψ∗(∇Y AF )X = −∇ψ∗Y ∇ψ∗ X ξ − 〈AF X, AY 〉η + ∇ψ∗∇Y X ξ,

which implies that

ψ∗(∇X AF )Y − ψ∗(∇Y AF )X = −∇ψ∗ X ∇ψ∗Y ξ + ∇ψ∗Y ∇ψ∗ X ξ + ∇ψ∗∇X Y ξ − ∇ψ∗∇Y X ξ

−〈AF Y, AX〉η + 〈AF X, AY 〉η.Since AF A is symmetric, we conclude that

ψ∗(∇X AF )Y − ψ∗(∇Y AF )X = −R(ψ∗ X, ψ∗Y )ξ. (55)

The symmetry of the AF A = AF A2 could be verified as follows⟨AF

∂

∂xi, A

∂

∂x j

⟩=

⟨AF A

∂

∂xi, A

∂

∂x j

⟩

= aki al

j

⟨AF

∂

∂xk,∂

∂xl

⟩= ak

i alj∂2 F

∂ηr∂ηs

∂ψr

∂ηk

∂ψ s

∂ηl

= aki al

j∂2 F

∂ηs∂ηr

∂ψ s

∂ηl

∂ψr

∂ηk= al

j aki

⟨AF

∂

∂xl,∂

∂xk

⟩=

⟨AF A

∂

∂x j, A

∂

∂xi

⟩

=⟨

AF∂

∂x j, A

∂

∂xi

⟩.

This concludes the proof of the proposition. ��

123


4 Rotationally symmetric examples

Let X ∈ (T M) be a closed conformal vector field. This means that there exists a functionϕ ∈ C∞(M) so that L〈·, ·〉 = 2ϕ〈·, ·〉. Moreover, the 1-form 〈X, ·〉 which is metricallyequivalent to X is closed. In this case, it follows that ∇ηX (y) = ϕ(y)η, for any (y, η) ∈ T M .

We then define a vector field Y ∈ (ψ∗T M) along the immersion ψ : M → M by

Y = 〈X, N 〉ξ − F X. (56)

We observe that Y is tangent to ψ(M). Indeed,

〈Y, N 〉 = 〈X, N 〉〈ξ, N 〉 − F〈X, N 〉 = 〈X, N 〉F − F〈X, N 〉 = 0.

Thus, given a tangent vector v in M , one obtains

(∇ψ∗vY )T =〈X, ∇ψ∗vN 〉ξ T +〈X, N 〉(∇ψ∗vξ)T −v(F ◦ (ψ, N ))X T − Fϕψ∗v

=−〈AX T , v〉ξ T+〈X, N 〉(∇ψ∗vξ)T−〈DF, (ψ∗v)h+(∇ψ∗vN )v〉T M X T −Fϕψ∗v

=−〈AX T , v〉ξ T +〈X, N 〉(∇ψ∗vξ)T − 〈χ,ψ∗v〉X T +〈ξ, ψ∗ Av〉X T −Fϕψ∗v,

where T indicates tangential projection. Then, taking traces, one has

divM Y = tr(∇Y )T = −〈AX T , ξ T 〉 + 〈X, N 〉divMξ − 〈χ, X T 〉 + 〈Aξ T , X T 〉 − nFϕ

= −nHF 〈X, N 〉 − 〈χ, X T 〉 − nFϕ.

Thus, by applying the Divergence Theorem, we prove the following anisotropic variant ofthe flux formula.

Theorem 4 Let X ∈ (T M) be a conformal closed vector field and� be a bounded domainin M whose boundary is oriented by a unit co-normal vector field ν pointing outward. Then

∫

�

nHF 〈X, N 〉 + 〈χ, X T 〉 + nFϕ =∫

F〈X, ν〉 − 〈X, N 〉〈ξ, ν〉, (57)

where HF is the anisotropic mean curvature of ψ .

We now consider an example of parametric Lagrangian that extends the functional F(ν3)

which has been intensively studied by Koiso and Palmer [3–5].Given a positive differentiable function f we define a Lagrangian F by

F(y, η) = |η| f

(〈X,

η

|η| 〉). (58)

In what follows, we will identify, with a slight abuse of notation, the vectors η and ηv and Xand Xv. We should recall also that the fibers of T M are totally geodesic with global normalcoordinates (ηk)n+1

k=1. Combining these facts, one verifies that

πv∗ DF = f (�)η

|η| + f ′(�)(

X −�η

|η|),

where we denoted

� =⟨

X,η

|η|⟩

T M.

123


Here, ′ denotes derivative with respect to the real variable �. Therefore

ξ = f (�)N + f ′(�)X T (59)

along ψ . It follows that

nHF = −divMξ=nH(

f (�)− f ′(�)�)+ f ′′(�)

⟨AX T , X T

⟩−n f ′(�)ϕ. (60)

Indeed, one has

〈∇ψ∗vξ, ψ∗w〉=ψ∗v[ f (�)]〈N , ψ∗w〉+ f (�)〈∇ψ∗vN , ψ∗w〉+ψ∗v[ f ′(�)]〈X T , ψ∗w〉+ f ′(�)〈∇ψ∗vX T , ψ∗w〉

= − f (�)〈Av,w〉+ f ′′(�)ψ∗v[�]〈X T , ψ∗w〉+ f ′(�)〈∇ψ∗vX, ψ∗w〉− f ′(�)〈X, N 〉〈∇ψ∗vN , ψ∗w〉

= − f (�)〈Av,w〉+ f ′′(�)(〈∇ψ∗vX, N 〉+〈X, ∇ψ∗vN 〉) 〈X T , ψ∗w〉

+ f ′(�)〈∇ψ∗vX, ψ∗w〉+ f ′(�)�〈Av,w〉= (

f ′(�)�− f (�)) 〈Av,w〉+ f ′′(�) (ϕ〈ψ∗v, N 〉−〈X, ψ∗ Av〉) 〈X T , ψ∗w〉

+ f ′(�)ϕ〈ψ∗v,ψ∗w〉= (

f ′(�)�− f (�)) 〈Av,w〉− f ′′(�)〈AX T , v〉〈X T , ψ∗w〉

+ f ′(�)ϕ〈ψ∗v,ψ∗w〉.Taking traces one has

divMξ = n( f ′(�)�− f (�))H − f ′′(�)⟨AX T , X T

⟩+ n f ′(�)ϕ.

