Hypersurfaces with Null Higher Order Anisotropic Mean Curvature

Abstract

Given a positive function 𝐹 on 𝕊^𝑛 which satisfies a convexity condition, for 1≤𝑟≤𝑛, we define for hypersurfaces in ℝ^𝑛+1 the 𝑟th anisotropic mean curvature function 𝐻_𝑟;𝐹, a generalization of the usual 𝑟th mean curvature function. We call a hypersurface anisotropic minimal if 𝐻_𝐹=𝐻_1;𝐹=0, and anisotropic 𝑟-minimal if 𝐻_𝑟+1;𝐹=0. Let 𝑊 be the set of points which are omitted by the hyperplanes tangent to 𝑀. We will prove that if an oriented hypersurface 𝑀 is anisotropic minimal, and the set 𝑊 is open and nonempty, then 𝑥(𝑀) is a part of a hyperplane of ℝ^𝑛+1. We also prove that if an oriented hypersurface 𝑀 is anisotropic 𝑟-minimal and its 𝑟th anisotropic mean curvature 𝐻_𝑟;𝐹 is nonzero everywhere, and the set 𝑊 is open and nonempty, then 𝑀 has anisotropic relative nullity 𝑛−𝑟.

1. Introduction

Let 𝐹∶𝕊^𝑛→ℝ⁺ be a smooth function which satisfies the following convexity condition: 𝐷²𝐹+𝐹𝐼_𝑥>0,∀𝑥∈𝕊^𝑛, (1) where 𝕊^𝑛 is the standard unit sphere in ℝ^𝑛+1, 𝐷²𝐹 denotes the intrinsic Hessian of 𝐹 on 𝕊^𝑛, 𝐼 denotes the identity on 𝑇_𝑥𝕊^𝑛, and >0 means that the matrix is positive definite. We consider the map 𝜙∶𝕊^𝑛⟶ℝ^𝑛+1,𝑥⟶𝐹(𝑥)𝑥+grad_𝕊𝑛𝐹_𝑥; (2) its image 𝑊_𝐹=𝜙(𝕊^𝑛) is a smooth, convex hypersurface in ℝ^𝑛+1 called the Wulff shape of 𝐹 (see [1–9]). When 𝐹≡1, the Wulff shape 𝑊_𝐹 is just 𝕊^𝑛.

Now let 𝑥∶𝑀→ℝ^𝑛+1 be a smooth immersion of an oriented hypersurface. Let 𝑁∶𝑀→𝕊^𝑛 denote its Gauss map. The map 𝜈=𝜙∘𝑁∶𝑀→𝑊_𝐹 is called the anisotropic Gauss map of 𝑥.

Let 𝑆_𝐹=−d𝜈. 𝑆_𝐹 is called the 𝐹-Weingarten operator, and the eigenvalues of 𝑆_𝐹 are called anisotropic principal curvatures. Let 𝜎_𝑟 be the elementary symmetric functions of the anisotropic principal curvatures 𝜅₁,𝜅₂,…,𝜅_𝑛: 𝜎_𝑟=_{𝑖1<⋯<𝑖𝑟}𝜅_𝑖1⋯𝜅_𝑖𝑟(1≤𝑟≤𝑛). (3) We set 𝜎₀=1. The 𝑟th anisotropic mean curvature 𝐻_𝑟;𝐹 is defined by 𝐻_𝑟;𝐹=𝜎_𝑟/𝐶^𝑟_𝑛, also see Reilly [10]. 𝐻_𝐹∶=𝐻_1;𝐹 is called the anisotropic mean curvature. When 𝐹≡1, 𝑆_𝐹 is just the Weingarten operator of hypersurfaces, and 𝐻_𝑟;𝐹 is just the 𝑟th mean curvature 𝐻_𝑟 of hypersurfaces which has been studied by many authors (see [11–14]). Thus, the 𝑟th anisotropic mean curvature 𝐻_𝑟;𝐹 generalizes the 𝑟th mean curvature 𝐻_𝑟 of hypersurfaces in the (𝑛+1)-dimensional Euclidean space ℝ^𝑛+1.

We say that 𝑥∶𝑀→ℝ^𝑛+1 is anisotropic 𝑟-minimal if 𝐻_𝑟+1;𝐹=0.

For 𝑝∈𝑀, we define 𝑣(𝑝)=dimker(𝑆_𝐹). We call 𝑣=min_𝑝∈𝑀𝑣(𝑝) the anisotropic relative nullity; it generalized the usual relative nullity.

For a smooth immersion 𝑥∶𝑀→ℚ_𝑐^𝑛+1 of a hypersurface into an (𝑛+1)-dimensional space form with constant sectional curvature 𝑐, we denote by 𝑊=ℚ_𝑐^𝑛+1−_𝑝∈𝑀ℚ^𝑛_𝑐𝑝, (4) where for every 𝑝∈𝑀, (ℚ^𝑛_𝑐)_𝑝 is the totally geodesic hypersurface of ℚ_𝑐^𝑛+1 tangent to 𝑥(𝑀) at 𝑥(𝑝). So, in the case of 𝑐=0, 𝑊 is the set of points which are omitted by the hyperplanes tangent to 𝑥(𝑀).

We will study immersion with 𝑊 nonempty. In this direction, Hasanis and Koutroufiotis (see [15]) proved the following.

Theorem 1. Let 𝑥∶𝑀→ℚ³_𝑐 be a complete minimal immersion with 𝑐≥0. If 𝑊 is nonempty, then 𝑥 is totally geodesic.

