Some Integral Formulas for the (r + 1)th Mean Curvature of a Closed Hypersurface

Abstract

By using the operator 𝐿_𝑟, we define the notions of rth order and rth type of a Euclidean hypersurface. By the use of these notions, we are able to obtain some sharp estimates of the (𝑟+1)th mean curvature for a closed hypersurface of the Euclidean space in terms of rth order.

1. Introduction

The study of submanifolds of finite type began in 1970, with Chen's attempts to find the best possible estimate of the total mean curvature of a compact submanifold of the Euclidean space and to find a notion of “degree” for Euclidean submanifolds [1, 2].

In algebraic geometry varieties are the main objects to study. Since an algebraic variety is defined by using algebraic equations, one can define the degree of an algebraic variety by its algebraic structure, and it is well known that the concept of degree plays a fundamental role in algebraic geometry [3].

On the other hand, in differential geometry, the main objects to study are Riemannian (sub) manifolds. According to Nash’s immersion Theorem, every Riemannian manifold can be realized as a submanifold of the Euclidean space via an isometric immersion [4], but there is no notion of degree for submanifolds of the Euclidean space in general.

So inspired by algebraic geometry, in 1970, Chen defined the notions of order and type for submanifolds of the Euclidean space by the use of the Laplace operator. After that, Chen was able to obtain some sharp estimates of the total mean curvature for compact submanifolds of the Euclidean space in terms of their orders. Moreover, he could introduce submanifolds and maps of finite type [1, 2].

On one hand finite-type submanifolds provides a natural way to exploit the spectral theory to study the geometry of submanifolds and smooth maps, in particular the Gauss map. On the other hand the techniques of the submanifold theory can be used in the study of spectral geometry via the study of finite-type submanifolds.

As is well known, the Laplace operator of a hypersurface 𝑀 immersed into ℝ^𝑛+1 is an (intrinsic) second-order linear elliptic differential operator which arises naturally as the linearized operator of the first variation of the mean curvature for normal variations of the hypersurface. From this point of view, the Laplace operator Δ can be seen as the first one of a sequence of 𝑛 operators 𝐿₀=Δ,𝐿₁,…,𝐿_𝑛−1, where 𝐿_𝑟 stands for the linearized operator of the first variation of the (𝑟+1)th mean curvature arising from normal variations of the hypersurface (see [5]). These operators are given by 𝐿_𝑟(𝑓)=𝑡𝑟(𝑃_𝑟∘∇²𝑓) for any 𝑓∈𝐶^∞(𝑀), where 𝑃_𝑟 denotes the 𝑟th Newton transformation associated to the second fundamental form of the hypersurface, and ∇²𝑓 is the hessian of 𝑓 (see the next section for details).

In contrast to the operator Δ, the operators 𝐿_𝑟 are not elliptic in general, but they still share some nice properties with Laplacian of 𝑀; moreover, under appropriate natural geometric hypotheses on the hypersurface, they are elliptic [6]. Therefore, from this point of view, it seems natural and interesting to generalize the definition of finite-type hypersurface by replacing Δ by the operator 𝐿_𝑟. Having this idea, for the first time in [7], the second author, inspired by a private communication with Alías, introduced such hypersurfaces and called them “𝐿_𝑟-finite type” hypersurfaces.

In this paper, by using the operator 𝐿_𝑟, we define the notions of 𝑟th order and 𝑟th type of a Euclidean hypersurface. Then we are able to obtain some sharp estimates of the (𝑟+1)th mean curvature for closed hypersurfaces of the Euclidean space in terms of their 𝑟th orders when 𝐿_𝑟 is elliptic. The paper generalizes the results of [8, 9].

2. Preliminaries

In this section, we recall some prerequisites about Newton transformations 𝑃_𝑟 and their associated second-order differential operators 𝐿_𝑟 from [10].

Consider an orientable isometrically immersed hypersurface 𝑥∶𝑀^𝑛→ℝ^𝑛+1 in the Euclidean space, with the Gauss map 𝑁. We denote by ∇⁰ and ∇ the Levi-Civita connections on ℝ^𝑛+1 and 𝑀, respectively. Then, the basic Gauss and Weingarten formulae of the hypersurface are written as ∇⁰_𝑋𝑌=∇_𝑋𝑌+⟨𝑆𝑋,𝑌⟩𝑁,𝑆𝑋=−∇⁰_𝑋𝑁 (2.1) for all tangent vector fields 𝑋,𝑌∈𝜒(𝑀), where 𝑆∶𝜒(𝑀)→𝜒(𝑀) is the shape operator of 𝑀 with respect to the Gauss map 𝑁. As is well known, 𝑆 defines a self-adjoint linear operator on tangent space 𝑇_𝑝𝑀, and its eigenvalues 𝜅₁(𝑝),…,𝜅_𝑛(𝑝) are called the principal curvatures of the hypersurface. Associated to the shape operator 𝑆, there are 𝑛 algebraic invariants given by 𝑠_𝑟𝑝=𝜎_𝑟𝜅₁𝑝,…,𝜅_𝑛𝑝,1≤𝑟≤𝑛, (2.2) where 𝜎_𝑟∶ℝ^𝑛→ℝ is the elementary symmetric function in ℝ^𝑛 given by 𝜎_𝑟𝑥₁,…,𝑥_𝑛=_{1≤𝑖1<⋯<𝑖𝑟≤𝑛}𝑥_𝑖1⋯𝑥_𝑖𝑟. (2.3) Observe that the characteristic polynomial of 𝑆 can be written in terms of the 𝑠_𝑟, 𝑠 as 𝑄_𝑆(𝑡)=det(𝑡𝐼−𝑆)=^𝑛_𝑟=0(−1)^𝑟𝑠_𝑟𝑡^𝑛−𝑟, (2.4) where 𝑠₀=1 by definition. The 𝑟th mean curvature 𝐻_𝑟 of the hypersurface is then defined by 𝑛𝑘𝐻_𝑟=𝑠_𝑟,0≤𝑟≤𝑛. (2.5) In particular, when 𝑟=1, 𝐻₁=1𝑛^𝑛_𝑖=1𝜅_𝑖=1𝑛𝑡𝑟(𝑆)=𝐻 (2.6) is nothing but the mean curvature of 𝑀, which is the main extrinsic curvature of the hypersurface. On the other hand, 𝐻_𝑛=𝜅₁⋯𝜅_𝑛 is called the Gauss-Kronecker curvature of 𝑀. A hypersurface with zero (𝑟+1)th mean curvature in ℝ^𝑛+1 is called 𝑟-minimal.

