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

Authors: Akram Mohammadpouri; S. M. B. Kashani
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

By using the operatorLr, 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.


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## Body

## 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 hypersurfaceM immersed into ℝn+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,…,Ln-1, where Lr 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 Lr(f)=tr(Pr∘∇2f) for any f∈C∞(M), where Pr 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 operatorΔ, the operators Lr 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 Lr. 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 “Lr-finite type” hypersurfaces.In this paper, by using the operatorLr, 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 Lr is elliptic. The paper generalizes the results of [8, 9].


## 2. Preliminaries

In this section, we recall some prerequisites about Newton transformationsPr and their associated second-order differential operators Lr from [10].Consider an orientable isometrically immersed hypersurfacex:Mn→ℝn+1 in the Euclidean space, with the Gauss map N. We denote by ∇0 and ∇ the Levi-Civita connections on ℝn+1 and M, respectively. Then, the basic Gauss and Weingarten formulae of the hypersurface are written as
(2.1)∇X0Y=∇XY+〈SX,Y〉N,SX=-∇X0N
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 TpM, and its eigenvalues κ1(p),…,κn(p) are called the principal curvatures of the hypersurface. Associated to the shape operator S, there are n algebraic invariants given by
(2.2)sr(p)=σr(κ1(p),…,κn(p)),1≤r≤n,
where σr:ℝn→ℝ is the elementary symmetric function in ℝn given by
(2.3)σr(x1,…,xn)=∑1≤i1<⋯<ir≤n‍xi1⋯xir.
Observe that the characteristic polynomial of S can be written in terms of the sr, s as
(2.4)QS(t)=det⁡(tI-S)=∑r=0n‍(-1)rsrtn-r,
where s0=1 by definition. The rth mean curvature Hr of the hypersurface is then defined by
(2.5)(nk)Hr=sr,0≤r≤n.
In particular, when r=1,
(2.6)H1=1n∑i=1n‍κi=1ntr(S)=H
is nothing but the mean curvature of M, which is the main extrinsic curvature of the hypersurface. On the other hand, Hn=κ1⋯κn is called the Gauss-Kronecker curvature of M. A hypersurface with zero (r+1)th mean curvature in ℝn+1 is called r-minimal.The classical Newton transformationsPr:χ(M)→χ(M) are defined inductively by
(2.7)P0=I,Pr=srI-S∘Pr-1=(nr)HrI-S∘Pr-1
for r=1,…,n where I denotes the identity of χ(M). Equivalently, we have
(2.8)Pr=∑j=0r‍(-1)jsr-jSj=∑j=0r‍(-1)j(nr-j)Hr-jSj.
Note that, by the Cayley-Hamilton Theorem stating that any operator T is annihilated by its characteristic polynomial, we have Pn=0 from (2.4).EachPr(p) is also a self-adjoint linear operator on the tangent space TpM which commutes with S(p). Indeed, S(p) and Pr(p) can be simultaneously diagonalized if {e1,…,en} are the eigenvectors of S(p) corresponding to the eigenvalues κ1(p),…,κn(p), respectively. Then they are also the eigenvectors of Pr(p) with corresponding eigenvalues given by
(2.9)μi,r(p)=∑1≤i1<⋯<ir≤n,ij≠i‍κi1(p)⋯κir(p),
for every r, 1≤i≤n. We have the following formulas for the Newton transformation Pr [5]:
(2.10)tr(Pr)=crHr,(2.11)tr(S∘Pr)=crHr+1,(2.12)tr(S2∘Pr)=(nr+1)(nH1Hr+1-(n-r-1)Hr+2),
where
(2.13)cr=(n-r)(nr)=(r+1)(nr+1).Associated to each Newton transformationPr, we consider the second-order linear differential operator Lr:C∞(M)→C∞(M) given by
(2.14)Lr(f)=tr(Pr∘∇2f).∇2f:χ(M)→χ(M) denotes the self-adjoint linear operator metrically equivalent to the Hessian of f and is given by
(2.15)〈∇2f(X),Y〉=〈∇X(∇f),Y〉,∀X,Y∈χ(M).
Let {e1,…,en} be a local orthonormal frame on M and observe that
(2.16)div⁡(Pr(∇f))=∑i=1n‍〈(∇eiPr)(∇f),ei〉+∑i=1n‍〈Pr(∇ei∇f),ei〉=〈div⁡Pr,∇f〉+Lr(f),
where div denotes the divergence operator on M.SincedivPr=0 (see [10]), as a consequence, from (2.16), one gets that
(2.17)Lr(f)=div⁡(Pr(∇f)).