Later, in [16], Alencar and Frensel extended the result above assuming an extra condition. They proved the following.

Theorem 2. Let 𝑥∶𝑀→ℚ_𝑐^𝑛+1 be an oriented, minimally immersed hypersurface. If  𝑊 is open and nonempty, then 𝑥 is totally geodesic.

In [17], Alencar and Batista studied hypersurfaces with null higher order mean curvature; they proved the following.

Theorem 3. Let 𝑀 be a complete and orientable Riemannian manifold and let 𝑥∶𝑀→ℚ_𝑐^𝑛+1 be an isometric immersion with 𝐻_𝑟+1=0 and 𝐻_𝑟≠0 everywhere, 𝑟≥1. If  𝑊 is open and nonempty, then the relative nullity 𝑣=𝑛−𝑟.

We note that, Alencar in [18] provides examples of nontotally geodesic minimal hypersurfaces in ℝ^2𝑛, 𝑛≥4, with nonempty 𝑊; in [17], Alencar and Batista provides examples of 1-minimal hypersurfaces with 𝐻₁≠0 everywhere in ℝ^2𝑛,  𝑛≥5, with nonempty 𝑊 but 𝑣≠𝑛−1. These examples show that it is necessary to add an extra hypothesis.

In this paper, we prove the anisotropic version of Theorems 2 and 3 for an immersion 𝑥∶𝑀→ℝ^𝑛+1. Explicitly, we prove the following two theorems.

Theorem 4. Let 𝑥∶𝑀→ℝ^𝑛+1 be an oriented, anisotropic minimally immersed hypersurface. If  𝑊 is open and nonempty, then 𝑥(𝑀) is a part of a hyperplane of ℝ^𝑛+1.

Theorem 5. Let 𝑥∶𝑀→ℝ^𝑛+1 be an oriented immersed hypersurface with 𝐻_𝑟+1;𝐹=0 and 𝐻_𝑟;𝐹≠0 everywhere, 𝑟≥1. If 𝑊 is open and nonempty, then the anisotropic relative nullity 𝑣=𝑛−𝑟.

2. Preliminaries

In this paper, we use the summation convention of Einstein and the following convention of index ranges unless otherwise stated: 1≤𝑖,𝑗,…≤𝑛;1≤𝛼,𝛽,…≤𝑛+1. (5)

We define 𝐹^∗∶ℝ^𝑛+1→ℝ to be 𝐹^∗𝑦=sup𝑦,𝑧𝐹(𝑧)∣𝑧∈ℝ^𝑛+1⧵{0}; (6) then 𝐹^∗ is a Minkowski norm on ℝ^𝑛+1. In fact, as proved in [19], 𝐹^∗∶ℝ^𝑛+1⧵{0}→ℝ is smooth and we have the following.

Proposition 6. (1)  𝐹^∗(𝑦)>0, for all 𝑦∈ℝ^𝑛+1⧵{0};
(2)  𝐹^∗(𝑡𝑦)=𝑡𝐹^∗(𝑦), for all 𝑦∈ℝ^𝑛+1, 𝑡>0;
(3)  𝐹^∗(𝑦+𝑧)≤𝐹^∗(𝑦)+𝐹^∗(𝑧), for all 𝑦,𝑧∈ℝ^𝑛+1, and the equality holds if and only if 𝑦=0, or 𝑧=0 or 𝑦=𝑘𝑧 for some 𝑘>0.
(4)  𝑊_𝐹={𝑦∈ℝ^𝑛+1∣𝐹^∗(𝑦)=1}.

We define 𝑔_𝛼𝛽𝑦=12𝜕²𝐹^∗2𝜕𝑦^𝛼𝜕𝑦^𝛽𝑦,𝑔_𝑦(𝑋,𝑌)=𝑔_𝛼𝛽𝑦𝑋^𝛼𝑌^𝛽, (7) where 𝑦∈ℝ^𝑛+1⧵{0} and 𝑋=(𝑋¹,𝑋²,…,𝑋^𝑛+1), 𝑌=(𝑌¹,𝑌²,…,𝑌^𝑛+1)∈𝑇_𝑦ℝ^𝑛+1≅ℝ^𝑛+1.

From the Euler’s theorem for homogeneous functions, we have 𝜕𝑔_𝛼𝛽𝜕𝑦^𝛾(𝑧)𝑧^𝛽=12𝜕³𝐹^∗2𝜕𝑦^𝛼𝜕𝑦^𝛽𝜕𝑦^𝛾(𝑧)𝑧^𝛽=0, (8) where 𝑧=(𝑧¹,𝑧²,…,𝑧^𝑛+1)∈ℝ^𝑛+1⧵{0}. Thus, 𝜕𝑔_𝑧(𝑋,𝑧)𝜕𝑦^𝛾=𝑔_𝛼𝛽(𝑧)𝜕𝑋^𝛼𝜕𝑦^𝛾𝑧^𝛽+𝑔_𝛼𝛾(𝑧)𝑋^𝛼𝜕𝑧^𝛽𝜕𝑦^𝛾, (9) where 𝑧=(𝑧¹,𝑧²,…,𝑧^𝑛+1)∈𝑇ℝ^𝑛+1 is nonzero everywhere and 𝑋=(𝑋¹,𝑋²,…,𝑋^𝑛+1)∈𝑇ℝ^𝑛+1.

As 𝐹^∗ is a Minkowski norm on ℝ^𝑛+1, the following lemma holds (see [20, 21]).