The classical Newton transformations 𝑃_𝑟∶𝜒(𝑀)→𝜒(𝑀) are defined inductively by 𝑃₀=𝐼,𝑃_𝑟=𝑠_𝑟𝐼−𝑆∘𝑃_𝑟−1=𝑛𝑟𝐻_𝑟𝐼−𝑆∘𝑃_𝑟−1 (2.7) for 𝑟=1,…,𝑛 where 𝐼 denotes the identity of 𝜒(𝑀). Equivalently, we have 𝑃_𝑟=^𝑟_𝑗=0(−1)^𝑗𝑠_𝑟−𝑗𝑆^𝑗=^𝑟_𝑗=0(−1)^𝑗𝑛𝐻𝑟−𝑗^𝑟−𝑗𝑆^𝑗. (2.8) Note that, by the Cayley-Hamilton Theorem stating that any operator 𝑇 is annihilated by its characteristic polynomial, we have 𝑃_𝑛=0 from (2.4).

Each 𝑃_𝑟(𝑝) is also a self-adjoint linear operator on the tangent space 𝑇_𝑝𝑀 which commutes with 𝑆(𝑝). Indeed, 𝑆(𝑝) and 𝑃_𝑟(𝑝) can be simultaneously diagonalized if {𝑒₁,…,𝑒_𝑛} are the eigenvectors of 𝑆(𝑝) corresponding to the eigenvalues 𝜅₁(𝑝),…,𝜅_𝑛(𝑝), respectively. Then they are also the eigenvectors of 𝑃_𝑟(𝑝) with corresponding eigenvalues given by 𝜇_𝑖,𝑟𝑝=_{1≤𝑖1<⋯<𝑖𝑟≤𝑛,𝑖𝑗≠𝑖}𝜅_𝑖1𝑝⋯𝜅_𝑖𝑟𝑝, (2.9) for every 𝑟, 1≤𝑖≤𝑛. We have the following formulas for the Newton transformation 𝑃_𝑟 [5]: 𝑡𝑟𝑃_𝑟=𝑐_𝑟𝐻_𝑟,𝑡𝑟𝑆∘𝑃_𝑟=𝑐_𝑟𝐻_𝑟+1,𝑡𝑟𝑆²∘𝑃_𝑟=𝑛𝑟+1𝑛𝐻₁𝐻_𝑟+1−(𝑛−𝑟−1)𝐻_𝑟+2, (2.10)(2.11)(2.12) where 𝑐_𝑟=()𝑛𝑟=()𝑛𝑛−𝑟𝑟+1𝑟+1. (2.13)

Associated to each Newton transformation 𝑃_𝑟, we consider the second-order linear differential operator 𝐿_𝑟∶𝐶^∞(𝑀)→𝐶^∞(𝑀) given by 𝐿_𝑟𝑓=𝑡𝑟𝑃_𝑟∘∇²𝑓. (2.14)∇²𝑓∶𝜒(𝑀)→𝜒(𝑀) denotes the self-adjoint linear operator metrically equivalent to the Hessian of 𝑓 and is given by ∇²𝑓(𝑋),𝑌=∇_𝑋∇𝑓,𝑌,∀𝑋,𝑌∈𝜒(𝑀). (2.15) Let {𝑒₁,…,𝑒_𝑛} be a local orthonormal frame on 𝑀 and observe that div𝑃_𝑟∇𝑓=^𝑛_𝑖=1∇_𝑒𝑖𝑃_𝑟∇𝑓,𝑒_𝑖+^𝑛_𝑖=1𝑃_𝑟∇_𝑒𝑖∇𝑓,𝑒_𝑖=div𝑃_𝑟,∇𝑓+𝐿_𝑟𝑓, (2.16) where div denotes the divergence operator on 𝑀.

Since div𝑃_𝑟=0 (see [10]), as a consequence, from (2.16), one gets that 𝐿_𝑟𝑓=div𝑃_𝑟∇𝑓. (2.17)

3. The 𝑟th Order and the 𝑟th Type of a Hypersurface

As mentioned in the introduction, there is no notion of degree for submanifolds of the Euclidean space in general. However, Chen could use the induced Riemannian structure on a submanifold to introduce a pair of well-defined numbers 𝑝 and 𝑞 associated with a submanifold (see [1] for the precise definition). 𝑝 is a natural number and 𝑞≥𝑝 or +∞. The pair [𝑝,𝑞] is called the order of the submanifold 𝑀; more precisely, 𝑝 is the lower order, and 𝑞 is the upper order of the submanifold. The submanifold is said to be of finite type if its upper order is finite, and it is of infinite type if its upper order is +∞ (see [1, 2] for details).

