Hypersurfaces with small extrinsic radius or large \lambda_{1} in Euclidean spaces

# Hypersurfaces with small extrinsic radius or large λ1 in Euclidean spaces

Erwann AUBRY, Jean-François GROSJEAN, Julien ROTH LJAD, Université de Nice Sophia-Antipolis, CNRS; 28 avenue Valrose, 06108 Nice, France Institut Élie Cartan (Mathématiques), Université Henri Poincaré Nancy I, B.P. 239, F-54506 Vandœuvre-les-Nancy cedex, France LAMA, Université Paris-Est - Marne-la-Vallée, 5 bd Descartes, Cité Descartes, Champs-sur-Marne, F-77454 Marne-la-Vallée
28th July 2019
###### Abstract.

We prove that hypersurfaces of which are almost extremal for the Reilly inequality on and have -bounded mean curvature () are Hausdorff close to a sphere, have almost constant mean curvature and have a spectrum which asymptotically contains the spectrum of the sphere. We prove the same result for the Hasanis-Koutroufiotis inequality on extrinsic radius. We also prove that when a supplementary bound on the second fundamental is assumed, the almost extremal manifolds are Lipschitz close to a sphere when , but not necessarily diffeomorphic to a sphere when .

###### Key words and phrases:
Mean curvature, Reilly inequality, Laplacian, Spectrum, pinching results, hypersurfaces
53A07, 53C21

## 1. Introduction

Sphere theorems in positive Ricci curvature are now a classical matter of study. The canonical sphere is the only manifold with which is extremal for the volume, the radius, the first non zero eigenvalue on functions or the diameter. Moreover, it was proved in [6, 7, 4] that manifolds with and volume close to are diffeomorphic and Gromov-Hausdorff close to the sphere. This stability result was extended in [14, 1], where it is proved that manifolds with have almost extremal volume if and only if they have almost extremal radius, if and only if they have almost extremal . Almost extremal diameter and almost extremal are also equivalent when ([9, 11]), but, as shown in [2, 13], it does not force the manifold to be diffeomorphic nor Gromov-Hausdorff close to . In this paper, we study the stability of three optimal geometric inequalities involving the mean curvature of Euclidean hypersurfaces, and whose equality case characterizes the Euclidean spheres (see Inequalities (1.1), (1.2) and 1.3 below). More precisely we study the metric and spectral properties of the hypersurfaces which almost realize the equality case. It completes the results of [5, 16].

Let be a closed, connected, isometrically immersed -manifold (. The first geometric inequality we are interested in is the following

 (1.1) ∥H∥2∥X−¯¯¯¯¯X∥2⩾1

where , is the volume of , is the mean curvature of the immersion and is the renormalized -norm on defined by . Equality holds in (1.1) if and only if is a sphere of radius and center (see section 2). From (1.1) we easily infer the Hasanis-Koutroufiotis inequality on extrinsic radius (i.e. the least radius of the balls of which contains )

 (1.2) ∥H∥2Rext⩾1

whose equality case also characterizes the sphere of radius and center . The last inequality is the well-known Reilly inequality

 (1.3) λ1⩽n∥H∥22

Here also, the extremal hypersurfaces are the spheres of radius . Let and be some reals. We will say that is almost extremal for Inequality (1.1) when it satisfies the pinching

We will say that is almost extremal for Inequality (1.2) when it satisfies the pinching

We will say that is almost extremal for Inequality (1.3) when it satisfies the pinching

###### Remark 1.1.

It derives from the proof of the three above geometric inequalities, given in section 2, that Pinching or Pinching imply Pinching . For that reason, Theorems 1.2, 1.7, 1.13 below are stated for hypersuraces satisfying Pinching but are obviously valid for Pinching or Pinching .

Our first result is that, when is bounded, almost extremal manifolds for one of the three Inequalities (1.1), (1.2) or (1.3) are Hausdorff close to an Euclidean sphere of radius and have almost constant mean curvature.

###### Theorem 1.2.

Let , and be some reals. There exist some positive functions and such that if satisfies and , then we have

 (1.4)

and there exist some positive functions and so that

 (1.5) ∥∥|H|−∥H∥2∥∥r⩽Cεβ∥H∥2       for any r∈[1,q).

We assume moreover that . For any   and , there exists  such that if satisfies (for ) and then for any , we have

 (1.6) ∣∣Vol(B(x,r∥H∥2)∩X(M))VolM−Vol(B(x,r∥H∥2)∩S)VolS∣∣⩽ηVol(B(x,r∥H∥2)∩S)VolS

where is the Euclidean ball with center and radius .

