#### Abstract

By using the operator , we define the notions of *r*th order and *r*th type of a Euclidean hypersurface. By the use of these notions, we are able to obtain some sharp estimates of the ()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 โdegreeโ 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 Nashโs 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 immersed into 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 can be seen as the first one of a sequence of operators , where stands for the linearized operator of the first variation of the th mean curvature arising from normal variations of the hypersurface (see [5]). These operators are given by for any , where denotes the th Newton transformation associated to the second fundamental form of the hypersurface, and is the hessian of (see the next section for details).

In contrast to the operator , the operators are not elliptic in general, but they still share some nice properties with Laplacian of ; 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 by the operator . Having this idea, for the first time in [7], the second author, inspired by a private communication with Alรญas, introduced such hypersurfaces and called them โ-finite typeโ hypersurfaces.

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

#### 2. Preliminaries

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

Consider an orientable isometrically immersed hypersurface in the Euclidean space, with the Gauss map . We denote by and the Levi-Civita connections on and , respectively. Then, the basic Gauss and Weingarten formulae of the hypersurface are written as for all tangent vector fields , where is the shape operator of with respect to the Gauss map . As is well known, defines a self-adjoint linear operator on tangent space , and its eigenvalues are called the principal curvatures of the hypersurface. Associated to the shape operator , there are algebraic invariants given by where is the elementary symmetric function in given by Observe that the characteristic polynomial of can be written in terms of the , as where by definition. The th mean curvature of the hypersurface is then defined by In particular, when , is nothing but the mean curvature of , which is the main extrinsic curvature of the hypersurface. On the other hand, is called the Gauss-Kronecker curvature of . A hypersurface with zero th mean curvature in is called -minimal.

The classical Newton transformations are defined inductively by for where denotes the identity of . Equivalently, we have Note that, by the Cayley-Hamilton Theorem stating that any operator is annihilated by its characteristic polynomial, we have from (2.4).

Each is also a self-adjoint linear operator on the tangent space which commutes with . Indeed, and can be simultaneously diagonalized if are the eigenvectors of corresponding to the eigenvalues , respectively. Then they are also the eigenvectors of with corresponding eigenvalues given by for every , . We have the following formulas for the Newton transformation [5]: where

Associated to each Newton transformation , we consider the second-order linear differential operator given by denotes the self-adjoint linear operator metrically equivalent to the Hessian of and is given by Let be a local orthonormal frame on and observe that where div denotes the divergence operator on .

Since (see [10]), as a consequence, from (2.16), one gets that

#### 3. The th Order and the th 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 and associated with a submanifold (see [1] for the precise definition). is a natural number and or . The pair is called the order of the submanifold ; more precisely, is the lower order, and 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 in the Euclidean space, with the Gauss map , and assume that, for a fixed , , is an elliptic differential operator on , the ring of all smooth real functions on .

It is well known that the eigenvalues of when it is elliptic (see [1], chapter 3, for properties of an elliptic operator) form a discrete infinite sequence Let be the eigenspace of with eigenvalue . Then is finite dimensional. Define an inner product on by where is the volume element of . Then is dense in (in sense). If we denote by the completion of , we have

For each function , let denote the projection of onto the subspace . Then we have the following spectral decomposition: Because is 1-dimensional, for any nonconstant function , there is a positive integer such that and where is a constant. If there are infinitely many 's which are nonzero, we put ; otherwise, there is an integer , such that and

Consider the set The smallest element of is called the lower th order of and is denoted by , and the supremum of is called the upper th order of and is denoted by . A function in is said to be of -finite type if is a finite set, that is, if its spectral decomposition contains only finitely many nonzero terms. Otherwise, is said to be of -infinite type. is said to be of - type if contains exactly elements.

For an isometrically immersed closed hypersurface in the Euclidean space , we put where is the -th component of . For each , we have For the isometric immersion , we put It is easy to see that and are independent of the choice of the Euclidean coordinate system on , and is a positive integer and is either or . Thus, and are well defined. Consequently, for each closed hypersurface in (or, more precisely, for each isometric immersion ), we have a pair associated with . We call the pair the order of the hypersurface .

By using the above notation, we have the following spectral decomposition of in vector form: We define by The immersion or the hypersurface is said to be of - type if contains exactly elements. Similarly, we can define the lower th order and the upper th order of the immersion.

The immersion is said to be of finite type if its upper th order is finite, and it is said to be of infinite type if its upper order is .

The following Lemma states that for an isometrically immersed closed orientable hypersurface , the constant vector in (3.11) is exactly the โcenter of massโ of in (i.e., , where is a chosen volume form of ).

Lemma 3.1. *Let be an isometric immersion of a closed orientable hypersurface into . Assume that is elliptic, for some . Then in (3.11) is the center of mass of in .*

*Proof. *Consider the decomposition
We have . If , then (2.17) and the Divergence Theorem imply that
Since is a constant vector in , we obtain from (3.13) and (3.14) that

This shows that is the center of mass of .

On the set of all -valued functions on which is a real vector space, we define an inner product on such space by for any two -valued functions on , where denotes the Euclidean inner product of for any . Then we have the following lemma.

Lemma 3.2. *For an isometric immersion of a closed orientable hypersurface into the components of the spectral decomposition (3.11) are mutually orthogonal, for example,
*

*Proof. *Since is self-adjoint with respect to the inner product (3.16), we have
Since , 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 -finite-type Euclidean hypersurfaces.