Consider an isometrically immersed closed orientable hypersurface 𝑥∶𝑀^𝑛→ℝ^𝑛+1 in the Euclidean space, with the Gauss map 𝑁, and assume that, for a fixed 𝑟, 1≤𝑟≤𝑛−1, 𝐿_𝑟 is an elliptic differential operator on 𝒞^∞(𝑀), the ring of all smooth real functions on 𝑀.

It is well known that the eigenvalues of −𝐿_𝑟 when it is elliptic (see [1], chapter 3, for properties of an elliptic operator) form a discrete infinite sequence 0=𝜆^𝐿𝑟₀<𝜆^𝐿𝑟₁<𝜆^𝐿𝑟₂<⋯↗∞. (3.1) Let 𝑉^𝐿𝑟_𝑘={𝑓∈𝒞^∞(𝑀)∶𝐿_𝑟𝑓+𝜆^𝐿𝑟_𝑘𝑓=0} be the eigenspace of −𝐿_𝑟 with eigenvalue 𝜆^𝐿𝑟_𝑘. Then 𝑉^𝐿𝑟_𝑘 is finite dimensional. Define an inner product (,) on 𝒞^∞(𝑀) by 𝑓,ℎ=_𝑀𝑓ℎ𝑑𝑀, (3.2) where 𝑑𝑀 is the volume element of 𝑀. Then ∑^∞_𝑘=0𝑉^𝐿𝑟_𝑘 is dense in 𝒞^∞(𝑀) (in 𝐿²− sense). If we denote by ⊕𝑉^𝐿𝑟_𝑘 the completion of ∑𝑉^𝐿𝑟_𝑘, we have 𝒞^∞(𝑀)=⊕_𝑘𝑉^𝐿𝑟_𝑘. (3.3)

For each function 𝑓∈𝒞^∞(𝑀), let 𝑓_𝑡 denote the projection of 𝑓 onto the subspace 𝑉^𝐿𝑟_𝑡(𝑡=0,1,2,…). Then we have the following spectral decomposition: 𝑓=^∞_𝑡=0𝑓_𝑡in𝐿²−sence. (3.4) Because 𝑉^𝐿𝑟₀ is 1-dimensional, for any nonconstant function 𝑓∈𝒞^∞(𝑀), there is a positive integer 𝑝≥1 such that 𝑓_𝑝≠0 and 𝑓−𝑓₀=_𝑡≥𝑝𝑓_𝑡, (3.5) where 𝑓₀∈𝑉^𝐿𝑟₀ is a constant. If there are infinitely many 𝑓_𝑡's which are nonzero, we put 𝑞=+∞; otherwise, there is an integer 𝑞,𝑞≥𝑝, such that 𝑓_𝑞≠0 and 𝑓−𝑓₀=^𝑞_𝑡=𝑝𝑓_𝑡. (3.6)

Consider the set 𝑇^𝐿𝑟_𝑓=𝑡∈𝑁∶𝑓_𝑡≢0. (3.7) The smallest element of 𝑇^𝐿𝑟_𝑓 is called the lower 𝑟th order of 𝑓 and is denoted by 𝑙.𝑜.^𝐿𝑟(𝑓), and the supremum of 𝑇^𝐿𝑟_𝑓 is called the upper 𝑟th order of 𝑓 and is denoted by 𝑢.𝑜.^𝐿𝑟(𝑓). A function 𝑓 in 𝒞^∞(𝑀) is said to be of 𝐿_𝑟-finite type if 𝑇^𝐿𝑟_𝑓 is a finite set, that is, if its spectral decomposition contains only finitely many nonzero terms. Otherwise, 𝑓 is said to be of 𝐿_𝑟-infinite type. 𝑓 is said to be of 𝐿_𝑟-𝑘 type if 𝑇^𝐿𝑟_𝑓 contains exactly 𝑘 elements.

For an isometrically immersed closed hypersurface 𝑥∶𝑀^𝑛→ℝ^𝑛+1 in the Euclidean space ℝ^𝑛+1, we put 𝑥=𝑥₁,…,𝑥_𝑛+1, (3.8) where 𝑥_𝐴 is the 𝐴-th component of 𝑥. For each 𝑥_𝐴, we have 𝑥_𝐴−𝑥_𝐴0=^𝑞𝐴_{𝑡=𝑝𝐴}𝑥_𝐴𝑡,𝐴=1,…,𝑛+1. (3.9) For the isometric immersion 𝑥∶𝑀→ℝ^𝑛+1, we put 𝑝=inf_𝐴𝑝_𝐴,𝑞=sup_𝐴𝑞_𝐴. (3.10) It is easy to see that 𝑝 and 𝑞 are independent of the choice of the Euclidean coordinate system on ℝ^𝑛+1, and 𝑝 is a positive integer and 𝑞 is either +∞ or 𝑞≥𝑝. Thus, 𝑝 and 𝑞 are well defined. Consequently, for each closed hypersurface 𝑀 in ℝ^𝑛+1 (or, more precisely, for each isometric immersion 𝑥∶𝑀^𝑛→ℝ^𝑛+1), we have a pair [𝑝,𝑞] associated with 𝑀. We call the pair [𝑝,𝑞] the order of the hypersurface 𝑀.