Lemma 7. For any 𝑦∈ℝ^𝑛+1⧵{0} and 𝑢∈ℝ^𝑛+1 one has 𝑔_𝑦𝑦,𝑧≤𝐹^∗𝑦𝐹^∗(𝑧), (10) and the equality holds if and only if there exists 𝑡≥0 such that 𝑧=𝑡𝑦.

Let 𝑥∶𝑀→ℝ^𝑛+1 be an oriented hypersurface in the Euclidean space ℝ^𝑛+1. Let 𝜈∶𝑀→𝑊_𝐹 denote its anisotropic Gauss map. Then for any 𝑝∈𝑀, 𝜈(𝑝) is perpendicular to 𝑥_∗(𝑇_𝑝𝑀) with respect to the inner product 𝑔_𝜈(𝑝) and 𝐹^∗(𝜈(𝑝))=1. Thus, we call 𝜈(𝑝) an anisotropic unit normal vector of 𝑇_𝑝𝑀.

3. A Connection on Hypersurfaces of Minkowski Space

Let 𝑥∶𝑀→ℝ^𝑛+1 be an oriented hypersurface in the Euclidean space ℝ^𝑛+1 and denote 𝜈∶𝑀→𝑊_𝐹 its anisotropic Gauss map.

Let ∇ be the standard connection on the (𝑛+1)-dimensional Euclidean space ℝ^𝑛+1. For vector fields 𝑋,𝑌 on 𝑀, we decompose ∇_𝑋𝑌 as the tangent part ∇_𝑋𝑌 and the anisotropic normal part II (𝑋,𝑌)𝜈 with respect to the inner product 𝑔_𝜈. That is, ∇_𝑋𝑌=∇_𝑋𝑌+II(𝑋,𝑌)𝜈, (11) where 𝑔_𝜈(∇_𝑋𝑌,𝜈)=0.

We also have the Weingarten formula: ∇_𝑋𝜈=−𝑆_𝐹𝑔𝑋,_𝜈𝑆_𝐹𝑋,𝑌=II(𝑋,𝑌), (12) where we have used (9).

It is easy to verify that ∇ is a torsion free connection on 𝑀 and II is a symmetric second order covariant tensor field on 𝑀. We call II the anisotropic second fundamental form.

Let {𝑒_𝑖}^𝑛_𝑖=1 be a local frame of 𝑀 and {𝜔^𝑖}^𝑛_𝑖=1 its dual frame. Let 𝑔_𝑖𝑗=𝑔_𝜈(𝑒_𝑖,𝑒_𝑗), ∇𝑒_𝑖=𝜔^𝑗_𝑖⊗𝑒_𝑗, II(𝑒_𝑖,𝑒_𝑗)=ℎ_𝑖𝑗, ℎ^𝑗_𝑖=𝑔^𝑗𝑘ℎ_𝑘𝑖, where (𝑔^𝑖𝑗) is the inverse matrix of (𝑔_𝑖𝑗). Then we have 𝑑𝑥=𝜔^𝑖𝑒_𝑖,𝑑𝑒_𝑖=𝜔^𝑗_𝑖𝑒_𝑗+ℎ_𝑖𝑗𝜔^𝑗𝜈,𝑑𝜈=−ℎ^𝑗_𝑖𝜔^𝑖𝑒_𝑗. (13)(14)(15)

Differentiating (13) and using (14), we get 𝑑𝜔^𝑖=𝜔^𝑗∧𝜔^𝑖_𝑗,ℎ_𝑖𝑗=ℎ_𝑗𝑖. (16)

Differentiating (14) and using (14)-(15), we get ℎ_𝑖𝑗𝑘=ℎ_𝑖𝑘𝑗,𝑑𝜔^𝑗_𝑖−𝜔^𝑘_𝑖∧𝜔^𝑗_𝑘1=−2𝑅^𝑗_𝑖𝑘𝑙𝜔^𝑘∧𝜔^𝑙, (17) where 𝑑ℎ_𝑖𝑗−ℎ_𝑖𝑘𝜔^𝑘_𝑗−ℎ_𝑘𝑗𝜔^𝑘_𝑖=ℎ_𝑖𝑗𝑘𝜔^𝑘, (18) and 𝑅^𝑗_𝑖𝑘𝑙=−𝑅^𝑗_𝑖𝑙𝑘=ℎ_𝑖𝑘ℎ^𝑗_𝑙−ℎ_𝑖𝑙ℎ^𝑗_𝑘.

Differentiating (15) and using (14), we get ℎ^𝑗_𝑖𝑘=ℎ^𝑗_𝑘𝑖, (19) where 𝑑ℎ^𝑗_𝑖+ℎ^𝑘_𝑖𝜔^𝑗_𝑘−ℎ^𝑗_𝑘𝜔^𝑘_𝑖=ℎ^𝑗_𝑖𝑘𝜔^𝑘. (20)

Note (ℎ^𝑗_𝑖) is the matrix of the 𝐹-Weingarten operator 𝑆_𝐹=−𝑑𝜈, its eigenvalues are called the anisotropic principal curvatures, and we denote them by 𝜅₁,…,𝜅_𝑛.