Theorem 1.2 generalizes and improves the main results of [5] and [16], where only the pinchings and for and were considered. The control on the mean curvature (Inequality (1.5)) and Inequality (1.6) are new, even under a bound on the mean curvature. Note that (1.6) says not only that goes near any point of the sphere (as was proven in [5, 16]) but also that the density of near each point of is close to .

###### Remark 1.3.

From Inequalities (1.4) and (1.6) we infer that almost extremal hypersurfaces for one of the three geometric inequalities (1.1), (1.2) or (1.3) converge in Hausdorff distance to a metric sphere of . As shown in Theorem 1.9, there is no Gromov-Hausdorff convergence if we do not assume a good enough bound on the second fundamental form.

###### Remark 1.4.

By Theorem 1.2, when (), Pinching implies Pinching for a constant . In other words, Pinchings and are equivalent (in bounded mean curvature) and are both implied by Pinching . However, we will see in Theorem 1.9 that Pinching (or ) does not imply Pinching .

###### Remark 1.5.

The constant tends to when or , but the same result can be proved with replaced by , where is a universal constant depending only on the dimension .

Inequality 1.4 follows from the following new pinching result on momenta.

###### Theorem 1.6.

Let be a real. There exists a constant such that for any isometrically immersed hypersurface of , we have

where .

In particular, this gives

Our next result shows that almost extremal hypersurfaces must satisfy strong spectral constraints. We denote the eigenvalues of the canonical sphere , the multiplicity of and (note that we have and ). We also denote the eigenvalues of counted with multiplicities.

###### Theorem 1.7.

Let , and be some reals. There exist some positive functions and such that if satisfies and then for any such that , the interval

 [(1−εα√mkCk)∥H∥22μk,(1+εα√mkCk)∥H∥22μk]

contains at least eigenvalues of counted with multiplicities.

Moreover, the previous intervals are disjoints and we get

 λi(M)⩽(1+εα√mkCk)∥H∥22λi(Sn)       for any i⩽σk−1,

and if then

###### Remark 1.8.

In the particular case of extremal hypersurfaces for Pinching , Theorem 1.7 implies that

 n∥H∥2p1+ε⩽λ1(M)⩽⋯⩽λn+1(M)⩽(1+C(n)εα)n∥H∥22

and so we must have the -first eigenvalues close to each other. Compare to positive Ricci curvature where close to implies close to , but we can have only eigenvalues close to for any (see [1]).

Note that Theorem 1.7 does not say that the spectrum of almost extremal hypersurfaces for Inequality (1.1) is close to the spectrum of an Euclidean sphere, but only that the spectrum of the sphere asymptotically appears in the spectrum of . Our next two results show that this inclusion is strict in general (we have normalized the mean curvature by for sake of simplicity and stands for the integral part of ).

###### Theorem 1.9.

For any integers there exists sequence of embedded hypersurfaces of diffeomorphic to spheres glued by connected sum along points, such that , , , , and for any we have

 λσ(Mj)→λE(σp)(Sn).

In particular, the have at least eigenvalues close to whereas its extrinsic radius is close to .

###### Theorem 1.10.

There exists sequence of immersed hypersurfaces of diffeomorphic to spheres glued by connected sum along great subsphere , such that , , , , and for any we have

 λσ(Mj)→λE(σ2)(Sn,d),

where is the sphere endowed with the singular metric, pulled-back of the canonical metric of by the map , where is identified with via the map . Note that has infinitely many eigenvalues that are not eigenvalues of .

###### Remark 1.11.

Theorem 1.9 shows that Pinching is not implied by Pinching nor Pinching , even under an upper bound on .

###### Remark 1.12.

It also shows that almost extremal manifolds are not necessarily diffeomorphic nor Gromov-Hausdorff close to a sphere. We actually prove that the can be constructed by gluing spheres along great subspheres with and with (see the last section of this article).

In [5] and [16] it has been proved that when the -norm of the second fundamental form is bounded above, then almost extremal hypersurfaces are Lipschitz close to a sphere of radius (which implies closeness of the spectra). In view of Theorem 1.9, we can wonder what stands when is bounded with .

###### Theorem 1.13.

Let , and be some reals. There exist some positive functions and such that if satisfies and , then the map

 F:M⟶1∥H∥2Snx⟼1∥H∥2Xx|Xx|

is a diffeomorphism and satisfies for any vector .

The structure of the paper is as follows: after preliminaries on the geometric inequalities for hypersurfaces in Section 2, we prove in Section 3 a general bound on extrinsic radius that depends on integral norms of the mean curvature (see Theorem 1.6). We prove Inequality (1.4) in Section 4 and Inequality (1.5) in Section 5. Theorem 1.13 is proven in Section 6. Section 7 is devoted to estimates on the trace on hypersurfaces of the homogeneous, harmonic polynomials of . These estimates are used in Section 8 to prove Theorem 1.7 and in section 9 to prove Inequality (1.6). We end the paper in section 10 by the constructions of Theorems 1.9 and 1.10.