By using the above notation, we have the following spectral decomposition of 𝑥 in vector form: 𝑥=𝑥₀+^𝑞_𝑡=𝑝𝑥_𝑡. (3.11) We define 𝑇^𝐿𝑟(𝑥) by 𝑇^𝐿𝑟_𝑥=𝑡∈𝑁∶𝑥_𝑡≢0. (3.12) The immersion 𝑥 or the hypersurface 𝑀 is said to be of 𝐿_𝑟-𝑘 type if 𝑇^𝐿𝑟_𝑥 contains exactly 𝑘 elements. Similarly, we can define the lower 𝑟th order and the upper 𝑟th order of the immersion.

The immersion 𝑥 is said to be of 𝐿_𝑟 finite type if its upper 𝑟th order 𝑞 is finite, and it is said to be of 𝐿_𝑟 infinite type if its upper order is +∞.

The following Lemma states that for an isometrically immersed closed orientable hypersurface 𝑥∶𝑀^𝑛→ℝ^𝑛+1, the constant vector 𝑥₀ in (3.11) is exactly the “center of mass” of 𝑀 in ℝ^𝑛+1 (i.e., ∫_𝑀𝑥𝑑𝑀/vol(𝑀), where 𝑑𝑀 is a chosen volume form of 𝑀).

Lemma 3.1. Let 𝑥∶𝑀^𝑛→ℝ^𝑛+1 be an isometric immersion of a closed orientable hypersurface 𝑀 into ℝ^𝑛+1. Assume that 𝐿_𝑟 is elliptic, for some 1≤𝑟≤𝑛−1. Then 𝑥₀ in (3.11) is the center of mass of 𝑀 in ℝ^𝑛+1.

Proof. Consider the decomposition 𝑥=^∞_𝑡=0𝑥_𝑡. (3.13) We have 𝐿_𝑟𝑥_𝑡+𝜆^𝐿𝑟_𝑡𝑥_𝑡=0. If 𝑡≠0, then (2.17) and the Divergence Theorem imply that _𝑀𝑥_𝑡1𝑑𝑀=−𝜆^𝐿𝑟_𝑡𝐿_𝑟𝑥_𝑡𝑑𝑀=0. (3.14) Since 𝑥₀ is a constant vector in ℝ^𝑛+1, we obtain from (3.13) and (3.14) that 𝑥₀=_𝑀𝑥𝑑𝑀(𝑀).vol (3.15)
This shows that 𝑥₀ is the center of mass of 𝑀.

On the set of all ℝ^𝑛+1-valued functions on 𝑀 which is a real vector space, we define an inner product on such space by (𝑣,𝑤)=_𝑀⟨𝑣,𝑤⟩𝑑𝑀 (3.16) for any two ℝ^𝑛+1-valued functions 𝑣,𝑤 on 𝑀, where ⟨𝑣,𝑤⟩(𝑥) denotes the Euclidean inner product of 𝑣(𝑥),𝑤(𝑥) for any 𝑥∈𝑀. Then we have the following lemma.

Lemma 3.2. For an isometric immersion 𝑥∶𝑀→ℝ^𝑛+1 of a closed orientable hypersurface 𝑀 into ℝ^𝑛+1 the components of the spectral decomposition (3.11) are mutually orthogonal, for example, 𝑥_𝑡,𝑥_𝑠=0,𝑡≠𝑠. (3.17)

Proof. Since 𝐿_𝑟 is self-adjoint with respect to the inner product (3.16), we have 𝜆^𝐿𝑟_𝑡𝑥_𝑡,𝑥_𝑠=−𝐿_𝑟𝑥_𝑡,𝑥_𝑠=−𝑥_𝑡,𝐿_𝑟𝑥_𝑠=𝜆^𝐿𝑟_𝑠𝑥_𝑡,𝑥_𝑠. (3.18) Since 𝜆^𝐿𝑟_𝑡≠𝜆^𝐿𝑟_𝑠, we obtain (3.17).

Before we give our main result and to facilitate the reader, we quote Theorem 3.3 from [11] and present Theorem 3.4 about 𝐿_𝑟-finite-type Euclidean hypersurfaces.

Theorem 3.3 (see [11]). Let 𝑥∶𝑀^𝑛→ℝ^𝑛+1 be an orientable connected hypersurface immersed into the Euclidean space, and let 𝐿_𝑟 be the linearized operator of the (𝑟+1)-th mean curvature of 𝑀, for some 𝑟=0,…,𝑛−1. Then, one has 𝐿_𝑟𝑥+𝜆𝑥=0 (3.19) for a real constant 𝜆 if and only if either 𝜆=0 and M is 𝑟-minimal in ℝ^𝑛+1 (i.e., 𝐻_𝑟+1=0 on 𝑀), or 𝜆≠0 and 𝑀 is an open piece of a round sphere 𝑆^𝑛(𝜚)⊂ℝ^𝑛+1 of radius 𝜚=(𝑐_𝑟/‖𝜆|)^(1/(𝑟+2)) centered at the origin of ℝ^𝑛+1, where 𝑐_𝑟=(𝑛−𝑟)(^𝑛_𝑟).

Theorem 3.4. There is no compact Euclidean hypersurface of 𝐿_𝑟-2 type with constant 𝐻_𝑟+1, when 𝐿_𝑟 is elliptic.