## 3. Therth 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 numbersp 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 hypersurfacex:Mn→ℝn+1 in the Euclidean space, with the Gauss map N, and assume that, for a fixed r, 1≤r≤n-1, Lr 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-Lr when it is elliptic (see [1], chapter 3, for properties of an elliptic operator) form a discrete infinite sequence
(3.1)0=λ0Lr<λ1Lr<λ2Lr<⋯↗∞.
Let VkLr={f∈𝒞∞(M):Lrf+λkLrf=0} be the eigenspace of -Lr with eigenvalue λkLr. Then VkLr is finite dimensional. Define an inner product (,) on 𝒞∞(M) by
(3.2)(f,h)=∫M‍fhdM,
where dM is the volume element of M. Then ∑k=0∞‍VkLr is dense in 𝒞∞(M) (in L2- sense). If we denote by ⊕^VkLr the completion of ∑‍VkLr, we have
(3.3)C∞(M)=⊕^kVkLr.For each functionf∈𝒞∞(M), let ft denote the projection of f onto the subspace VtLr(t=0,1,2,…). Then we have the following spectral decomposition:
(3.4)f=∑t=0∞‍ft(inL2-sence).
Because V0Lr is 1-dimensional, for any nonconstant function f∈𝒞∞(M), there is a positive integer p≥1 such that fp≠0 and
(3.5)f-f0=∑t≥p‍ft,
where f0∈V0Lr is a constant. If there are infinitely many ft's which are nonzero, we put q=+∞; otherwise, there is an integer q,q≥p, such that fq≠0 and
(3.6)f-f0=∑t=pq‍ft.Consider the set(3.7)TfLr={t∈N:ft≢0}.
The smallest element of TfLr is called the lower rth order of f and is denoted by l.o.Lr(f), and the supremum of TfLr 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 Lr-finite type if TfLr is a finite set, that is, if its spectral decomposition contains only finitely many nonzero terms. Otherwise, f is said to be of Lr-infinite type. f is said to be of Lr-k type if TfLr contains exactly k elements.For an isometrically immersed closed hypersurfacex:Mn→ℝn+1 in the Euclidean space ℝn+1, we put
(3.8)x=(x1,…,xn+1),
where xA is the A-th component of x. For each xA, we have
(3.9)xA-(xA)0=∑t=pAqA‍(xA)t,A=1,…,n+1.
For the isometric immersion x:M→ℝn+1, we put
(3.10)p=inf⁡A{pA},q=sup⁡A{qA}.
It is easy to see that p and q are independent of the choice of the Euclidean coordinate system on ℝn+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 ℝn+1 (or, more precisely, for each isometric immersion x:Mn→ℝn+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 ofx in vector form:
(3.11)x=x0+∑t=pq‍xt.
We define TLr(x) by
(3.12)TxLr={t∈N:xt≢0}.
The immersion x or the hypersurface M is said to be of Lr-k type if TxLr contains exactly k elements. Similarly, we can define the lower rth order and the upper rth order of the immersion.The immersionx is said to be of Lr finite type if its upper rth order q is finite, and it is said to be of Lr infinite type if its upper order is +∞.The following Lemma states that for an isometrically immersed closed orientable hypersurfacex:Mn→ℝn+1, the constant vector x0 in (3.11) is exactly the “center of mass” of M in ℝn+1 (i.e., ∫M‍xdM/vol(M), where dM is a chosen volume form of M).Lemma 3.1.