Now, differentiating F again with respect to vertical vector fields, we get

Proposition 1 The tensor D2 F |V×V has eigenvalues given by

μ‖ = f ′′(�) 1

|η|(|X |2 −�2) + 1

|η|(

f (�)− f ′(�)�)

(61)

and

μ⊥ = 1

|η|(

f (�)− f ′(�)�)

(62)

corresponding to multiplicities 1 and n − 1. The eigenvector corresponding to μ|| is

X −�η

|η| (63)

whereas the other eigenvectors are perpendicular to this direction with respect to the Sasakimetric. We point out that f should be chosen in such a way that μ‖ > 0 and μ⊥ > 0 inT M − {0}.

Moreover, one has

〈DF(y, η), ζ h〉T M = f ′(�)〈∇ζ X, η〉 = f ′(�)ϕ〈η, ζ 〉. (64)

for any (y, ζ ) ∈ T M . It follows that if X is parallel then F is horizontally constant. IndeedX is parallel if and only if ϕ = 0. In particular, in this case we have χ = 0.

123


We conclude from (64) that

〈χ, X T 〉 = f ′(�)ϕ〈N , X T 〉 = 0. (65)

From now on, we consider the particular case of the warped product M = P ×h R, where Pis a complete manifold endowed with a rotationally invariant metric 〈·, ·〉P and h : R → R

is a positive smooth function. If t stands for the natural Euclidean coordinate in the factorR, then X = h(t) ∂

∂t is a conformal vector field. In this case, we fix, for some R > 0, polarcoordinates (r, θ) ∈ [0, R)×S

n−1 in P centered at a point o ∈ P such that the warped metricin M may be written as

dt2 + h2(t)(dr2 + g2(r)dθ2) (66)

where g2(r) = 〈 ∂∂r ,

∂∂r 〉P and dθ2 represents the canonical metric in S

n−1. Notice that

g(0) = 0 and dgdr (0) = 1.

An isometric immersion ψ : M → P ×h R describes a rotationally invariant surface if itis equivariant with respect to the action of SO(n) in P ×h R fixing the line {o} × R. Such asurface is parameterized in terms of coordinates (t, r, θ) by

(s, θ) → (t (s), r(s), θ) , s ∈ (0, S), (67)

for some S > 0. At each point of ψ(M), the tangent space is generated by the vector fields

ψ∗∂

∂θ i= ∂

∂θ i

∣∣∣ψ

(68)

and

ψ∗∂

∂s= r

∂

∂r+ t

∂

∂t, (69)

where · indicates derivatives with respect to s. if s is the arc-lenght of the profile curve, themetric induced in ψ(M) is

ds2 + h2(t (s))g2(r(s))dθ2. (70)

In this case,

N = t

h

∂

∂r− hr

∂

∂t. (71)

defines a unit normal vector field along ψ .

Proposition 2 The anisotropic mean curvature of a rotationally invariant hypersurface ψ :M → P ×h R with respect to the parametric Lagrangian (58) is given by

nHF =μ‖(

h(t r −r t)+ dh

dtt2r + dh

dtr

)− (n−1)

hμ⊥

(1

g

dg

drt−h

dh

dtr

)−n f ′ϕ, (72)

where

μ‖ = f ′′(−h2r)h2 t2 + ( f (−h2r)+ f ′(−h2r)h2r)

and

μ⊥ = f (−h2r)+ f ′(−h2r)h2r

are the eigenvalues of the form D2 F |V×V , calculated at points of the cross-section

ϕ(x) = (ψ(x), N (ψ(x))), x ∈ M,

123


of ψ∗T M, where N is the unit normal vector field fixed in (71).

Proof Using (70) and (71), we easily see that the Weingarten map A = −∇N of ψ hasprincipal values given by

k‖ = h(t r − r t)+ dh

dtt2r + dh

dtr (73)

and

k⊥ = − 1

h

(1

g

dg

drt − h

dh

dtr

)(74)

with multiplicities respectively equal to 1 and n − 1.On the other hand, since X = h ∂

∂t we have

� = h

⟨∂

∂t, N

⟩= −h2r

from what follows that

μ‖ = f ′′(−h2r)h2(1 − h2r2)+ ( f (−h2r)+ f ′(−h2r)h2r),

and

μ⊥ = f (−h2r)+ f ′(−h2r)h2r .

Moreover

X T = X −�N = ht

(t∂

∂t+ r

∂

∂r

)= htψ∗

∂

∂s.

Therefore

〈AX T , X T 〉 = h2 t2⟨

A∂

∂s,∂

∂s

⟩= h2 t2k|| = (|X |2 −�2) k||.

Replacing these expressions in (60) we obtain

nHF = nHμ⊥+ f ′′(|X |2−�2)k||−n f ′ϕ= ( f ′′(|X |2−�2)+μ⊥)k||+(n − 1)μ⊥k⊥−n f ′ϕ=μ||k||+(n − 1)μ⊥k⊥ − n f ′ϕ,

what concludes the proof of the proposition. ��In terms of the angle ϑ between ∂

∂r and ψ∗ ∂∂s , we reduce (72) to the first-order system

r = 1

hcosϑ, (75)

t = sin ϑ, (76)

ϑ =(

1+(n−1)μ⊥

μ||

) dhdt

hcosϑ−(n−1)

μ⊥

μ||dgdr

gsin ϑ− n

μ||

(HF + f ′ dh

dt

). (77)

If h′ = 0 the vector field X is parallel. In this case, translations along the vertical lines andreflections through the totally geodesic leaves P × {t}, t ∈ R, take a solution of the system(75)–(77) into another.