Proof. If 𝑀 is of 𝐿_𝑟-2 type, by using (3.11), the position vector field 𝑥 of 𝑀 in ℝ^𝑛+1 has the following spectral decomposition: 𝑥−𝑥₀=𝑥_𝑝+𝑥_𝑞,𝐿_𝑟𝑥_𝑝+𝜆_𝑝𝑥_𝑝=0,𝐿_𝑟𝑥_𝑞+𝜆_𝑞𝑥_𝑞=0,forsome𝑥₀∈ℝ^𝑛+1,𝑝,𝑞∈𝑁, (3.20) so 𝐿²_𝑟𝑥=−𝜆_𝑝+𝜆_𝑞𝐿_𝑟𝑥−𝜆_𝑝𝜆_𝑞𝑥−𝑥₀. (3.21) From [10], we also have 𝐿_𝑟𝑥=𝑐_𝑟𝐻_𝑟+1𝐿𝑁,_2𝑟𝑥=−𝑐_𝑟𝑛𝐻𝑟+1^𝑟+1𝑛𝐻₁𝐻_𝑟+1−(𝑛−𝑟−1)𝐻_𝑟+2𝑁. (3.22)(3.23)
The formula (3.23) holds since 𝐻_𝑟+1 is a nonzero constant, see [10]. Therefore, by using (3.21), (3.22), and (3.23), we obtain that −𝑐_𝑟𝑛𝐻𝑟+1^𝑟+1𝑛𝐻₁𝐻_𝑟+1−(𝑛−𝑟−1)𝐻_𝑟+2𝑁=𝑐_𝑟𝜆_𝑝+𝜆_𝑞𝐻_𝑟+1𝑁−𝜆_𝑝𝜆_𝑞𝑥−𝑥₀. (3.24)
Since 𝜆_𝑝𝜆_𝑞≠0, from (3.24), we have 𝑥−𝑥₀ that is normal to 𝑀 at every point of 𝑀. So ⟨𝑥−𝑥₀,𝑥−𝑥₀⟩ is a positive constant. In this case, 𝑀 is an open piece of 𝑆^𝑛 centered at 𝑥₀, by Theorem 3.3, 𝑀 is of 𝐿_𝑟-1 type, which is not.

4. The 𝑟th Order and the (𝑟+1)th Mean Curvature

In this section, we will relate the notion of the 𝑟th order of a Euclidean hypersurface with the (𝑟+1)th mean curvature. In particular, we will obtain some sharp estimates of the (𝑟+1)th mean curvature for a closed hypersurface of the Euclidean space in terms of rth order of the hypersurface when 𝐿_𝑟 is elliptic. In the following we will state several results from [11, 12] which guarantee the ellipticity of 𝐿_𝑟.

A classical theorem of Hadamard [12] gives three equivalent conditions on a closed connected hypersurface 𝑀^𝑛 immersed into the Euclidean space ℝ^𝑛+1 which imply that 𝑀 is a convex hypersurface (i.e., 𝑀 is embedded in ℝ^𝑛+1 and is the boundary of a convex body).

Theorem 4.1 (Hadamard Theorem, see [12]). Let 𝑥∶𝑀^𝑛→ℝ^𝑛+1 be a closed connected hypersurface immersed into the Euclidean space. The following assertions are equivalent.(i)The second fundamental form is definite at every point of 𝑀.(ii)𝑀 is orientable, and its Gauss map is a diffeomorphism onto 𝑆^𝑛.(iii)The Gauss-Kronecker curvature never vanishes on 𝑀.
Moreover, any of the above conditions implies that 𝑀 is a convex hypersurface.

Here we observe that the convexity of a hypersurface in ℝ^𝑛+1 is closely related to its Ricci curvature.

Theorem 4.2 (see [11]). Let 𝑥∶𝑀^𝑛→ℝ^𝑛+1 be a closed connected hypersurface immersed into the Euclidean space. The following assertion is equivalent to any of the assertions ((𝑖)−(𝑖𝑖𝑖)) in Hadamard theorem, and therefore it also implies that 𝑀 is a convex hypersurface(iv) The Ricci curvature of 𝑀 is positive everywhere on 𝑀.

Corollary 4.3 (see [11]). Let 𝑥∶𝑀^𝑛→ℝ^𝑛+1 be a closed connected hypersurface isometrically immersed into the Euclidean space. If 𝑀 has positive Ricci curvature, then each operator 𝐿_𝑟 on 𝐶^∞(𝑀) is elliptic, and each 𝑟-th mean curvatures of 𝑀 is positive.

In [8] by using the concept of order, Chen obtained the following best possible lower bound of total mean curvature for a closed Euclidean hypersurface.

Theorem 4.4 (see [8]). Let 𝑥∶𝑀→ℝ^𝑛+1 be an orientable closed connected hypersurface isometrically immersed into the Euclidean space. Then, one has _𝑀𝐻²𝜆𝑑𝑀≥^𝑝𝑛vol(𝑀), (4.1) where 𝑝 is the lower order of 𝑀, and vol(𝑀) denotes the 𝑛-dimensional volume of 𝑀. Equality holds if and only if 𝑀 is a round sphere in ℝ^𝑛+1.

Now we establish the corresponding result for the operator 𝐿_𝑟 (since 𝐿₀=Δ, taking 𝑟=0, we recover Theorem 4.4).