Throughout the paper we adopt the notation that is function greater than which depends on , , , . These functions will always be of the form . But it eases the exposition to disregard the explicit nature of these functions. The convenience of this notation is that even though might change from line to line in a calculation it still maintains these basic features.

## 2. Preliminaries

Let be a closed, connected, isometrically immersed -manifold (. If denotes a local normal vector field of in , the second fundamental form of associated to is and the mean curvature is , where and are the Euclidean connection and inner product on .

Any function on gives rise to a function on which, for more convenience, will be also denoted subsequently. An easy computation gives the formula

 (2.1) ΔF=nHdF(ν)+Δ0F+∇0dF(ν,ν),

where denotes the Laplace-Beltrami operator of and is the Laplace-Beltrami operator of . Applied to or , Formula 2.1 gives the following

 (2.2) ΔXi=nHνi,        ⟨ΔX,X⟩=nH⟨ν,X⟩
 (2.3) 12Δ|X|2=nH⟨ν,X⟩−n,        ∫MH⟨ν,X⟩dv=VolM

These formulas are fundamental to control the geometry of hypersurfaces by their mean curvature.

### 2.1. A rough bound on geometry

The integrated Hsiung formula (2.3) and the Cauchy-Schwarz inequality give the following

 (2.4) ∫MH⟨ν,X⟩dvVolM=1⩽∥H∥2∥∥X−¯¯¯¯¯X∥∥2=∥H∥2infu∈Rn+1∥X−u∥2

This inequality is optimal since satisfies

 ∥H∥2∥∥X−¯¯¯¯¯X∥∥2=1

if and only if is a sphere of radius and center . Indeed, in this case and are everywhere colinear, hence the differential of the function is zero on . Equality (2.3) then implies that is constant on equal to .

### 2.2. Hasanis-Koutroufiotis inequality on extrinsic radius

We set the extrinsic Radius of , i.e. the least radius of the balls of which contain . Then Inequality (2.4) gives

 ∥H∥2Rext⩾∥H∥2∥X−¯¯¯¯¯X∥2 =∥H∥2infu∈Rn+1∥X−u∥2 (2.5) ∥H∥2Rext ⩾1

and when , we have equality in (2.4), i.e. is a sphere of radius .

### 2.3. Reilly inequality on λ1

We translate so that . By the min-max principle and Equality (2.2), we have

 λ1n∥X∥22⩽∫M⟨X,ΔX⟩dvnVolM=∫MH⟨ν,X⟩dvVolM=1=(∫MH⟨ν,X⟩dvVolM)2

where is the first nonzero eigenvalue of . Combined with Inequality (2.4), we get the Reilly inequality

 (2.6) λ1⩽nVolM∫MH2dv.

Here also, equality in the Reilly inequality gives equality in 2.4 and so it characterizes the sphere of radius .

## 3. Upper bound on the extrinsic radius

In this section we prove Theorem 1.6.

###### Proof.

We translate such that . We set . We have , hence, using the Sobolev inequality (see [12])

 (3.1) ∥f∥nn−1⩽K(n)(VolM)1n(∥df∥1+∥Hf∥1)

we get for any

We set and , where and (i.e. ). The previous inequality gives

 (∥φ∥ap+1∥φ∥∞)ap+1νp+1⩽(K(n)(VolM)1n(apq−1q+1∥φ∥∞+∥H∥q))nνp+1(n−1)(∥φ∥ap∥φ∥∞)apνp

Since then and converges to and we have

hence we have

We set . If then we get the result since we have

where we have used that . We infer the result from the equality . If , we get immediately the desired inequality of the Theorem from the above expression of and the fact that . ∎

## 4. Proof of Inequality (1.4)

Let be an isometrically immersed hypersurface of . We can, up to translation, assume that . By the Hölder inequality and Pinching , we have , hence

 1−∥X∥1∥X∥2⩽((1+ε)pp−2−1)⩽pp−222p−2ε

On the other hand applying Inequality (3.1) to we get

 (4.1) 1⩽K(n)(VolM)1n∥H∥1

And combining the two above inequalities with Theorem 1.6 and we get (1.4). More precisely we have .

###### Remark 4.1.

Combining (4.1) with Inequality (1.4) we get

 (4.2) ∥X∥∞⩽C(p,q,n)Aγ/n(VolM)1/n
###### Lemma 4.2.