Letx:Mn→ℝn+1 be an isometric immersion of a closed orientable hypersurface M into ℝn+1. Assume that Lr is elliptic, for some 1≤r≤n-1. Then x0 in (3.11) is the center of mass of M in ℝn+1.Proof.
Consider the decomposition(3.13)x=∑t=0∞‍xt.
We have Lrxt+λtLrxt=0. If t≠0, then (2.17) and the Divergence Theorem imply that
(3.14)∫M‍xtdM=-1λtLrLrxtdM=0.
Since x0 is a constant vector in ℝn+1, we obtain from (3.13) and (3.14) that
(3.15)x0=∫M‍xdMvol(M).
This shows thatx0 is the center of mass of M.On the set of allℝn+1-valued functions on M which is a real vector space, we define an inner product on such space by
(3.16)(v,w)=∫M‍〈v,w〉dM
for any two ℝn+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 immersionx:M→ℝn+1 of a closed orientable hypersurface M into ℝn+1 the components of the spectral decomposition (3.11) are mutually orthogonal, for example,
(3.17)(xt,xs)=0,t≠s.Proof.
SinceLr is self-adjoint with respect to the inner product (3.16), we have
(3.18)λtLr(xt,xs)=-(Lrxt,xs)=-(xt,Lrxs)=λsLr(xt,xs).
Since λtLr≠λsLr, we obtain (3.17).Before we give our main result and to facilitate the reader, we quote Theorem3.3 from [11] and present Theorem 3.4 about Lr-finite-type Euclidean hypersurfaces.Theorem 3.3 (see [11]).
Letx:Mn→ℝn+1 be an orientable connected hypersurface immersed into the Euclidean space, and let Lr be the linearized operator of the (r+1)-th mean curvature of M, for some r=0,…,n-1. Then, one has
(3.19)Lrx+λx=0
for a real constant λ if and only if either λ=0 and M is r-minimal in ℝn+1 (i.e., Hr+1=0 on M), or λ≠0 and M is an open piece of a round sphere Sn(ϱ)⊂ℝn+1 of radius ϱ=(cr/∥λ|)(1/(r+2)) centered at the origin of ℝn+1, where cr=(n-r)(nr).Theorem 3.4.
There is no compact Euclidean hypersurface ofLr-2 type with constant Hr+1, when Lr is elliptic.Proof.
IfM is of Lr-2 type, by using (3.11), the position vector field x of M in ℝn+1 has the following spectral decomposition:
(3.20)x-x0=xp+xq,Lrxp+λpxp=0,Lrxq+λqxq=0,forsomex0∈Rn+1,p,q∈N,
so
(3.21)Lr2x=-(λp+λq)Lrx-λpλq(x-x0).
From [10], we also have
(3.22)Lrx=crHr+1N,(3.23)Lr2x=-cr(nr+1)Hr+1(nH1Hr+1-(n-r-1)Hr+2)N.
The formula (3.23) holds since Hr+1 is a nonzero constant, see [10]. Therefore, by using (3.21), (3.22), and (3.23), we obtain that
(3.24)-cr(nr+1)Hr+1(nH1Hr+1-(n-r-1)Hr+2)N=cr(λp+λq)Hr+1N-λpλq(x-x0).
Sinceλpλq≠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 Sn centered at x0, by Theorem 3.3, M is of Lr-1 type, which is not.