We have 𝑛 invariants, the elementary symmetric function 𝜎_𝑟 of the anisotropic principal curvatures: 𝜎_𝑟=_{𝑖1<⋯𝑖𝑟}𝜅_𝑖1⋯𝜅_𝑖𝑛(1≤𝑟≤𝑛). (21) For convenience, we set 𝜎₀=1. The 𝑟th anisotropic mean curvature 𝐻_𝑟;𝐹 is defined by 𝐻_𝑟;𝐹=𝜎_𝑟𝐶^𝑟_𝑛,𝐶^𝑟_𝑛=𝑛!()!.𝑟!𝑛−𝑟 (22)

Using the characteristic polynomial of 𝑆_𝐹, 𝜎_𝑟 is defined by det𝑡𝐼−𝑆_𝐹=^𝑛_𝑟=0(−1)^𝑟𝜎_𝑟𝑡^𝑛−𝑟. (23)

So, we have 𝜎_𝑟=1𝑟!_{𝑖1,…,𝑖𝑟;𝑗1,…,𝑗𝑟}𝛿^{𝑗1⋯𝑗𝑟}_{𝑖1⋯𝑖𝑟}ℎ^𝑖1_𝑗1⋯ℎ^𝑖𝑟_𝑗𝑟, (24) where 𝛿^{𝑗1⋯𝑗𝑟}_{𝑖1⋯𝑖𝑟} is the usual generalized Kronecker symbol; that is, 𝛿^{𝑗1⋯𝑗𝑟}_{𝑖1⋯𝑖𝑟} equals +1 (resp., −1) if 𝑖₁⋯𝑖_𝑟 are distinct and (𝑗₁⋯𝑗_𝑟) is an even (resp., odd) permutation of (𝑖₁⋯𝑖_𝑟) and in other cases it equals zero.

Definition 8. Let 𝑓∶𝑀→ℝ be a smooth function. One defines the gradient (with respect to the induced metric 𝑔_𝜈 on 𝑀) grad𝑓 of the function 𝑓 by 𝑔_𝜈grad𝑓,𝑋=𝑋𝑓, (25) where 𝑋 is any smooth vector field on 𝑀.
Define 𝑓_𝑖 by 𝑑𝑓=𝑓_𝑖𝜔^𝑖; then grad𝑓=𝑔^𝑖𝑗𝑓_𝑗𝑒_𝑖. (26)

We define 𝑑𝑉=𝑒₁,…,𝑒_𝑛,𝜈𝜔¹∧⋯∧𝜔^𝑛, (27) where |𝑒₁,…,𝑒_𝑛,𝜈| is the determinant of the matrix (𝑒₁,…,𝑒_𝑛,𝜈). Then 𝑑𝑉 is a volume element on 𝑀.

Definition 9. Let 𝑋 be a smooth vector field on 𝑀. One defines the divergence (with respect to the volume element 𝑑𝑉) div𝑋 by 𝑑{𝑖(𝑋)𝑑𝑉}=(div𝑋)𝑑𝑉, where (𝑖(𝑋)𝑑𝑉)𝑌₁,…,𝑌_𝑛−1≡𝑑𝑉𝑋,𝑌₁,…,𝑌_𝑛−1,∀𝑌₁,…,𝑌_𝑛−1∈𝒳(𝑀). (28)

Lemma 10. Let 𝑋=𝑋^𝑖𝑒_𝑖; then div𝑋=𝑋^𝑖_𝑖, where 𝑑𝑋^𝑖+𝑋^𝑗𝜔^𝑖_𝑗=𝑋^𝑖_𝑗𝜔^𝑗. (29)

Proof. By (14), (15), we get 𝑑𝑒₁,…,𝑒_𝑛,𝜈=𝜔^𝑖_𝑖𝑒₁,…,𝑒_𝑛,𝜈. (30) From the definition of 𝑖(𝑋), we have 𝑖(𝑋)𝑑𝑉=_𝑖(−1)^𝑖+1𝑋^𝑖𝑒₁,…,𝑒_𝑛,𝜈𝜔¹∧⋯∧𝜔^𝑖∧⋯∧𝜔^𝑛. (31) So, 𝑑{𝑖(𝑋)𝑑𝑉}=_𝑖(−1)^𝑖+1𝑑𝑋^𝑖∧𝑒₁,…,𝑒_𝑛,𝜈𝜔¹∧⋯∧𝜔^𝑖∧⋯∧𝜔^𝑛+_𝑖(−1)^𝑖+1𝑋^𝑖𝑑𝑒₁,…,𝑒_𝑛,𝜈∧𝜔¹∧⋯∧𝜔^𝑖∧⋯∧𝜔^𝑛+_𝑗<𝑖(−1)^𝑖+𝑗𝑋^𝑖𝑒₁,…,𝑒_𝑛,𝜈𝑑𝜔^𝑗∧𝜔¹∧⋯∧𝜔^𝑗∧⋯∧𝜔^𝑖∧⋯∧𝜔^𝑛+_𝑗>𝑖(−1)^𝑖+𝑗+1𝑋^𝑖𝑒₁,…,𝑒_𝑛,𝜈𝑑𝜔^𝑗∧𝜔¹∧⋯∧𝜔^𝑖∧⋯∧𝜔^𝑗∧⋯∧𝜔^𝑛=𝑋^𝑖_𝑖𝑑𝑉. (32)

4. 𝐿_𝑟;𝐹 Operator for Hypersurfaces

We introduce the Newton transformation defined by 𝑃_𝑟=𝜎_𝑟𝐼−𝜎_𝑟−1𝑆_𝐹+⋯+(−1)^𝑟𝑆^𝑟_𝐹,𝑟=0,…,𝑛; (33) then 𝑃₀=𝐼,𝑃_𝑛=0,𝑃_𝑟=𝜎_𝑟𝐼−𝑃_𝑟−1𝑆_𝐹. (34)