Theorem 4.5. Let 𝑥∶𝑀→ℝ^𝑛+1 be an orientable closed connected hypersurface isometrically immersed into the Euclidean space. Assume that 𝐿_𝑟 is elliptic, for some 1≤𝑟≤𝑛−1. Then, one has _𝑀𝐻²_𝑟+1𝜆𝑑𝑀≥^𝐿𝑟𝑝𝑐_𝑟𝑀𝐻_𝑟𝑑𝑀, (4.2) and equality holds if and only if 𝑀 is a round sphere in ℝ^𝑛+1. In particular, if 𝑀 is embedded in ℝ^𝑛+1, then _𝑀𝐻²_𝑟+1𝑑𝑀≥(𝑛+1)²𝜆^𝐿𝑟₁²𝑐_𝑟²vol(Ω)²(𝑀),vol (4.3) where equality holds if and only if 𝑀 is a round sphere in ℝ^𝑛+1. Here, vol(𝑀) denotes the 𝑛-dimensional volume of 𝑀, and Ω is the compact domain in ℝ^𝑛+1 bounded by 𝑀, and vol(Ω) denotes its (𝑛+1)-dimensional volume of Ω.

Proof. We will follow the techniques introduced by Chen (Theorem 4.4) in our context, we generalize some properties of Δ and 𝐻, respectively, to 𝐿_𝑟 and 𝐻_𝑟+1. Since 𝐿_𝑟𝑥=𝑐_𝑟𝐻_𝑟+1𝑁 (see [10]), and 𝑥=𝑥₀+^𝑞_𝑡=𝑝𝑥_𝑡,𝐿_𝑟𝑥_𝑡+𝜆^𝐿𝑟_𝑡𝑥_𝑡=0, (4.4) by using the inner product on the set of all ℝ^𝑛+1-valued functions on 𝑀, defined by (3.16), we have 𝑐_𝑟²_𝑀𝐻²_𝑟+1𝑑𝑀=𝑐_𝑟²𝐻_𝑟+1𝑁,𝐻_𝑟+1𝑁=𝐿_𝑟𝑥,𝐿_𝑟𝑥=^𝑞_𝑡=𝑝𝜆^𝐿𝑟_𝑡²𝑥_𝑡². (4.5) Since, by Lemma 3.1, 𝑥₀ is the center of mass of 𝑀, we have the well-known Minkowski formula [13] as follows: 𝐻_𝑟+𝐻_𝑟+1𝑥−𝑥₀,𝑁𝑑𝑀=0,𝑟=0,…,𝑛−1, (4.6) so we get that 𝑐_𝑟𝑀𝐻_𝑟𝑑𝑀=−𝑐_𝑟𝑥−𝑥₀,𝐻_𝑟+1𝑁=−𝑥−𝑥₀,𝐿_𝑟𝑥=^𝑞_𝑡=𝑝𝜆^𝐿𝑟_𝑡𝑥_𝑡². (4.7) Thus, by (4.5) and (4.7), we find that 𝑐_𝑟²_𝑀𝐻²_𝑟+1𝑑𝑀−𝑐_𝑟𝜆^𝐿𝑟_𝑝𝑀𝐻_𝑟𝑑𝑀=^𝑞_𝑡=𝑝+1𝜆^𝐿𝑟_𝑡𝜆^𝐿𝑟_𝑡−𝜆^𝐿𝑟_𝑝𝑥_𝑡²≥0. (4.8) Therefore, we obtain (4.2). Moreover, equality holds if and only if 𝐿_𝑟𝑥+𝜆^𝐿𝑟_𝑝𝑥=0, (4.9) Which, by Theorem 3.3, means that 𝑀 is a round sphere. If 𝑀 is embedded in ℝ^𝑛+1, by formula (15) of [11], we have 𝜆^𝐿𝑟₁≤𝑐_𝑟(𝑛+1)²vol(𝑀)vol(Ω)²_𝑀𝐻_𝑟𝑑𝑀, (4.10) where vol(𝑀) denotes the 𝑛-dimensional volume of 𝑀, and Ω is the compact domain in ℝ^𝑛+1 bounded by 𝑀, and vol(Ω) denotes its (𝑛+1)-dimensional volume of Ω. So (4.3) is obtained by (4.2) and (4.10) easily.

By applying Theorem 4.2, Corollary 4.3, and Theorem 4.5 to positively Ricci curved hypersurfaces in ℝ^𝑛+1, we have the following Corollary.

Corollary 4.6. Let 𝑥∶𝑀→ℝ^𝑛+1 be a closed connected hypersurface of the Euclidean space with positive Ricci curvature, and let Ω be the convex body in ℝ^𝑛+1 bounded by 𝑀. Then for every 𝑟=1,…,𝑛−1, it follows that _𝑀𝐻²_𝑟+1𝜆𝑑𝑀≥^𝐿𝑟𝑝𝑐_𝑟𝑀𝐻_𝑟𝑑𝑀,_𝑀𝐻²_𝑟+1𝜆𝑑𝑀≥^𝐿𝑟12𝑐_𝑟²(𝑛+1)²vol(Ω)²(𝑀),vol (4.11) where equalities hold if and only if 𝑀 is a round sphere in ℝ^𝑛+1. Here, vol(𝑀) denotes the 𝑛-dimensional volume of 𝑀, and vol(Ω) denotes its (𝑛+1)-dimensional volume of Ω.

By using the concept of order Chen in [9], we obtained the following best possible upper bound of total mean curvature for closed Euclidean hypersurface.

