Recognizing shape via 1st eigenvalue, mean curvature and upper curvature bound
Let be a closed immersed hypersurface lying in a contractible ball of the ambient -manifold . We prove that, by pinching Heintze-Reilly’s inequality via sectional curvature upper bound of , 1st eigenvalue and mean curvature of , not only is Hausdorff close to a geodesic sphere in , but also the “enclosed” ball is close to be of constant curvature, provided with a uniform control on the volume and mean curvature of . We raise a conjecture for to be a diffeomorphic sphere, and give positive partial answers for several special cases, one of which is as follows: If in addition, the renormalized norm () of ’s 2nd fundamental form admits a uniform upper bound, then is an embedded diffeomorphic sphere, almost isometric to , and intrinsically -close to a round sphere of constant curvature.
The isoperimetric inequality in the Euclidean plane has a long history, and has many generalizations both on Riemannian surfaces and higher dimensional manifolds (e.g. , ). One of those such that “equality implies rigidity” was founded around 1950’s, as follows.
However, not only few generalization of (1.1) with similar rigidity are known to hold naturally on higher dimensional manifolds, but also other rigidity phenomena with respect to the upper curvature bound are rarely studied.
In contrast, nowadays rigidity results and their quantitative version (e.g. [14, 15], , , [11, 12], , , etc.) under curvature bounded from below have been extensively studied and provided fundamental tools in study of Riemannian manifolds and their limit geometry under Gromov-Hausdorff topology.
In this paper we prove a quantitative rigidity for a domain (resp. an immersed closed hypersurface) in a contractible neighborhood on a complete Riemannian manifold to be a geodesic ball (resp. geodesic sphere) of constant curvature , where is upper curvature bound of ambient space.
Our starting point is an observation on Heintze’s result . An open ball is called geodesic contractible if for any point , there is a unique radial minimal geodesic from to ; i.e., is no more than injectivity radius of .
Let be an immersed, oriented and connected compact hypersurface without boundary in a geodesic contractible ball of , where the sectional curvature of , for some . Let denote ’s volume and the mean curvature. If the first non-zero eigenvalue of Laplace-Beltrami operator on satisfies one of the following conditions,
if , and
then is an embedded geodesic sphere, and the enclosed ball is of constant curvature .
Recall that in [23, Theorem 2.1] Heintze proved that a submanfold (including higher codimension) lying a convex ball of satisfies
Equality holds in (1.4) if and only if is minimally immersed in some geodesic sphere of .
The observation in Theorem 1.1 is that, if equality holds in 1.4, then the norm of Jacobi fields along radial geodesic starting from spherical center to is that of constant curvature . Moreover, the convexity of can be weakened to be geodesic contractible; see §2.
An interesting consequence of Theorem 1.1 is that, any interior perturbation (no matter large or small) of a ball of constant curvature with radius has to raise up interior curvature. This fact can be seen from Gauss-Bonnet theorem when . But in high dimension it is hard to see without using (1.4) that involves upper curvature bound.
Our main result is a quantitative version of Theorem 1.1 via pinching (1.3), which in particular, implies that any interior perturbation of a ball of constant curvature cannot be large, if it raises up in a small amount of curvature.
As working for a class of manifolds, certain uniform geometric bounds are usually required. We will work under the assumptions that ambient space admits a bounded sectional curvature , the volume and mean curvature of immersed hypersurface satisfy the following rescaling invariant bound
For the case that either and volume of admits a uniform upper bound or , by Lemma 3.6 below, volume of extrinsic ball on admits a uniform lower bound relatively,
Hence we may view (1.5) as an extrinsically relative non-collapsing condition; it can be compared with volume comparison under lower bounded Ricci curvature.
Let be the -norm of mean curvature vector of . Let be the usual -sine function (see (2.1) below), be its inverse function. Let the volume of -sphere of radius in . Throughout the paper we view for , and use to denote a positive function on that converges to as with other quantities fixed.
Let be an integer , , and let be a complete Riemannian manifold with . Let be an immersed closed, connected and oriented hypersurface in a geodesic contractible ball . If , we further assume
If satisfies (1.5) for , and
holds with , then
is -Hausdorff-close to a geodesic sphere , where and is a positive constant;
is -close to a ball of constant curvature for any ;
where , and do not depend on when .
From the proof, is center of mass of in with respect to an appropriate variation of distance function; see §2.4.
The conclusion of Main Theorem is only known before in space forms (, , ). (M2) reveals a substantially new phenomena on Riemannian manifolds; a contractible domain can be recognized to be a ball of almost constant curvature by “hearing” the 1st eigenvalue of its boundary (or any immersed hypersurface close to its boundary), maximum of ’s mean curvature and interior curvature’s upper bound.
Due to that is known to be true only for  or on a hyperbolic space for  (it fails for if is hyperbolic, cf. ), the pinching condition (1.8) is the best one can expect at present for submanifolds whose ambient space changes curvature sign.
It is natural to ask
Whether in Main Theorem is an embedded diffeomorphic sphere?