Lemma 11. The matrix of 𝑃_𝑟 is given by: 𝑃_𝑟^𝑗_𝑖=1𝛿𝑟!^{𝑗1⋯𝑗𝑟𝑗𝑖1⋯𝑖𝑟𝑖}ℎ^𝑖1_𝑗1⋯ℎ^𝑖𝑟_𝑗𝑟. (35)

Proof. We prove Lemma 11 inductively. For 𝑟=0, it is easy to check that (35) is true.
We can check directly 𝛿^{𝑗1⋯𝑗𝑞}_{𝑖1⋯𝑖𝑞}=𝛿^𝑗1_𝑖1𝛿^𝑗2_𝑖1⋯𝛿^𝑗𝑞−1_𝑖1𝛿^𝑗𝑞_𝑖1𝛿^𝑗1_𝑖2𝛿^𝑗2_𝑖2⋯𝛿^𝑗𝑞−1_𝑖2𝛿^𝑗𝑞_𝑖2𝛿⋮⋮⋱⋮⋮_{𝑗1𝑖𝑞−1}𝛿^𝑗2_𝑖𝑞−1⋯𝛿^𝑗𝑞−1_𝑖𝑞−1𝛿^𝑗𝑞_𝑖𝑞−1𝛿^𝑗1_𝑖𝑞𝛿^𝑗2_𝑖𝑞⋯𝛿^𝑗𝑞−1_𝑖𝑞𝛿^𝑗𝑞_𝑖𝑞. (36)
Assume that (35) is true for 𝑟=𝑘, we only need to show that it is also true for 𝑟=𝑘+1. For 𝑟=𝑘+1, using (24) and (36), we have ()=1RHSof35(𝑘+1)!_{𝑖1,…,𝑖𝑘+1;𝑗1,…,𝑗𝑘+1}𝛿^{𝑗1⋯𝑗𝑘+1𝑗}_{𝑖1⋯𝑖𝑘+1𝑖}ℎ^𝑗1_𝑖1⋯ℎ^𝑗𝑘+1_𝑖𝑘+1=1()!𝛿𝑘+1^𝑗1𝑖1𝛿^𝑗2_𝑖1⋯𝛿^𝑗𝑘+1_𝑖1𝛿^𝑗_𝑖1𝛿^𝑗1_𝑖2𝛿^𝑗2_𝑖2⋯𝛿^𝑗𝑘+1_𝑖2𝛿^𝑗_𝑖2𝛿⋮⋮⋱⋮⋮_{𝑗1𝑖𝑘+1}𝛿^𝑗2_𝑖𝑘+1⋯𝛿^𝑗𝑘+1_𝑖𝑘+1𝛿^𝑗_𝑖𝑘+1𝛿^𝑗1_𝑖𝛿^𝑗2_𝑖⋯𝛿^𝑗𝑘+1_𝑖𝛿^𝑗_𝑖ℎ^𝑗1_𝑖1⋯ℎ^𝑗𝑘+1_𝑖𝑘+1=1()!𝛿𝑘+1^𝑗𝑖𝛿^{𝑗1⋯𝑗𝑘+1}_{𝑖1⋯𝑖𝑘+1}−𝛿^𝑗𝑘+1_𝑖𝛿^{𝑗1⋯𝑗𝑘𝑗}_{𝑖1⋯𝑖𝑘𝑖𝑘+1}+⋯ℎ^𝑗1_𝑖1⋯ℎ^𝑗𝑘+1_𝑖𝑘+1=𝜎_𝑘+1𝛿^𝑗_𝑖−1()!𝛿𝑘+1^{𝑗𝑘+1𝑖}𝛿^{𝑗1⋯𝑗𝑘𝑗}_{𝑖1⋯𝑖𝑘𝑖𝑘+1}ℎ^𝑗1_𝑖1⋯ℎ^𝑗𝑘+1_𝑖𝑘+1+⋯=𝜎_𝑘+1𝛿^𝑗_𝑖−𝑃_𝑘^𝑖𝑘+1_𝑖ℎ^𝑗_𝑖𝑘+1=𝑃_𝑘+1^𝑗_𝑖. (37)

Lemma 12. For each 𝑟, one has(a) (𝑃_𝑟)^𝑗_𝑖𝑗=0;(b) Trace(𝑃_𝑟𝑆_𝐹)=(𝑟+1)𝜎_𝑟+1;(c) Trace(𝑃_𝑟)=(𝑛−𝑟)𝜎_𝑟;(d) Trace(𝑃_𝑟𝑆²_𝐹)=𝜎₁𝜎_𝑟+1−(𝑟+2)𝜎_𝑟+2.