From now on, we proceed with a qualitative analysis of the system (75)–(77). We beginstating the following consequence of the Theorem 4 which gives a first integral to (75)–(77).

123


Proposition 3 Supppose that HF is constant. The expression

I = μ⊥hngn−1 t +(

nHF + n f (h)dh

dt

)hn−1

r∫

0

gn−1(τ ) dτ (78)

is a first integral to the system (75)–(77). In particular, if ∂M = ∅, then I = 0.

Proof We consider the domain � ⊂ ψ(M) bounded between two leaves P × {t (s)} andP × {t (s′)} with s ≤ s′. We denote by D and D′ the geodesic discs bounded by ψ(M) ∩P × {t (s)} and bt ψ(M)∩ P × {t (s′)}, respectively, chosen in a such a way that�∪ D ∪ D′bounds an oriented cycle in P ×h R. Thus, the Divergence Theorem implies that∫

�

nHF 〈X, N 〉 + nFϕ = −∫

D

nHF

⟨X,− ∂

∂t

⟩+ nF

(〈X,− ∂

∂t〉)ϕ

−∫

D′nHF

⟨X,

∂

∂t

⟩+ nF

(〈X,

∂

∂t〉)ϕ

=∫

D

nHF h − n f (−h)dh

dt−

∫

D′nHF h + n f (h)

dh

dt.

Applying (78) we get∫

�

nHF 〈X, N 〉 + nFϕ = −∫

F

⟨X, ψ∗

∂

∂s

⟩− 〈X, N 〉

⟨ξ, ψ∗

∂

∂s

⟩

+∫

′F

⟨X, ψ∗

∂

∂s

⟩− 〈X, N 〉

⟨ξ, ψ∗

∂

∂s

⟩,

where = ∂D and ′ = ∂D′. Gathering these expressions, we conclude that∫

D

nHF − n f (−h)dh

dt+

∫

F

⟨X, ψ∗

∂

∂s

⟩− 〈X, N 〉

⟨ξ, ψ∗

∂

∂s

⟩

=∫

D′nHF h + n f (h)

dh

dt+

∫

′F

⟨X, ψ∗

∂

∂s

⟩− 〈X, N 〉

⟨ξ, ψ∗

∂

∂s

⟩.

However, one has ⟨X, ψ∗

∂

∂s

⟩= ht, 〈X, N 〉 = −h2r .

Moreover, along the geodesic circles and ′ one has F = f (〈X, N 〉) = f (−h2r) and⟨ξ, ψ∗

∂

∂s

⟩= f ′h

⟨∂

∂t, ψ∗

∂

∂s

⟩= f ′ht .

We conclude that∫

F

⟨X, ψ∗

∂

∂s

⟩− 〈X, N 〉

⟨ξ, ψ∗

∂

∂s

⟩

=∫

ht(

f (−h2r)+ f ′(−h2r)h2r) =

∫

htμ⊥ = htμ⊥||.

123


Taking in account that |D| = hn−1∫ r

0 gn−1(τ )dτ |Sn−1| and || = hn−1gn−1|Sn−1|, thisends the proof of the proposition. ��

The following theorems in this section assure the existence of rotationally invariant hyper-surfaces with constant anisotropic mean curvature. In particular, we prove, under someassumptions, the existence of topological spheres with constant anisotropic mean curva-ture which are the analogs of Wulff shapes for the case of the general ambient product spaceswe are considering.

Theorem 5 Suppose that there exist t0 ∈ R and r0 ∈ (0, R) such that

nHF = −n f (h(t0))dh

dt(t0)− gn−1(r0)∫ r0

0 gn−1(τ ) dτf (0)h(t (0)). (79)

Then there exists a rotationally invariant sphere with constant anisotropic mean curvatureHF with respect to the Lagrangian (58) in the warped product P ×h R.

Proof We fix HF �= 0 and I = 0. Under this choice, (78) reduces to

μ⊥hngn−1 t = −(

nHF + n f (h)dh

dt

)hn−1

r∫

0

gn−1(τ ) dτ

Since h �= 0 we have

μ⊥hgn−1 t = −(

nHF + n f (h)dh

dt

) r∫

0

gn−1(τ ) dτ (80)

Notice that g(0) = 0 and dgdr (0) = 1 imply that

limr→0

∫ r0 gn−1(τ ) dτ

gn−1(r)= 1

n − 1limr→0

g(r)dgdr (r)

= 0.

Since g(r) > 0 in (0, R), we may write (80) in the following form

μ⊥ht = −(

nHF + n f (h)dh

dt

) ∫ r0 gn−1(τ ) dτ

gn−1(r), (81)

for r ∈ [0, R). Since μ⊥h �= 0 we conclude that t → 0 as r → 0.Since r = 0 if and only if t = 1 we also deduce from (81) that r = 0 if and only if

f (0)h = −(

nHF + n f (h)dh

dt

) ∫ r00 gn−1(τ ) dτ

gn−1(r0).

In other words, if we suppose that there exist r0 ∈ (0, R) and t0 ∈ R such that

nHF = −n f (h(t0))dh

dt(t0)− gn−1(r0)∫ r0

0 gn−1(τ ) dτf (0)h(t (0)).

then we assure the existence of a solution of the system (75)–(77) with initial conditionst (0) = t0, r(0) = r0 and ϑ(0) = π

2 . In the interval [0, r0) on which r �= 0 this solution isgiven by the graph of t = t (r) where

123


(dt

dr

)2

=h2(nHF + n f (h) dh

dt )2(∫ r

0 gn−1(τ ) dτgn−1(r)

)2

μ⊥2h2 − (nHF + n f (h) dhdt )

2(∫ r

0 gn−1(τ ) dτgn−1(r)

)2 . (82)

Similarly, we consider a solution of (75)–(77) with initial conditions t (0) = t0, r(0) = r0

and ϑ(0) = −π2 . These two solutions together describe a curve which touches the axis

r = 0 orthogonally. Rotating this curve around the axis r = 0 we obtain a closed rotationallyinvariant hypersurface with constant anisotropic mean curvature HF and diffeomorphic witha sphere.