## 4. Therth Order and the (r+1)th Mean Curvature

In this section, we will relate the notion of therth 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 Lr is elliptic. In the following we will state several results from [11, 12] which guarantee the ellipticity of Lr.A classical theorem of Hadamard [12] gives three equivalent conditions on a closed connected hypersurface Mn immersed into the Euclidean space ℝn+1 which imply that M is a convex hypersurface (i.e., M is embedded in ℝn+1 and is the boundary of a convex body).Theorem 4.1 (Hadamard Theorem, see [12]).
Letx:Mn→ℝn+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 ofM.(ii)
M is orientable, and its Gauss map is a diffeomorphism onto Sn.(iii)
The Gauss-Kronecker curvature never vanishes onM.
Moreover, any of the above conditions implies thatM is a convex hypersurface.Here we observe that the convexity of a hypersurface inℝn+1 is closely related to its Ricci curvature.Theorem 4.2 (see [11]).
Letx:Mn→ℝn+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 ofM is positive everywhere on M.Corollary 4.3 (see [11]).
Letx:Mn→ℝn+1 be a closed connected hypersurface isometrically immersed into the Euclidean space. If M has positive Ricci curvature, then each operator Lr 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]).
Letx:M→ℝn+1 be an orientable closed connected hypersurface isometrically immersed into the Euclidean space. Then, one has
(4.1)∫M‍H2dM≥λpnvol(M),
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 ℝn+1.Now we establish the corresponding result for the operatorLr (since L0=Δ, taking r=0, we recover Theorem 4.4).Theorem 4.5.
Letx:M→ℝn+1 be an orientable closed connected hypersurface isometrically immersed into the Euclidean space. Assume that Lr is elliptic, for some 1≤r≤n-1. Then, one has
(4.2)∫M‍Hr+12dM≥λpLrcr∫M‍HrdM,
and equality holds if and only if M is a round sphere in ℝn+1. In particular, if M is embedded in ℝn+1, then
(4.3)∫M‍Hr+12dM≥(n+1)2(λ1Lr)2(cr)2vol(Ω)2vol(M),
where equality holds if and only if M is a round sphere in ℝn+1. Here, vol(M) denotes the n-dimensional volume of M, and Ω is the compact domain in ℝn+1 bounded by M, and vol(Ω) denotes its (n+1)-dimensional volume of Ω.Proof.
We will follow the techniques introduced by Chen (Theorem4.4) in our context, we generalize some properties of Δ and H, respectively, to Lr and Hr+1. Since Lrx=crHr+1N (see [10]), and
(4.4)x=x0+∑t=pq‍xt,Lrxt+λtLrxt=0,
by using the inner product on the set of all ℝn+1-valued functions on M, defined by (3.16), we have
(4.5)(cr)2∫M‍Hr+12dM=(cr)2(Hr+1N,Hr+1N)=(Lrx,Lrx)=∑t=pq‍(λtLr)2‖xt‖2.
Since, by Lemma 3.1, x0 is the center of mass of M, we have the well-known Minkowski formula [13] as follows:
(4.6)∫‍(Hr+Hr+1〈x-x0,N〉)dM=0,r=0,…,n-1,
so we get that
(4.7)(cr)∫M‍HrdM=-(cr)(x-x0,Hr+1N)=-(x-x0,Lrx)=∑t=pq‍λtLr‖xt‖2.
Thus, by (4.5) and (4.7), we find that
(4.8)(cr)2∫M‍Hr+12dM-(cr)λpLr∫M‍HrdM=∑t=p+1q‍λtLr(λtLr-λpLr)‖xt‖2≥0.
Therefore, we obtain (4.2). Moreover, equality holds if and only if
(4.9)Lrx+λpLrx=0,
Which, by Theorem 3.3, means that M is a round sphere. If M is embedded in ℝn+1, by formula (15) of [11], we have
(4.10)λ1Lr≤cr(n+1)2vol(M)vol(Ω)2∫M‍HrdM,
where vol(M) denotes the n-dimensional volume of M, and Ω is the compact domain in ℝn+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 Theorem4.2, Corollary 4.3, and Theorem 4.5 to positively Ricci curved hypersurfaces in ℝn+1, we have the following Corollary.Corollary 4.6.
Letx:M→ℝn+1 be a closed connected hypersurface of the Euclidean space with positive Ricci curvature, and let Ω be the convex body in ℝn+1 bounded by M. Then for every r=1,…,n-1, it follows that
(4.11)∫M‍Hr+12dM≥λpLrcr∫M‍HrdM,∫M‍Hr+12dM≥(λ1Lr)2(cr)2(n+1)2vol(Ω)2vol(M),