Proof. (a) Noting (𝑗,𝑗_𝑟) is skew symmetric in 𝛿^{𝑗1⋯𝑗𝑟𝑗}_{𝑖1⋯𝑖𝑟𝑖} and (𝑗,𝑗_𝑟) is symmetric in ℎ^𝑖1_𝑗1⋯ℎ_𝑗𝑟^𝑖𝑟_𝑗 (from (19), we have _𝑗𝑃_𝑟^𝑗_𝑖𝑗=1(𝑟−1)!_{𝑖1,…,𝑖𝑟;𝑗1,…,𝑗𝑟;𝑗}𝛿^{𝑗1⋯𝑗𝑟𝑗}_{𝑖1⋯𝑖𝑟𝑖}ℎ^𝑖1_𝑗1⋯ℎ_𝑗𝑟^𝑖𝑟_𝑗=0. (38) (b) Using (35) and (24), we have Trace𝑃_𝑟𝑆_𝐹=_𝑖𝑗𝑃_𝑟^𝑗_𝑖ℎ^𝑖_𝑗=1𝑟!_{𝑖1,…,𝑖𝑟;𝑗1,…,𝑗𝑟;𝑖,𝑗}𝛿^{𝑗1⋯𝑗𝑟𝑗}_{𝑖1⋯𝑖𝑟𝑖}ℎ^𝑖1_𝑗1⋯ℎ^𝑖𝑟_𝑗𝑟ℎ^𝑖_𝑗=(𝑟+1)𝜎_𝑟+1. (39) (c) Using (b) and the definition of 𝑃_𝑟, we have Trace𝑃_𝑟=tr𝜎_𝑟𝐼−tr𝑃_𝑟−1𝑆_𝐹=𝑛𝜎_𝑟−𝑟𝜎_𝑟=(𝑛−𝑟)𝜎_𝑟. (40) (d) Using (b) and the definition of 𝑃_𝑟+1, we have Trace𝑃_𝑟𝑆²_𝐹=Trace𝜎_𝑟+1𝑆_𝐹−Trace𝑃_𝑟+1𝑆_𝐹=𝜎₁𝜎_𝑟+1−(𝑟+2)𝜎_𝑟+2. (41)

Remark 13. When 𝐹=1, Lemma 12 was a well-known result (e.g., see Barbosa and Colares [22], or Reilly [23]).

Lemma 14. One has 𝜎_𝑟𝑘=_𝑖,𝑗𝑃_𝑟−1^𝑗_𝑖ℎ_𝑗^𝑖_𝑘. (42)

Proof. From the definition of 𝜎_𝑟, we have the following calculation: 𝜎_𝑟𝑘=1𝑟!_{𝑖1,…,𝑖𝑟;𝑗1,…,𝑗𝑟}𝛿^{𝑗1⋯𝑗𝑟}_{𝑖1⋯𝑖𝑟}ℎ^𝑗1_𝑖1⋯ℎ^𝑗𝑟_𝑖𝑟𝑘=1(𝑟−1)!_{𝑖1,…,𝑖𝑟;𝑗1,…,𝑗𝑟}𝛿^{𝑗1⋯𝑗𝑟}_{𝑖1⋯𝑖𝑟}ℎ^𝑗1_𝑖1⋯ℎ_𝑖𝑟^𝑗𝑟_𝑘=_{𝑖𝑟,𝑗𝑟}𝑃_𝑟−1^𝑗𝑟_𝑖𝑟ℎ^𝑗𝑟_𝑖𝑟𝑘=_𝑖,𝑗𝑃_𝑟−1^𝑗_𝑖ℎ^𝑗_𝑖𝑘. (43)

We define an operator 𝐿_𝑟;𝐹∶𝐶^∞(𝑀)→𝐶^∞(𝑀) by 𝐿_𝑟;𝐹𝑓=div𝑃_𝑟∇𝑓. (44)

In the sequel, we will need the following lemma. Item (a) is essentially the content of Lemma 1.1 and Equation (1.3) in [24], while item (b) is quoted as Proposition  1.5 in [25].

Lemma 15. Let 𝑥∶𝑀→ℝ^𝑛+1 be an oriented hypersurface, and 0≤𝑟≤𝑛−1, 𝑝∈𝑀.(a)If 𝜎_𝑟+1(𝑝)=0, then 𝑃_𝑟 is semidefinite at 𝑝;(b)if 𝜎_𝑟+1(𝑝)=0 and 𝜎_𝑟+2(𝑝)≠0, then 𝑃_𝑟 is definite at 𝑝.

Another important result is as follows (see [26]).

Lemma 16. Let 𝑥∶𝑀→ℝ^𝑛+1 be an oriented hypersurface, and 𝑝∈𝑀.(a)For 1≤𝑟≤𝑛, one has 𝐻²_𝑟;𝐹≥𝐻_{𝑟−1;𝐹}𝐻_𝑟+1;𝐹. Moreover, if equality happens for 𝑟=1 or for some 1<𝑟<𝑛, with 𝐻_𝑟+1;𝐹≠0 in this case, then 𝑝 is an anisotropic umbilical point (i.e. 𝜅₁(𝑝)=𝜅₂(𝑝)=⋯=𝜅_𝑛(𝑝));(b)if, for some 1≤𝑟<𝑛, one has 𝐻_𝑟;𝐹=𝐻_𝑟+1;𝐹=0, then 𝐻_𝑗;𝐹=0 for all 𝑟≤𝑗≤𝑛. In particular, at most 𝑟−1 of the anisotropic principal curvatures are different from zero.

The result below is standard, so we omit the proof.

Lemma 17. Let 𝑥∶𝑀→ℝ^𝑛+1 be an oriented hypersurface. The operator 𝐿_𝑟;𝐹 associated to the immersion 𝑥 is elliptic if and only if 𝑃_𝑟 is positive definite.

Definition 18. Let 𝑓∶𝑀→ℝ be a smooth function. The Laplacian Δ𝑓 is defined by Δ𝑓∶=𝐿_0;𝐹𝑓=div(grad𝑓).

It is easy to see that Δ is an elliptic differential operator.

Definition 19. Let 𝑥∶𝑀→ℝ^𝑛+1 be an immersed hypersurface and 𝜈 its anisotropic unit normal vector field. The function 𝑢∶=𝑔_𝜈(𝑥,𝜈) is called the support function of the immersion 𝑥.