Theorem 3.3 (see [11]). *Let be an orientable connected hypersurface immersed into the Euclidean space, and let be the linearized operator of the -th mean curvature of , for some . Then, one has
**
for a real constant if and only if either and M is -minimal in (i.e., on ), or and is an open piece of a round sphere of radius centered at the origin of , where .*

Theorem 3.4. *There is no compact Euclidean hypersurface of - type with constant , when is elliptic.*

*Proof. *If is of - type, by using (3.11), the position vector field of in has the following spectral decomposition:
so
From [10], we also have

The formula (3.23) holds since is a nonzero constant, see [10]. Therefore, by using (3.21), (3.22), and (3.23), we obtain that

Since , from (3.24), we have that is normal to at every point of . So is a positive constant. In this case, is an open piece of centered at , by Theorem 3.3, is of - type, which is not.

#### 4. The th Order and the th Mean Curvature

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

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

Theorem 4.1 (Hadamard Theorem, see [12]). *Let 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 .*(ii)* is orientable, and its Gauss map is a diffeomorphism onto .*(iii)*The Gauss-Kronecker curvature never vanishes on .**Moreover, any of the above conditions implies that is a convex hypersurface.*

Here we observe that the convexity of a hypersurface in is closely related to its Ricci curvature.

Theorem 4.2 (see [11]). *Let be a closed connected hypersurface immersed into the Euclidean space. The following assertion is equivalent to any of the assertions in Hadamard theorem, and therefore it also implies that is a convex hypersurface*(iv)* The Ricci curvature of is positive everywhere on .*

Corollary 4.3 (see [11]). *Let be a closed connected hypersurface isometrically immersed into the Euclidean space. If has positive Ricci curvature, then each operator on is elliptic, and each -th mean curvatures of 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 be an orientable closed connected hypersurface isometrically immersed into the Euclidean space. Then, one has
**
where is the lower order of , and denotes the -dimensional volume of . Equality holds if and only if is a round sphere in .*

Now we establish the corresponding result for the operator (since , taking , we recover Theorem 4.4).

Theorem 4.5. *Let be an orientable closed connected hypersurface isometrically immersed into the Euclidean space. Assume that is elliptic, for some . Then, one has
**
and equality holds if and only if is a round sphere in . In particular, if is embedded in , then
**
where equality holds if and only if is a round sphere in . Here, denotes the -dimensional volume of , and is the compact domain in bounded by , and denotes its -dimensional volume of .*

*Proof. *We will follow the techniques introduced by Chen (Theorem 4.4) in our context, we generalize some properties of and , respectively, to and . Since (see [10]), and
by using the inner product on the set of all -valued functions on , defined by (3.16), we have
Since, by Lemma 3.1, is the center of mass of , we have the well-known Minkowski formula [13] as follows:
so we get that
Thus, by (4.5) and (4.7), we find that
Therefore, we obtain (4.2). Moreover, equality holds if and only if
Which, by Theorem 3.3, means that is a round sphere. If is embedded in , by formula (15) of [11], we have
where denotes the -dimensional volume of , and is the compact domain in bounded by , and denotes its -dimensional volume of . 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 , we have the following Corollary.

Corollary 4.6. *Let be a closed connected hypersurface of the Euclidean space with positive Ricci curvature, and let be the convex body in bounded by . Then for every , it follows that
**
where equalities hold if and only if is a round sphere in . Here, denotes the -dimensional volume of , and denotes its -dimensional volume of .*

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 be an orientable closed connected isometrically immersed hypersurface into the Euclidean space. Then, one has
**
where is the upper order of , and denotes the -dimensional volume of . Equality holds if and only if is a round sphere in .*

Now we establish the corresponding result for the operator (since , taking , we recover Theorem 4.7).

Theorem 4.8. *Let be an orientable closed connected isometrically immersed hypersurface of the Euclidean space. Assume that is elliptic on , for some . Then, one has
**
and equality holds if and only if is a round sphere in .*

*Proof. *We will follow the techniques introduced by Chen (Theorem 4.7) in our context, we generalize some properties of and , respectively, to and . Let be an orientable closed connected hypersurface in the Euclidean space with Gauss map . From [10], we have
Formula (4.14) implies that
Furthermore, from (4.4), (4.5), and (4.7), we have
Assume that . We put
Then we have
where equality holds if and only if is either of - type or of - type.

Combining (4.15), (4.17), and (4.18), we find that
By Proposition 3.1 of [14], we have the following equation:
Since is elliptic, it follows from (2.16) that the Newton transformation is positive definite, so (4.20) implies that . On the other hand, by using (2.12), we have
We suppose that , and we show that is positive.

For every , one has the following inequalities (see, for instance, [15, Theorems 51 and 144]):
Since each for , this is equivalent to
And these inequalities imply that
So by using (4.22) and (4.24), we get that is positive.

Combining (4.19), (4.20), and (4.21) and Schwartz's inequality, we get that
Hence, we obtain that
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 is either of - type or of - type, and is constant. So, by Theorems 3.3 and 3.4, is a round sphere.

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

Corollary 4.9. *Let be a closed connected hypersurface of the Euclidean space with positive Ricci curvature. Then for every the following inequality holds:
**
where equality holds if and only if is a round sphere in .*

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

Corollary 4.10. *Let be a closed connected hypersurface of the Euclidean space with positive Ricci curvature. If is constant, then
**
and equality holds if and only if is a round sphere in .*