We remark that if r �= 0 in [0, R), we have a solution of (75)–(77) with initial conditionst (0) = 0, r(0) = 0 and ϑ(0) = 0. This curve generates a disc with nonzero constantanisotropic mean curvature HF . This example is complete when g(r) > 0 for r ∈ (0,∞).

This finishes the proof of the theorem. ��Corollary 2 Suppose that there exist t0 ∈ R and r0 ∈ (0,∞) such that either

HF =(

f (e−t0)− 1

r0

)e−t0 (83)

or

nHF = −n f (sinh(t0)) cosh(t0)− sinn−1(r0)∫ r00 sinn−1(τ ) dτ

f (0) sinh(t (0)), (84)

then there exists a rotationally invariant sphere with constant anisotropic mean curvatureHF with respect to the Lagrangian (58) in the hyperbolic space H

n+1.

Proof We obtain (83) and (84) from (79) by considering the warped product models for Hn+1

given by dt2 + e−2t (dr2 + r2dθ2) and dt2 + sinh2 t (dr2 + sin2 rdθ2), respectively. ��Remark 1 Results similar to Corollary 2 may be stated for ambient spaces as Euclideanspheres and complex and quaternionic hyperbolic spaces as well as to Riemannian productof the form M

n(κ)× R, where Mn(κ) is a simply connected n-dimensional space form with

constant curvature κ .

Corollary 3 Let B = {(y, t) ∈ P × R : t ≥ 0} be the halfspace above the totally geodesichypersurface P ×{0} in P ×R. Let D

n be a disc in Rn+1. Then there exist compact anisotropic

capillary immersions ψ : (Dn, Sn−1) → (B, ∂B) with constant anisotropic mean curvature.

Proof We observe that given a rotationally invariant sphere S in P × R there exists t0 ∈ R

such that the 〈ξ, ∂∂t 〉 = 0 along S ∩ P × {t0}. Indeed, since

ξ = f (�)N + f ′(�)X T = (f (�)− f ′(�)�

)N + f ′(�) ∂

∂t

with � = 〈N , ∂∂t 〉, it follows that � = f (1) ∂

∂t [respectively, � = − f (−1) ∂∂t ] at the

maximum (resp., minimum) point of t |S . Hence, 〈ξ, ∂∂t 〉varies from− f (−1) < 0 to f (1) > 0

and then there exists t0 ∈ (minS t,maxS t) such that 〈ξ, ∂∂t 〉|t0 = 0.

Therefore, Theorem 2 implies that the components of S above and below P ×{t0} are bothanisotropic capillary hypersurfaces with constant anisotropic mean curvature. Moreover, thecomponent of S above P × {t0} is a graph over the geodesic disc bounded by S ∩ P × {t0}in P × {t0}. This finishes the proof of the corollary. ��

123


Now, we focus on the particular case Mn(κ)×R, that is, with h(t) = 1, for t ∈ R. Hence,

we have g(r) = snκ (r) is the solution of the initial value problem u′′ + κu = 0, u(0) = 0,u′(0) = 1. Recall that r ∈ (0, π√

κ) for κ > 0 and r ∈ (0,∞) for κ ≤ 0. We also denote

csκ (r) = ddr snκ (r). In this case, we are able to classify the rotationally invariant surfaces

with constant anisotropic mean curvature when n = 2.In the statement of the theorem below, we denote f0 = f (0).

Theorem 6 The complete rotationally invariant surfaces isometrically immersed in M2(κ)×

R with constant anisotropic mean curvature HF are classified according to the parametersHF and I as: (i) totally geodesic leaves, if HF = I = 0; (i i) embedded annuli, if HF = 0 andI �= 0; (i i i) embedded spheres, if HF �= 0, I = 0 and 4H2

F > −κ f 20 ; (i i i − bis) embedded

discs, if HF �= 0, I = 0 and 4H2F ≤ −κ f 2

0 ; (iv) Periodic, cylindrically bounded embeddedsurfaces, if I HF < 0, 4H2

F > −κ f 20 and f 2

0 + 4HF I − κ I 2 > 0; (v) a catenoid cousin typesurface if I HF < 0, 4H2

F = −κ f 20 and 2HF < κ I ; (vi) cylinders over geodesic circles in

M2(κ)× {0} if HF �= 0, I �= 0, 4H2

F > −κ f 20 and f 2

0 + 4HF I − κ I 2 = 0; (vi i) periodic,cylindrically bounded immersed surfaces if I HF > 0, 4H2

F > −κ f 20 , f 2

0 +4HF I −κ I 2 > 0and 4HF < κ I and 2HF �= κ I ; (vi i i) tori, if I HF > 0, 4H2

F > −κ f 20 , f 2

0 +4HF I−κ I 2 > 0and 2HF = κ I .

Proof (i) If HF = 0 and I = 0, then (78) implies that t = 0 from what follows that t = a forsome constant a ∈ R. This gives a solution of (75)–(77) of the form r(s) = ±s, t (s) = aandϑ(s) = 0, π . We conclude in this case that the resulting rotationally invariant surfaceis contained in M

2(κ)× {a}.(ii) If HF = 0 and I �= 0 we have from (78) that

I = μ⊥snκ (r) t (85)

and then t �= 0 and r �= 0. Supposing that t > 0 and combining (70) and (85) yield(

dt

dr

)2

= I 2

(μ⊥)2sn2κ (r)− I 2 . (86)

The denominator in (86) vanishes when drdt = 0, that is, for r = r0 given by

sn2κ (r0) = I 2

(μ⊥)2. (87)

Notice that μ⊥ = f (0) = f0 when (87) holds. Therefore

sn2κ (r0) = I 2

f 20

. (88)

If κ > 0, then (88) admits a solution when I 2 ≤ f 20 /κ . If κ ≤ 0, there always exists an

unique solution.For κ ≤ 0, we conclude that the solution of (75)–(77) with HF = 0 and I �= 0 and initialconditions r(0) = r0, t (0) = 0, ϑ(0) = ±π

2 is given as the bi-graph of the function

t (r) = ±r∫

r0

I√(μ⊥)2sn2

κ (τ )− I 2dτ, r > r0. (89)

Therefore the corresponding rotationally invariant surface is complete and diffeomorphicwith an annulus.