Next, we compute 𝐿_𝑟;𝐹𝑢 for the support function 𝑢=𝑔_𝜈(𝑥,𝜈).

Differentiating the decomposition 𝑥=𝑔^𝑖𝑗𝑔_𝜈𝑥,𝑒_𝑖𝑒_𝑗+𝑢𝜈, (45) we obtain 𝑑𝑥=𝑑𝑔^𝑖𝑗𝑔_𝜈𝑥,𝑒_𝑗𝑒_𝑖+𝑔^𝑖𝑗𝑔_𝜈𝑥,𝑒_𝑗𝑑𝑒_𝑖+(𝑑𝑢)𝜈+𝑢𝑑𝜈. (46) So, from (13), (14), and (15) we have 𝜔^𝑖𝑒_𝑖=𝑑𝑔^𝑖𝑗𝑔_𝜈𝑥,𝑒_𝑗+𝑔^𝑘𝑗𝑔_𝜈𝑥,𝑒_𝑗𝜔^𝑖_𝑘−𝑢ℎ^𝑖_𝑗𝜔^𝑗𝑒_𝑖+𝑑𝑢+𝑔^𝑗𝑘𝑔_𝜈𝑥,𝑒_𝑗ℎ_𝑖𝑘𝜔^𝑖𝜈. (47) Thus, we get 𝑑𝑢=−𝑔^𝑗𝑘𝑔_𝜈𝑥,𝑒_𝑗ℎ_𝑖𝑘𝜔^𝑖,𝑑𝑔^𝑖𝑗𝑔_𝜈𝑥,𝑒_𝑗+𝑔^𝑘𝑗𝑔_𝜈𝑥,𝑒_𝑗𝜔^𝑖_𝑘−𝑢ℎ^𝑖_𝑗𝜔^𝑗=𝜔^𝑖. (48) Denote 𝑢^𝑖, (𝑔^𝑖𝑗𝑔_𝜈(𝑥,𝑒_𝑗))_𝑘, 𝑢^𝑖_𝑗 by grad𝑢=𝑢^𝑖𝑒_𝑖,𝑔^𝑖𝑗𝑔_𝜈𝑥,𝑒_𝑗𝑘𝜔^𝑘=𝑑𝑔^𝑖𝑗𝑔_𝜈𝑥,𝑒_𝑗+𝑔^𝑘𝑗𝑔_𝜈𝑥,𝑒_𝑗𝜔^𝑖_𝑘,𝑢^𝑖_𝑗𝜔^𝑗=𝑑𝑢^𝑖+𝑢^𝑗𝜔^𝑖_𝑗, (49) respectively. Then we have (using (19) the following 𝑢^𝑖=−𝑔^𝑖𝑙ℎ_𝑘𝑙𝑔^𝑗𝑘𝑔_𝜈𝑥,𝑒_𝑗=−ℎ^𝑖_𝑘𝑔^𝑘𝑙𝑔_𝜈𝑥,𝑒_𝑙,𝑔^𝑖𝑘𝑔_𝜈𝑥,𝑒_𝑘𝑗=𝛿^𝑖_𝑗+ℎ^𝑖_𝑗𝑔_𝜈(),𝑢𝑥,𝜈_𝑖𝑗=−ℎ^𝑖_𝑘𝑗𝑔^𝑘𝑙𝑔_𝜈𝑥,𝑒_𝑙−ℎ^𝑖_𝑘𝑔^𝑘𝑙𝑔_𝜈𝑥,𝑒_𝑙𝑗=−ℎ_𝑗^𝑖_𝑘𝑔^𝑘𝑙𝑔_𝜈𝑥,𝑒_𝑙−ℎ^𝑖_𝑗−ℎ^𝑖_𝑘ℎ^𝑘_𝑗𝑢. (50) By using Lemmas 12 and 14, we get 𝐿_𝑟;𝐹𝑢=𝑃_𝑟^𝑗_𝑖𝑢^𝑖_𝑗=−𝑃_𝑟^𝑗_𝑖ℎ_𝑗^𝑖_𝑘𝑔^𝑘𝑙𝑔_𝜈𝑥,𝑒_𝑙−𝑃_𝑟^𝑗_𝑖ℎ^𝑖_𝑗−𝑃_𝑟^𝑗_𝑖ℎ^𝑖_𝑘ℎ^𝑘_𝑗𝑢=−𝜎_𝑟+1𝑘𝑔^𝑘𝑙𝑔_𝜈𝑥,𝑒_𝑙−𝑃_𝑟^𝑗_𝑖ℎ^𝑖_𝑗−𝑃_𝑟^𝑗_𝑖ℎ^𝑖_𝑘ℎ^𝑘_𝑗𝑢=−𝑔_𝜈∇𝜎_𝑟+1,𝑥−(𝑟+1)𝜎_𝑟+1−𝜎₁𝜎_𝑟+1−(𝑟+2)𝜎_𝑟+2𝑢. (51) Thus, we proved the following lemma.