where equalities hold if and only if M is a round sphere in ℝn+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]).
Letx:M→ℝn+1 be an orientable closed connected isometrically immersed hypersurface into the Euclidean space. Then, one has
(4.12)∫M‍H2dM≤λqnvol(M),
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 ℝn+1.Now we establish the corresponding result for the operatorLr (since L0=Δ, taking r=0, we recover Theorem 4.7).Theorem 4.8.
Letx:M→ℝn+1 be an orientable closed connected isometrically immersed hypersurface of the Euclidean space. Assume that Lr is elliptic on M, for some 1≤r≤n-1. Then, one has
(4.13)∫M‍Hr+12dM≤λqLrcr∫M‍HrdM,
and equality holds if and only if M is a round sphere in ℝn+1.Proof.
We will follow the techniques introduced by Chen (Theorem4.7) in our context, we generalize some properties of Δ and H, respectively, to Lr and Hr+1. Let x:Mn→ℝn+1 be an orientable closed connected hypersurface in the Euclidean space with Gauss map N. From [10], we have
(4.14)Lr(Hr+1N)=-(nr+1)Hr+1∇Hr+1-2(S∘Pr)(∇Hr+1)+((LrHr+1)N-Hr+1tr(S2∘Pr)N).
Formula (4.14) implies that
(4.15)〈Lr(Hr+1N),Hr+1N〉=Hr+1LrHr+1-Hr+12tr(S2∘Pr).
Furthermore, from (4.4), (4.5), and (4.7), we have
(4.16)cr2∫M‍Hr+12dM=∑t=pq‍(λtLr)2‖xt‖2,cr2∫M‍〈Lr(Hr+1N),Hr+1N〉dM=-∑t=pq‍(λtLr)3‖xt‖2,cr∫M‍HrdM=∑t=pq‍λtLr‖xt‖2.
Assume that q≤∞. We put
(4.17)Λ=-cr2∫M‍〈Lr(Hr+1N),Hr+1N〉dM-cr2(λpLr+λqLr)∫M‍Hr+12dM+crλpLrλqLr∫M‍HrdM.
Then we have
(4.18)Λ=∑t=p+1q-1‍(λtLr-λpLr)(λtLr-λqLr)‖xt‖2≤0,
where equality holds if and only if M is either of Lr-1 type or of Lr-2 type.
Combining (4.15), (4.17), and (4.18), we find that
(4.19)-cr2∫M‍Hr+1LrHr+1dM+cr2∫M‍Hr+12tr(S2∘Pr)dM-cr2(λpLr+λqLr)∫M‍Hr+12dM+crλpLrλqLr∫M‍HrdM≤0.
By Proposition 3.1 of [14], we have the following equation:
(4.20)-∫M‍Hr+1LrHr+1dM=∫M‍〈Pr∇Hr+1,∇Hr+1〉dM.
Since Lr is elliptic, it follows from (2.16) that the Newton transformation Pr is positive definite, so (4.20) implies that -∫M‍Hr+1LrHr+1dM≥0. On the other hand, by using (2.12), we have
(4.21)Hr+12tr(S2∘Pr)=Hr+14Hr+{n(nr+1)H1Hr+13Hr-(n-r-1)(nr+1)Hr+2Hr+12Hr-Hr+14}Hr.
We suppose that B={n(nr+1)H1Hr+13Hr-(n-r-1)n(nr+1)Hr+2Hr+12Hr-Hr+14}/Hr, and we show that B is positive.
For every1≤j≤n, one has the following inequalities (see, for instance, [15, Theorems 51 and 144]):
(4.22)Hj-1Hj+1≤Hj2.
Since each Hj>0 for j=1,…,r, this is equivalent to
(4.23)Hr+1Hr≤HrHr-1≤⋯H2H1≤H1.
And these inequalities imply that
(4.24)H1Hr≥Hr+1.
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
(4.25)0≥cr2∫M‍〈Pr∇Hr+1,∇Hr+1〉dM+cr3∫M‍Hr+14HrdM+cr3∫M‍BdM-cr2(λpLr+λqLr)∫M‍Hr+12dM+crλpLrλqLr∫M‍HrdM≥cr2∫M‍〈Pr∇Hr+1,∇Hr+1〉dM+cr3(∫M‍Hr+12dM)2∫M‍HrdM+cr3∫M‍BdM-cr2(λpLr+λqLr)∫M‍Hr+12dM+crλpLrλqLr∫M‍HrdM.
Hence, we obtain that
(4.26)0≥cr∫M‍HrdM∫M‍〈Pr∇Hr+1,∇Hr+1〉dM+∫M‍HrdM∫M‍BdM+(cr∫M‍Hr+12dM-λpLr∫M‍HrdM)(cr∫M‍Hr+12dM-λqLr∫M‍HrdM)
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 Lr-1 type or of Lr-2 type, and Hr+1 is constant. So, by Theorems 3.3 and 3.4, M is a round sphere.By applying Theorem4.2, Corollary 4.3, and Theorem 4.8 to positively Ricci curved hypersurfaces in ℝn+1, we have the following Corollary.Corollary 4.9.
Letx:M→ℝn+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:
(4.27)∫M‍Hr+12dM≤λqLrcr∫M‍HrdM,
where equality holds if and only if M is a round sphere in ℝn+1.An immediate consequence of Corollaries4.6 and 4.9 is the following.Corollary 4.10.
Letx:M→ℝn+1 be a closed connected hypersurface of the Euclidean space ℝn+1 with positive Ricci curvature. If Hr+1 is constant, then
(4.28)λpLr≤crHr+12Hr≤λqLr,
and equality holds if and only if M is a round sphere in ℝn+1.



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*Source: 784028-2012-08-09.xml*