123


For κ > 0, we suppose without loss of generality that r0 ≤ π2√κ

. If r0 <π

2√κ

then the

solution (89) is initially defined in the interval [r0,π√κ

r0] and then extended to a perodic

curve by reflections through the lines t = t (r0) and t = t ( π√κ

− r0). If r0 = π2√κ

thenthe solution is the vertical line r = r0.

(iii) If HF �= 0 and I = 0, we write (78) as

μ⊥snκ (r) sin ϑ + 4HF sn2κ

( r

2

)= 0, (90)

what implies that

μ⊥ csκ( r

2

)snκ

( r2

) sin ϑ = −2HF . (91)

Hence considering without loss of generality that HF < 0 we conclude that sin ϑ > 0.Moreover (91) implies that ϑ = π

2 at r = r0, where r0 is given by

csκ (r0)

snκ (r0)= −2HF

f0. (92)

Therefore, fixed HF �= 0 and I = 0, we have a solution of the system (75)–(77) withinitial conditions r(0) = r0, t (0) = 0, ϑ(0) = π

2 . Such a solution could be describednon-parametrically as the graph of the function t = t (r) given by

(dt

dr

)2

= 4H2F sn2

κ

( r2

)(μ⊥)2cs2

κ

( r2

) − 4H2F sn2

κ

( r2

) , t (r0) = 0, (93)

from what follows that dtdr → 0 when r → 0. We conclude that the solution could be extended

to r = 0 with sin ϑ → 0 as r → 0. Hence, the solution could be extended by reflectionthrough the axis r = 0 and then it generates a rotationally invariant surface diffeomorphicwith a sphere.

Critical values for the coordinate r occur at r = r0 given by (92). If κ ≥ 0, then (92) alwayshas solutions. However, when κ < 0, this is the case only if we suppose that 4H2

F > −κ f 20 .

Moreover, if 4H2F ≤ −κ f 2

0 , then we necessarily have κ < 0 and in this case the corre-sponding rotationally invariant surfaces are diffeomorphic to discs.(iv)− −(v) If HF �= 0 and I �= 0, we have from (78) that

μ⊥ tsnκ (r) = I − 4HF sn2κ

( r

2

). (94)

Critical values for the coordinate r occur when t = 1. Then r = r± are extreme values of rif and only if

(4H2F + κ f 2

0 ) sn4κ

(r±2

)− (2HF I + f 2

0 ) sn2κ

(r±2

)+ I 2

4= 0. (95)

If 4H2F �= −κ f 2

0 and

f 20 + 4HF I − κ I 2 ≥ 0 (96)

then the roots of (95) are given by

sn2κ

(r±2

)=

2HF I + f 20 ± f0

√f 20 + 4HF I − κ I 2

8H2F + 2κ f 2

0

. (97)

123


However we easily verifies that the right-hand side in (97) is non-negative if and only if4H2

F ≥ −κ f 20 . Hence the solutions for which 4H2

F < −κ f 20 have no critical values for r . This

implies that if these solutions are complete then the curves they define have points arbitrarilyclose to the axis r = 0. However, this contradicts (94). Therefore, we should suppose fromnow on that 4H2

F ≥ −κ f 20 .

We conclude that if 4H2F > −κ f 2

0 and

f 20 + 4HF I − κ I 2 > 0, (98)

then there exist two distinct critical values r = r− and r = r+ for the coordinate r . Thisimplies in particular that there exists a solution of the system (75)–(77) with initial conditionsr(0) = r+, t (0) = 0, ϑ(0) = π

2 which could be also described as the graph of the function

t (r) =r−∫

r

I − 4HF sn2κ (τ2 )√

(μ⊥)2 sn2κ (τ )− (I − 4HF sn2

κ (τ2 ))

2dτ, r < r+. (99)

This curve could be extended through reflection around the lines t = t (r−) e t = t (r+) andthen by iterated reflections in order to produce a complete and periodic solution of (75)–(77)with period given by

T = 2

r−∫

r+

I − 4HF sn2κ (τ2 )√

(μ⊥)2 sn2κ (τ )− (I − 4HF sn2

κ (τ2 ))

2dτ (100)

Suppose that HF < 0 and I > 0. Hence (94) implies that t > 0 and r > 0. We conclude thatthe solution is given as the graph of (99) over the whole real line. Moreover this embeddedcurve has inflection points which correspond to points where ϑ = 0. At these points, it holdsthat

μ⊥k⊥ = 2HF (101)

what is equivalent to

− μ⊥ csκ (r)t = 2HF snκ (r). (102)

Hence replacing (94) into (102) one concludes that inflection points of the curve occur forvalues of r satisfying

(1 − 2κsn2

κ

( r

2

)) (I − 4HF sn2

κ

( r

2

))= −2HF sn2

κ (r)

or, equivalently,

(4HF − 2κ I )sn2κ

( r

2

)= −I.

The existence of such points in the interval [r−, r+] is due to the fact that k‖ > 0 at r = r−and k‖ < 0 at r = r+.

Finally, if 4H2F = −κ f 2

0 , then we necessarily have κ < 0. In this case, there exists anunique solution r = r0 of (95) if and only if 2HF I > − f 2

0 , that is, if and only if 2HF < κ I .The solution is then given by

sn2κ

(r0

2

)= I 2

4(2HF I + f 20 ).

123


These examples of rotationally invariant surfaces are the analogs of the so-called cousincatenoids in the theory of constant mean curvature surfaces in H

3(κ).(vi) Still considering the case when HF �= 0 and I �= 0, a trivial solution to the system

(75)–(77) is given by r = r0, for some fixed r0 ∈ (0, π√κ) if κ > 0 or r0 ∈ (0,+∞) if κ ≤ 0.