Lemma 20. For 0≤𝑟≤𝑛−1, one has the following 𝐿_𝑟;𝐹𝑢=−𝑔_𝜈∇𝜎_𝑟+1,𝑥−(𝑟+1)𝜎_𝑟+1−𝜎₁𝜎_𝑟+1−(𝑟+2)𝜎_𝑟+2𝑢. (52)

Remark 21. Recall 𝜎₁=𝑛𝐻_𝐹 and |II|²=𝜎²₁−2𝜎₂; let 𝑟=0 in (52); we get Δ𝑢=−𝑛𝐻_𝐹+𝑔_𝜈grad𝐻_𝐹,𝑥−|II|²𝑢. (53)

5. Proof of Theorems 4 and 5

We fix a point 𝑜∈𝑊 as the origin of ℝ^𝑛+1. Without loss of generality, we assume, for each 𝑝∈𝑀, 𝜈(𝑝) is the anisotropic unit normal vector of 𝑥(𝑀) at 𝑥(𝑝) such that ⟨𝑥(𝑝),𝜈(𝑝)⟩_𝜈(𝑝)>0 (otherwise we consider the function −𝑢 instead). This gives an orientation to 𝑀; indeed, the component of the position vector 𝑥 perpendicular (with respect to the inner product 𝑔_𝜈) to 𝑀 defines a never zero, anisotropic normal, vector field on 𝑀, such that the support function 𝑢=⟨𝑥(𝑝),𝜈(𝑝)⟩_𝜈(𝑝) is positive on 𝑀.

5.1. Proof of Theorem 4

Since 𝑥 is anisotropic minimal, from (53) we get Δ𝑢=−|II|²𝑢≤0,on𝑀. (54) Let 𝑢_∗=inf_𝑀𝑢. We claim that 𝑢_∗ is attained at some point 𝑥₀∈𝑀. Consider a sequence {𝑥_𝑘}⊂𝑀 such that 𝑢(𝑥_𝑘)→𝑢_∗ as 𝑘→+∞. To each 𝑥_𝑘 we associate 𝑦_𝑘=𝑢(𝑥_𝑘)𝜈(𝑥_𝑘); then 𝑦_𝑘∈𝑇_𝑥𝑘𝑀. Since ‖𝑦_𝑘‖_ℝ𝑛+1=𝑢(𝑥_𝑘)‖𝜈(𝑥_𝑘)‖_ℝ𝑛+1 is bounded, there exists a subsequence, which again we call {𝑦_𝑘}, such that 𝑦_𝑘→𝑦₀ for some 𝑦₀∈ℝ^𝑛+1. Since ⋃_𝑝∈𝑀𝑇_𝑝𝑀 is closed and {𝑦_𝑘}⊂_𝑝∈𝑀𝑇_𝑝𝑀 we deduce that 𝑦₀∈𝑇_𝑥0𝑀 for some 𝑥₀∈𝑀. Thus, by the continuity of 𝐹^∗ and Lemma 7, 𝑢_∗=lim_𝑘→+∞𝑢𝑥_𝑘=lim_𝑘→+∞𝐹^∗𝑦_𝑘=𝐹^∗𝑦₀≥𝑔_𝜈(𝑥0)𝑦₀,𝜈𝑥₀=𝑢𝑥₀, (55) so 𝑢^∗=𝑢(𝑥₀) as needed. Now, from the usual maximum principle 𝑢 is constant, 𝑢=𝑢_∗=𝑢(𝑥₀)>0. From (54) we then have II≡0 and 𝑥 is totally geodesic.

5.2. Proof of Theorem 5

Since 𝐻_𝑟+1;𝐹=0, from Lemma 20 we get 𝐿_𝑟;𝐹𝑢=(𝑟+2)𝜎_𝑟+2𝑢. (56) Using Lemma 15(a) we have that 𝑃_𝑟 is semidefinite. Since 𝐻_𝑟;𝐹 does not vanish, we have that 𝐻_𝑟;𝐹 is positive or negative, because 𝑐(𝑟)𝐻_𝑟;𝐹=Trace(𝑃_𝑟), where 𝑐(𝑟)=(𝑛−𝑟)𝐶^𝑟_𝑛. Now we use Lemma 16 and obtain the following: 0=𝐻²_𝑟+1;𝐹≥𝐻_𝑟;𝐹𝐻_𝑟+2;𝐹. (57) Using the information above, we claim that 𝐻_𝑟+2;𝐹≡0.

Case  (i) (𝐻_𝑟;𝐹>0). In this case, 𝑃_𝑟 is positive definite, and 𝐿_𝑟;𝐹 is elliptic by Lemma 17. Using (57) we conclude that 𝐻_𝑟+2;𝐹≤0. Whereas from (56) we have 𝐿_𝑟;𝐹𝑢≤0. (58) Following exactly the proof as in Theorem 4, we conclude that 𝑢 is constant, 𝑢=𝑢_∗=𝑢(𝑥₀)>0. From (56) we then have 𝐻_𝑟+2;𝐹≡0.

Case  (ii) (𝐻_𝑟;𝐹<0). In this case, 𝑃_𝑟 is negative definite, and −𝐿_𝑟;𝐹 is elliptic by Lemma 17. Using (57) we conclude that 𝐻_𝑟+2;𝐹≥0. Whereas from (56) we have −𝐿_𝑟;𝐹𝑢≤0. (59)

Now, following exactly the proof as in Theorem 4, we conclude that 𝑢 is constant, 𝑢=𝑢_∗=𝑢(𝑥₀)>0. From (56) we then have 𝐻_𝑟+2;𝐹≡0.

Thus we conclude that 𝐻_𝑟+2;𝐹≡0. Now, we use Lemma 16(b) to conclude that 𝐻_𝑗;𝐹=0 for 𝑗≥𝑟+1 and so that 𝑣≥𝑛−𝑟. Since 𝐻_𝑟;𝐹 does not change sign we have that 𝑣=𝑛−𝑟.