For any if is satisfied, then there exist some positive functions , and so that the vector field satisfies

 (4.3) ∥Z∥r⩽C(p,q,n)(1+A)βεα(q−r)r(q−2)    for any r∈[2,q).
###### Proof.

By the Hölder inequality we have for any

 ∥Z∥r⩽∥Z∥q(r−2)r(q−2)q∥Z∥2(q−r)r(q−2)2⩽(1+∥X∥∞∥H∥q)q(r−2)r(q−2)∥Z∥2(q−r)r(q−2)2\default@cr⩽(1+∥X∥∞∥H∥q)qq−2∥Z∥2(q−r)r(q−2)2

By remark 4.1, we have . Then

 ∥Z∥r⩽C(p,q,n)Aβ∥Z∥2(q−r)r(q−2)2

Moreover by integrating the Hsiung-Minkowsky formula (2.3) we have

 ∥Z∥22 =1−2VolM∫MH⟨ν,X⟩dv+∥HX∥22⩽−1+∥H∥22∥X∥2∞.

which, by Inequality (1.4), gives . ∎

## 5. Proof of Inequality (1.5)

Since we have , Inequality gives us

and so

By Inequalities (1.4), this gives

 ∥∥H2−∥H∥22∥∥1 =∥H∥42(∥∥H2∥H∥42−|X|2∥∥1+∥∥|X|2−1∥H∥22∥∥1) ⩽∥H∥42(∥∥Hν∥H∥22+X∥∥2∥∥X−Hν∥H∥22∥∥2+CAγ/nεα∥H∥2(∥X∥2+1∥H∥2)) ⩽∥H∥42(√3ε∥X∥2(∥Hν∥H∥22∥2+∥X∥2)+CAγ/nεα1∥H∥22) ⩽CAγ/nεα∥H∥22

Hence we have  . Moreover  we have  . Hence by the Hölder inequality, for any we have

 ∥∥|H|−∥H∥2∥∥r ⩽(∥∥|H|−∥H∥2∥∥1)q−rr(q−1)(∥∥|H|−∥H∥2∥∥q)q(r−1)r(q−1) ⩽C(p,q,r,n)Aβ(q,r,n)εα(q−r)r(q−1)∥H∥2

## 6. Proof of the theorem 1.13

Let be a unit vector and put where is the tangential projection of on . For small enough we have from (1.4) and then the application is well defined. We have (see [5]), hence for any

 (6.1) ||dFx(u)|2−1| ⩽1|X|2∣∣ ∣∣1∥H∥22−|X|2∣∣ ∣∣+1∥H∥22⟨u,X⟩2|X|4 ⩽Cεα|X|2∥H∥22+∥ψ∥2∞∥H∥22|X|4

Now an easy computation using 1.4 shows that . Now using the Sobolev inequality 3.1 and the fact that (see 4.1), we have

 ∥ψ∥2α2αnn−1 ⩽K(n)(VolM)1/n(2αCAβ∥H∥2∥B∥q+2αn+∥H∥q∥ψ∥∞)∥ψ∥2α−1(2α−1)qq−1 ⩽K(n)(2αCAβ(VolM)1/n+A1/n∥ψ∥∞)∥ψ∥2α−1(2α−1)qq−1

And similarly to the proof of the theorem 1.6 we obtain

 ∥ψ∥∞⩽C(q,n)((VolM)1/nCAβ∥ψ∥∞+A1/n)γ∥ψ∥2

And using the fact that and we get that is

 ∥ψ∥γ+1∞⩽CAβ∥ψ∥2∥H∥γ2

Now since and we deduce that . And reporting this in (6.1) and using (1.4) with the fact that we get .

## 7. Homogeneous, harmonic polynomials of degree k

Let be the space of homogeneous, harmonic polynomials of degree on . Note that induces on the spaces of eigenfunctions of associated to the eigenvalues with multiplicity (see [3]).

On the space , we define the following inner product

 (P,Q)Sn:=1VolSn∫SnPQdvcan\ ,

where denotes the element volume of the sphere with its standard metric. On the other hand the inner product on will be defined by

 (f,g)=∫MfgdvVolM    % for f,g∈C∞(M).

In this section we give some estimates on harmonic homogeneous polynomials needed subsequently. We set an arbitrary orthonormal basis of . Remind that for any and any , we have and .

###### Lemma 7.1.

For any , we have .

###### Proof.

For any , is a quadratic form on whose trace is given by . Since for any and any such that we have and since is an isometry of , we have . Now

and so . We conclude by homogeneity of the . ∎

As an immediate consequence, we have the following lemma.

###### Lemma 7.2.

For any , we have

 mk∑i=1(dxPi(u