In this case, we have ϑ = ±π2 and t (s) = ±s. This example correponds to a cylinder over a

geodesic circle in M2(κ)× {0} centered at o with radius r0.

It is worth to remark that these cylinders correspond to the case 4H2F > −κ f 2

0 when Iattains the equality in (98), that is, when

f 20 + 4HF I − κ I 2 = 0.

(vi i) − −(vi i i) Now, we deal with the case when HF < 0 and I < 0. In this case, onehas

μ⊥ tsnκ (r)− I = −4HF sn2κ

( r

2

). (103)

The points r = r0 where t = 0 are given by

sn2κ

(r0

2

)= I

4HF.

This equation has solutions either if κ ≤ 0 or if κ > 0 and 4HF ≤ κ I . The examplesin this case correspond to solutions of the system (75)–(77) with initial conditions r(0) =r0, t (0) = 0, ϑ(0) = 0, π .

We first consider the case 4HF = κ I when we necessarily have κ > 0. In this case (103)becomes

μ⊥ tsnκ (r) = I cs2κ

( r

2

).

Thus t = 0 only at the zeroes of the function r → csκ ( r2 ). This implies that in this case

one has r0 = π√κ

. We conclude that if 4HF = κ I the curve defined by the solution reaches

orthogonally the line r = π√κ

. Hence, this curve generates a rotationally invariant sphere.

Now, if 2HF = κ I we have r0 = π2κ . In this case, the curve reaches orthogonally the

axis r = π2√κ

and its reflection through this axis gives a closed curve which generates arotationally invariant tori.

Now, in all of the remaining cases it is possible to reproduce the same analysis as in items(iv)–(v) above. This permits to conclude the solutions of the system (75)–(77) are in thesecases complete and periodic curves obtained by successive reflections through the verticallines where the coordinate r attains critical values.

It follows from the fact that the coordinate t has critical values that this time the curvesare no longer embedded. However, we could also verify that these curves have no inflectionpoints. Indeed, since 2HF = μ‖k‖ +μ⊥k⊥, it follows that k‖ = 0 at points of such a solutionif and only if

μ⊥csκ (r)t = −4HF snκ (r). (104)

However, whereas the right-hand side in (104) is strictly positive, we had just verified thatthe left-hand side is zero at critical points of the coordinate t . We then conclude that the curvehas no inflection points in this case. ��

123


5 Graphs with constant anisotropic mean curvature

Throughout this section, we consider M as the product manifold P × R defined in Sect. 4.Recall that in this case the vector field X = ∂

∂t is a parallel vector field in M .Now, we investigate the existence of graphs with constant anisotropic mean curvature in

M . This requires to establish some preliminary facts which have also independent interest.As before, we still consider an isometric immersion ψ : M → M . Moreover, the

Lagrangian F we fix in this section is that one defined in (58).

Proposition 4 The second order differential operators �F and L F in M given respectivelyby

�F u = tr(A∗F∇2u) (105)

and

Lu = �F u − divA∗F · ∇u, (106)

for u ∈ C2(M), are elliptic in M.

Proof Given an arbitrary function u in M we compute in local coordinates

�F u = div(A∗F∇u) =

((A∗

F )ij u

j)

;i = (A∗F )

ij u

j;i + (A∗

F )ij;i u

j

= tr(A∗F · ∇2u)+ divA∗

F · ∇u,

what implies that both�F and L have symbol (AF )ij which is positive-definite by definition.

��We apply this proposition for some fundamental geometric functions.

Proposition 5 Let h = t |M . Then

Lh = nHF 〈X, N 〉, (107)

where N is the unit normal vector field along ψ .

Proof We compute

∇h = ∇t − 〈∇t, N 〉N = X − 〈X, N 〉N .

Thus, given a local orthonormal frame field {ei }ni=1 in M , one has

�F h = divM (A∗F∇h) = divM (A∗

F X T )

= 〈∇ei A∗F X T , ei 〉 =

⟨(∇ei A∗

F )XT , ei

⟩+

⟨A∗

F∇ei X T , ei

⟩

= divMA∗F · X T +

⟨∇ei X T ,A∗

F ei

⟩

= divMA∗F · X T − ⟨∇ei 〈X, N 〉N ,A∗

F ei⟩

= divMA∗F · X T − 〈X, N 〉〈∇ei N ,A∗

F ei 〉= divMA∗

F · X T + 〈X, N 〉 tr(A∗F A)

Therefore

Lh = nHF 〈X, N 〉. (108)

This finishes the proof. ��

123


Proposition 6 If the anisotropic mean curvature HF of ψ is constant, then the function� = 〈X, N 〉 satisfies

�F�+ tr(AAF )�+ μ⊥RicM (N , N )� = 0. (109)

where

μ⊥ = f (�)− f ′(�)�. (110)

Proof Let {ei }ni=1 be a local orthonormal frame in M such that ∇ei = 0 at an arbitrarily

fixed point. We then compute

∇ei 〈N , X〉 = −〈Aei , X T 〉 = −〈ei , AX T 〉. (111)

Therefore

∇〈N , X〉 = −AX T .

Thus,

�F 〈N , X〉 = divM(A∗

F∇〈N , X〉) = −divM (A∗F AX T ) = −

∑i

⟨∇ei (A∗

F AX T ), ei

⟩

= −∑

i

⟨∇ei (AF X T ), ei

⟩= −

∑i

⟨(∇ei AF )X

T , ei

⟩−

∑i

⟨AF∇ei X T , ei

⟩

= −∑

i

⟨(∇X T AF )ei , ei

⟩ + ∑i

⟨R(ei , X T )ξ, ei

⟩−

∑i

⟨AF∇ei X T , ei

⟩

= −∑

i

⟨∇X T AF ei , ei⟩ + ∑

i

⟨AF∇X T ei , ei

⟩ + ∑i


⟩

−∑

i

⟨∇ei X T , AF ei

⟩,

where we used the anisotropic Codazzi’s equation (54).Supposing that ∇ei = 0 at the point we are doing calculations, it follows that

�F 〈N , X〉 = −∑

i

X T 〈AF ei , ei 〉 +∑

i

⟨AF ei ,∇X T ei

⟩ + ∑i

⟨AF∇X T ei , ei

⟩

+∑

i


⟩−

∑i

⟨∇ei X T , AF ei

⟩

= −X T (nHF )+∑

i


⟩+

∑i

〈X, N 〉 ⟨∇ei N , AF ei⟩

= −X T (nHF )+∑

i


⟩− tr(AF A)〈X, N 〉.