Theorem 4.7 (see [9]). Let 𝑥∶𝑀→ℝ^𝑛+1 be an orientable closed connected isometrically immersed hypersurface into the Euclidean space. Then, one has _𝑀𝐻²𝜆𝑑𝑀≤^𝑞𝑛vol(𝑀), (4.12) where 𝑞 is the upper order of 𝑀, and vol(𝑀) denotes the 𝑛-dimensional volume of 𝑀. Equality holds if and only if 𝑀 is a round sphere in ℝ^𝑛+1.

Now we establish the corresponding result for the operator 𝐿_𝑟 (since 𝐿₀=Δ, taking 𝑟=0, we recover Theorem 4.7).

Theorem 4.8. Let 𝑥∶𝑀→ℝ^𝑛+1 be an orientable closed connected isometrically immersed hypersurface of the Euclidean space. Assume that 𝐿_𝑟 is elliptic on 𝑀, for some 1≤𝑟≤𝑛−1. Then, one has _𝑀𝐻²_𝑟+1𝜆𝑑𝑀≤^𝐿𝑟𝑞𝑐_𝑟𝑀𝐻_𝑟𝑑𝑀, (4.13) and equality holds if and only if 𝑀 is a round sphere in ℝ^𝑛+1.

Proof. We will follow the techniques introduced by Chen (Theorem 4.7) in our context, we generalize some properties of Δ and 𝐻, respectively, to 𝐿_𝑟 and 𝐻_𝑟+1. Let 𝑥∶𝑀^𝑛→ℝ^𝑛+1 be an orientable closed connected hypersurface in the Euclidean space with Gauss map 𝑁. From [10], we have 𝐿_𝑟𝐻_𝑟+1𝑁𝑛=−𝐻𝑟+1^𝑟+1∇𝐻_𝑟+1−2𝑆∘𝑃_𝑟∇𝐻_𝑟+1+𝐿_𝑟𝐻_𝑟+1𝑁−𝐻_𝑟+1𝑡𝑟𝑆²∘𝑃_𝑟𝑁. (4.14) Formula (4.14) implies that 𝐿_𝑟𝐻_𝑟+1𝑁,𝐻_𝑟+1𝑁=𝐻_𝑟+1𝐿_𝑟𝐻_𝑟+1−𝐻²_𝑟+1𝑡𝑟𝑆²∘𝑃_𝑟. (4.15) Furthermore, from (4.4), (4.5), and (4.7), we have 𝑐²_𝑟𝑀𝐻²_𝑟+1𝑑𝑀=^𝑞_𝑡=𝑝𝜆^𝐿𝑟_𝑡²𝑥_𝑡²,𝑐²_𝑟𝑀𝐿_𝑟𝐻_𝑟+1𝑁,𝐻_𝑟+1𝑁𝑑𝑀=−^𝑞_𝑡=𝑝𝜆^𝐿𝑟_𝑡³𝑥_𝑡²,c_𝑟𝑀𝐻_𝑟𝑑𝑀=^𝑞_𝑡=𝑝𝜆^𝐿𝑟_𝑡𝑥_𝑡². (4.16) Assume that 𝑞≤∞. We put Λ=−𝑐²_𝑟𝑀𝐿_𝑟𝐻_𝑟+1𝑁,𝐻_𝑟+1𝑁𝑑𝑀−𝑐²_𝑟𝜆^𝐿𝑟_𝑝+𝜆^𝐿𝑟_𝑞𝑀𝐻²_𝑟+1𝑑𝑀+𝑐_𝑟𝜆^𝐿𝑟_𝑝𝜆^𝐿𝑟_𝑞𝑀𝐻_𝑟𝑑𝑀. (4.17) Then we have Λ=^𝑞−1_𝑡=𝑝+1𝜆^𝐿𝑟_𝑡−𝜆^𝐿𝑟_𝑝𝜆^𝐿𝑟_𝑡−𝜆^𝐿𝑟_𝑞𝑥_𝑡²≤0, (4.18) where equality holds if and only if 𝑀 is either of 𝐿_𝑟-1 type or of 𝐿_𝑟-2 type.
Combining (4.15), (4.17), and (4.18), we find that −𝑐²_𝑟𝑀𝐻_𝑟+1𝐿_𝑟𝐻_𝑟+1𝑑𝑀+𝑐²_𝑟𝑀𝐻²_𝑟+1𝑡𝑟𝑆²∘𝑃_𝑟𝑑𝑀−𝑐²_𝑟𝜆^𝐿𝑟_𝑝+𝜆^𝐿𝑟_𝑞𝑀𝐻²_𝑟+1𝑑𝑀+𝑐_𝑟𝜆^𝐿𝑟_𝑝𝜆^𝐿𝑟_𝑞𝑀𝐻_𝑟𝑑𝑀≤0. (4.19) By Proposition 3.1 of [14], we have the following equation: −_𝑀𝐻_𝑟+1𝐿_𝑟𝐻_𝑟+1𝑑𝑀=_𝑀𝑃_𝑟∇𝐻_𝑟+1,∇𝐻_𝑟+1𝑑𝑀. (4.20) Since 𝐿_𝑟 is elliptic, it follows from (2.16) that the Newton transformation 𝑃_𝑟 is positive definite, so (4.20) implies that −∫_𝑀𝐻_𝑟+1𝐿_𝑟𝐻_𝑟+1𝑑𝑀≥0. On the other hand, by using (2.12), we have 𝐻²_𝑟+1𝑡𝑟𝑆²∘𝑃_𝑟=𝐻⁴_𝑟+1𝐻_𝑟+𝑛(^𝑛_𝑟+1)𝐻₁𝐻³_𝑟+1𝐻_𝑟−(𝑛−𝑟−1)(^𝑛_𝑟+1)𝐻_𝑟+2𝐻²_𝑟+1𝐻_𝑟−𝐻⁴_𝑟+1𝐻_𝑟. (4.21) We suppose that 𝐵={𝑛(^𝑛_𝑟+1)𝐻₁𝐻³_𝑟+1𝐻_𝑟−(𝑛−𝑟−1)𝑛(^𝑛_𝑟+1)𝐻_𝑟+2𝐻²_𝑟+1𝐻_𝑟−𝐻⁴_𝑟+1}/𝐻_𝑟, and we show that 𝐵 is positive.
For every 1≤𝑗≤𝑛, one has the following inequalities (see, for instance, [15, Theorems 51 and 144]): 𝐻_𝑗−1𝐻_𝑗+1≤𝐻²_𝑗. (4.22) Since each 𝐻_𝑗>0 for 𝑗=1,…,𝑟, this is equivalent to 𝐻_𝑟+1𝐻_𝑟≤𝐻_𝑟𝐻_𝑟−1𝐻≤⋯²𝐻₁≤𝐻₁. (4.23) And these inequalities imply that 𝐻₁𝐻_𝑟≥𝐻_𝑟+1. (4.24) So by using (4.22) and (4.24), we get that 𝐵 is positive.
Combining (4.19), (4.20), and (4.21) and Schwartz's inequality, we get that 0≥𝑐²_𝑟𝑀𝑃_𝑟∇𝐻_𝑟+1,∇𝐻_𝑟+1𝑑𝑀+𝑐³_𝑟𝑀𝐻⁴_𝑟+1𝐻_𝑟𝑑𝑀+𝑐³_𝑟𝑀𝐵𝑑𝑀−𝑐²_𝑟𝜆^𝐿𝑟_𝑝+𝜆^𝐿𝑟_𝑞𝑀𝐻²_𝑟+1𝑑𝑀+𝑐_𝑟𝜆^𝐿𝑟_𝑝𝜆^𝐿𝑟_𝑞𝑀𝐻_𝑟𝑑𝑀≥𝑐²_𝑟𝑀𝑃_𝑟∇𝐻_𝑟+1,∇𝐻_𝑟+1𝑐𝑑𝑀+^3𝑟∫_𝑀𝐻²_𝑟+1𝑑𝑀²∫_𝑀𝐻_𝑟𝑑𝑀+𝑐³_𝑟𝑀𝐵𝑑𝑀−𝑐²_𝑟𝜆^𝐿𝑟_𝑝+𝜆^𝐿𝑟_𝑞𝑀𝐻²_𝑟+1𝑑𝑀+𝑐_𝑟𝜆^𝐿𝑟_𝑝𝜆^𝐿𝑟_𝑞𝑀𝐻_𝑟𝑑𝑀. (4.25) Hence, we obtain that 0≥𝑐_𝑟𝑀𝐻_𝑟𝑑𝑀_𝑀𝑃_𝑟∇𝐻_𝑟+1,∇𝐻_𝑟+1𝑑𝑀+_𝑀𝐻_𝑟𝑑𝑀_𝑀+𝑐𝐵𝑑𝑀_𝑟𝑀𝐻²_𝑟+1𝑑M−𝜆^𝐿𝑟_𝑝𝑀𝐻_𝑟𝑑𝑀𝑐_𝑟𝑀𝐻²_𝑟+1𝑑𝑀−𝜆^𝐿𝑟_𝑞𝑀𝐻_𝑟𝑑𝑀 (4.26) By inequalities (4.2) and (4.26), we obtain (4.13). If in (4.13) the equality holds, then all the inequalities in (4.17) through (4.26) have to be equalities. Thus, we find that 𝑀 is either of 𝐿_𝑟-1 type or of 𝐿_𝑟-2 type, and 𝐻_𝑟+1 is constant. So, by Theorems 3.3 and 3.4, 𝑀 is a round sphere.