Since HF is constant we have

�F 〈N , X〉 + tr(AAF )〈X, N 〉 =∑

i


⟩.

123


However,∑

i


⟩=

∑i

⟨R(ei , X)ξ, ei

⟩ − ∑i

〈X, N 〉 ⟨R(ei , N )ξ, ei

⟩

=∑

i

⟨R(ei , ξ)X, ei

⟩ − 〈X, N 〉RicM (N , ξ)

= −〈X, N 〉RicM (N , ξ).

where we used the fact that X is a parallel vector field. Therefore, since

RicM (N , ξ) = RicM

(N , f (�)N + f ′(�)X T

)= (

f (�)− f ′(�)�)

RicM (N , N ).

we conclude that

�F 〈N , X〉+tr(AAF )〈X, N 〉+( f (�)− f ′(�)�)RicM (N , N )〈X, N 〉=0. (112)

This finishes the proof of the proposition. ��Recall that we are supposing that the Riemannian metric in P is rotationally invariant.

More precisely, we suppose that there exist a point o ∈ P and a radius R > 0 for which themetric in P is written as

dr2 + g2(r)dθ2,

for a certain function g : [0, R) → R, in terms of polar coordinates (r, θ) ∈ [0, R) × Sn−1

defined in the geodesic ball BR(o) ⊂ P . Here dθ2 denotes the usual metric in Sn−1.

Theorem 5 assures the existence of a rotationally invariant spheres Sr with constantanisotropic mean curvature HF such that Sr intersects P×{0} orthogonally along the geodesicsphere ∂Br (o) provided that

n|HF | = μ⊥(0) |∂Br (o)||Br (o)| .

These spheres are natural barriers for estimating the height of hypersurfaces with constantanisotropic mean curvature.

Proposition 7 Let ψ : M → M be an isometric immersion with constant anisotropic meancurvature HF . Suppose thatψ(∂M) is contained in the geodesic ball Br (o) ⊂ P with r < Rgiven by

|∂Br (o)||Br (o)| = 1

μ⊥(0)n|HF |, (113)

then

|t |M | ≤ maxSr

t. (114)

Proof Notice that (113) implies that the anisotropic mean curvature of the graph and of thesphere Sr coincide. Hence the conclusion follows from the comparison principle. ��

In terms of the notation fixed above, we state the following existence result which may beregarded as an extension of the classical Serrin’s result ([7]) to the case of anisotropic meancurvature.

123


Theorem 7 Let F : T M −{0} → R be the parametric Lagrangian defined in (58). Supposethat RicM ≥ 0. Let � be an open bounded domain contained in Br (o) ⊂ P with r given in(113). Suppose that the boundary = ∂� is a C2,α submanifold of P. Let K be the cylinderover , that is,

K = {(x, t) : x ∈ , t ∈ R} (115)

and denote its anisotropic mean curvature by HK . Suppose that HK < 0. Then, given aconstant HF such that

|HF | < μ⊥

μ⊥(0)HK , (116)

there exists a unique graph in M with constant anisotropic mean curvature HF and boundary.

Proof The graph of a function u ∈ C2,α(�) is parameterized by the map

ψ(x) = (x, u(x)), x ∈ �. (117)

We denote the graph of u by M .It is a well-known fact that the Weingarten map of ψ is given by

Aij =

(δik − ui

W

u j

W

) ∇ Pk ∇ P

j u

W, (118)

where we denote

W =√

1 + |∇ P u|2,Here, ∇ P denotes the Riemannian connection in P induced from its embedding in P × R.This Weingarten map is calculated with respect to the orientation fixed by

N = 1

W

(X − ∇ P u

). (119)

We conclude that the graph has constant anisotropic mean curvature HF if u satisfies thefollowing elliptic partial differential equation

(A∗F )

ij

(δ jk − u j

W

u j

W

) ∇ Pk ∇ P

i u

W= nHF , (120)

where the matrix (A∗F )

ji has positive eigenvalues given by

μ|| = f ′′(�) |∇u|2W 2 + f (�)− f ′(�)�

and

μ⊥ = f (�)− f ′(�)�. (121)

Notice that

� = 〈X, N 〉 = 1

W> 0. (122)

It follows that the graph of u has constant anisotropic mean curvature HF and boundary if usatisfies the Dirichlet problem associated to the equation (120) with boundary data u| = 0.

123


The standard method to solve this Dirichlet problem is the Continuity Method. Thisinvolves to consider the one-parameter family of Dirichlet problems Pτ , τ ∈ [0, 1], given by

(A∗F )

ij

(δ jk − u j

W

u j

W

) ∇ Pk ∇ P

i u

W= nτHF in �, u| = 0. (123)

Then we must prove that the set I of τ ∈ [0, 1] for which Pτ has a C2,α solution is the wholeinterval [0, 1].

First of all, we observe that the function u = 0 solves the problem P0. Moreover, since(120) do not depend explicitly on u, one could verify that it satisfies the maximum andcomparison principles for quasilinear second order elliptic equations. This assures that I isopen.

Finally, if we establish a priori C0 and C1 estimates for solutions of Pτ which are inde-pendent of the parameter τ , then we will be able to prove that I is also closed. If it is thecase, this implies that I = [0, 1] as we want to prove.

Now, we obtain the aforementioned a priori estimates. It suffices to present the proof ofthe estimates for τ = 1. From now on, denote the graph of u by M .

The hypothesis on the Ricci tensor and the fact that � = 1W > 0 along the graph (see

122) imply that � is a superharmonic. Indeed one has μ|| > 0 and

trAF A = trA∗F A2 = (A∗

F )ij A j

k Aki = Ai

k(A∗F )i j A j

k ≥ 0.

Then it follows from (109) that

�� ≤ 0.

Let x ∈ a point where � attains its minimum. Notice that

minM� ≥ min

� ≥ �(x).

If ν is the unit conormal direction along pointing inwards M , then

ν(�) ≥ 0.

However, since ν = X T

|X T | along , it follows that

〈X, ν〉〈Aν, ν〉 = 〈AX T , ν〉 = −〈∇�, ν〉 ≤ 0.

Moreover one has

A∗Fν = μ||ν.

Therefore

〈X, ν〉〈AFν, ν〉 = 〈X, ν〉〈A∗F Aν, ν〉 = 〈X, ν〉〈Aν,A∗

Fν〉 = 〈AX T ,A∗Fν〉 = −〈∇�,A∗

Fν〉= −A∗

Fν(�) ≤ 0.

We may suppose without loss of generality that the graph M is contained in the half-spacet ≥ 0. Indeed suppose by contradiction that there exist points of M in both halfspaces t > 0and t < 0 determined by P × {0}. Then denote t− = minM t < 0 and t+ = maxM t > 0.However observe that the comparison principle may be applied to (123). Hence comparingM with the totally geodesic hypersurfaces P × {t±}, it follows that the anisotropic meancurvature of M calculated with respect to the orientation given by (119) is nonpositive atpoints of M where t = t+ and it is nonnegative at points of M where t = t−. In this case

123


we conclude that HF = 0. Since ∂M = ⊂ P it follows that M = � what contradicts theassumption that M has points in both halfspaces t > 0 and t < 0. We conclude that M isentirely contained in only one of the halfspaces t ≥ 0 and t ≤ 0.

Since M ⊂ P × {t ≥ 0} we have 〈X, ν〉 ≥ 0. Then, we conclude that

〈AFν, ν〉 ≤ 0.

Now we observe that the anisotropic Weingarten map A0 of the cylinder

K = {(x, t) : x ∈ , t ∈ R} (124)

over is given by A0 = −∇ξ0, where ξ0 is the vector field given by

ξ0 = f (0)η + f ′(0)X. (125)

Here, η is the unit normal vector field along ⊂ P pointing outwards �. This vector fieldis translated along the vertical lines in K .

Expression (125) may be deduced as follows. One has

�∣∣K = 〈X, η〉 = 0.

Then given a tangent vector v in T K orthogonal to X one has

A0v = −∇vξ0 = − f (0)∇vη + f ′(0)∇vX = f (0)Av,

where A is the Weingarten map of ⊂ P . Here we used the fact that X is a parallel vectorfield what implies that ∇vX = 0. Moreover

A0 X = f (0)∇Xη = 0

since η is parallel along the flow lines of X .We could determine A0 also remarking that the eigenvalues of A∗

F at the points of T Kare μ|| = f ′′(0)+ f (0) and μ⊥ = f (0). Then in matricial terms

A0 =[

f ′′(0)+ f (0) 00 f (0)In−1

] [0 00 A

]. (126)

Now, we consider a tangent orthonormal frame at Tx M formed by ν(x) and an orthonormallocal frame e1, . . . , en−1 for . Then, we calculate at x

nHF = 〈AFν, ν〉 −n−1∑i=1

⟨∇ei ξ, ei⟩.

However, since ξ = f (�)N + f ′(�)X T along , one has

ξ = 〈ξ, X〉X + 〈ξ, η〉η.Hence, we obtain

nHF = 〈AFν, ν〉 −n−1∑i=1

⟨∇ei η, ei⟩ 〈ξ, η〉 = 〈AFν, ν〉 −

n−1∑i=1

⟨∇ei η, ei⟩ 〈ξ, η〉

what implies that

nHF = 〈AFν, ν〉 + (n − 1)tr A〈ξ, η〉 = 〈AFν, ν〉 + 1

f (0)nHK 〈ξ, η〉 ≤ 1

f (0)nHK 〈ξ, η〉,

123


where

HK = 1

ntrA0 (127)

is the anistropic mean curvature of the cylinder.However,

〈ξ, η〉 = ⟨f (�)N + f ′(�)(X −�N ), η

⟩ = ( f (�)− f ′(�)�)〈N , η〉 = μ⊥〈N , η〉 ≥ 0,

where we used that 〈N , η〉 ≥ 0 which follows from the fact that � is a graph. It results that

HF ≤ μ⊥

f (0)HK 〈N , η〉. (128)

Notice that since M ⊂ P × {t ≥ 0} the comparison principle implies HF ≤ 0. Recall thatHF is calculated with respect to the orientation given by the normal vector field defined in(119). Moreover, HK < 0 by hypothesis.

Therefore, squaring both sides in (128), we obtain

H2F ≥ μ⊥2

f (0)2〈N , η〉2 H2

K = (1 −�2)μ⊥2

f (0)2H2

K ,

Since f (0) corresponds to the eigenvalue μ⊥ of the cylinder we denote it by μ⊥0 . Then in

terms of this notation we conclude that

minM�2 ≥ �2(x) ≥

μ⊥2

μ⊥20

H2K − H2

F

μ⊥2

μ⊥20

H2K

.

Therefore (116) implies that the gradient is uniformly bounded in M ∪ since

� = 1√1 + |∇ P u|2 .

Moreover a height estimate follows from Proposition 7 since ⊂ Br (o).These uniform height and gradient estimates imply the existence of a solution to the

Dirichlet problem. The uniqueness of solutions follows from a simple application of themaximum principle which is consequence of the expression (107). This finishes the proof ofthe theorem. ��Acknowledgements Jorge Lira was partially supported by CNPq and FUNCAP. Marcelo Melo was partiallysupported by CAPES.

References

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Hypersurfaces with constant anisotropic mean curvature in Riemannian manifolds