By applying Theorem 4.2, Corollary 4.3, and Theorem 4.8 to positively Ricci curved hypersurfaces in ℝ^𝑛+1, we have the following Corollary.

Corollary 4.9. Let 𝑥∶𝑀→ℝ^𝑛+1 be a closed connected hypersurface of the Euclidean space with positive Ricci curvature. Then for every 𝑟=1,…,𝑛−1 the following inequality holds: _𝑀𝐻²_𝑟+1𝜆𝑑𝑀≤^𝐿𝑟𝑞𝑐_𝑟𝑀𝐻_𝑟𝑑𝑀, (4.27) where equality holds if and only if 𝑀 is a round sphere in ℝ^𝑛+1.

An immediate consequence of Corollaries 4.6 and 4.9 is the following.

Corollary 4.10. Let 𝑥∶𝑀→ℝ^𝑛+1 be a closed connected hypersurface of the Euclidean space ℝ^𝑛+1 with positive Ricci curvature. If 𝐻_𝑟+1 is constant, then 𝜆^𝐿𝑟_𝑝≤𝑐_𝑟𝐻²_𝑟+1𝐻_𝑟≤𝜆^𝐿𝑟_𝑞, (4.28) and equality holds if and only if 𝑀 is a round sphere in ℝ^𝑛+1.