In general the mean curvature is too weak to determine the topology of a submanifold. At present we do not know if in Main Theorem could be very twisted or not.
Our next result answers the above problem under an additional assumption on the 2nd fundamental form. Let be the normalized norm of the 2nd fundamental form of .
Let the assumptions be as in Main Theorem. If in addition, for and ,
Then for ,
is embedded, diffeomorphic and -almost isometric to a geodesic sphere , where is a positive constant;
is - close to a round sphere of constant curvature with ;
where , and do not depend on when .
The -closeness of metric tensors follows from -regularity [42, Theorem 2.35] under -integral Ricci curvature bound with and -non-collapsing condition.
We point it out that the bound on 2nd fundamental form is not a necessary condition for to be a diffeomorphic sphere; we prove that a non-spherical satisfying (1.8) cannot be constructed via any reflective gluing operations with a small neck between spheres, and any small neck with a rotational symmetric neighborhood cannot appear on ; see §6.
According the above, we propose the following conjecture.
In Main Theorem is an embedded diffeomorphic sphere, provided with or other weaker conditions.
What earlier known about pinching (1.8) is very restrictive when the ambient space is a Riemannian manifold. In  Grosjean and Roth proved Theorem 1.4 under some technical assumptions such that the hypersurface was required to be contained in a small geodesic ball of radius , where coincides with the pinching error in (1.8). Thus in their case approaches to a point as , and (M2) in Main Theorem is trivially satisfied.
Under bound of 2nd fundamental form (1.9), results corresponding to Theorem 1.4 has been recently studied and known in space forms; see ,  and . But Conjecture 1.5 is open for hypersurfaces in a space form.
Note that by Main Theorem, now we know that, roughly speaking, pinching (1.8) for hypersurfaces satisfying (1.5) essentially can happen only in space forms, as long as the ambient space around has trivial topology and bounded geometry.
The lower curvature bound is not essential in Main Theorem. According to our proof it is possible that Main Theorem holds under conditions weaker than (e.g., an integral Ricci curvature lower bound); cf. Remark 4.6.
Under our setting , one cannot expect that sectional curvature of is pointwise close to . It is not difficult to construct a warped product manifold where is its slice, and pinching condition (1.8) holds with arbitrary small , but there are points in the enclosed domain by where sectional curvature is arbitrarily away from .
Otherwise, a counterexample can be easily constructed via smoothing a cylinder glued with one cap at , where the geometry of boundary at and interior curvature bound does not change as varies.
Cheeger-Colding’s quantitative rigidity for warped products says that if an annulus with respect to some distance satisfies the criteria of warped-product function in the -sense, then the annulus away from boundary is Gromov-Hausdorff close to a metric warped product. In particular, for a ball around which Ricci curvature , if the area of is close to that in -space form of constant curvature, then is Gromov-Hausdorff close to a ball of constant curvature . For such rigidity and comparison results under lower bounded Ricci curvature, the appearance of cut points makes no problem.
Due to natural geometric restrictions, the criteria in Main Theorem generally fails when contains nontrivial topology via cut points. The connected sum of a flat torus with a flat disk via a think neck of non-positive curvature glued around center of provides a counterexample.
Though the ambient space for pinching condition (1.8) is simpler than those with respect to Ricci curvature’s lower bound (e.g., Cheeger-Colding’s almost rigidity above), our “boundary condition” via pinching (1.8) is less restrictive, i.e., Main Theorem holds for an immersed hypersurface , which could be twisted very much and a priori lie far away from a level set of a warping function, though it turns out to be Hausdorff close to a sphere finally.
By the discussion on (1.5) above Main Theorem, typical examples not covered by Main Theorem contain the boundary of a -neighborhood of a high co-dimensional submanifold (not a point) with .
Let us point out the main ideas and difficulty in proving Main Theorem.
In order to prove the Hausdorff closeness (M1), we improve estimates in  to prove that lies in a small neighborhood of a geodesic sphere , where is the center of mass of in . Then based on an observation in , via contacting a “standard” sphere-tori to , it is not difficult (see §3) to show that is also near .
In this part, we follow the main ideas in ; i.e, first to transform pinching condition (1.8) into an pinching (4.2) on position vector ; then to apply Moser iteration to bound . A careful geometric analysis calculation involving out-radius is required in order to improve corresponding estimates in  and drop a technical assumption in . This is done in §4.
By (M1), a naive approach for (M2) is arguing by contradiction. Up to a rescaling, there is a sequence of pairs converging to a limit in Gromov-Hausdorff topology, where is -Riemannian manifold and is a geodesic sphere of radius in . One may guess the pinching condition (1.8) can be passed to the limit pair with zero pinching error, such that rigidity for the limit may follow from similar arguments as Theorem 1.1.
According to , by passing to a subsequence, converges to in measured Gromov-Hausdorff topology, where and are the restricted distance from the ambient space respectively, is the Riemann-Lebesgue measure and is its limit measure. By Fukaya’s observation in , .
If coincides with Hausdorff measure of , then it is easy to apply similar arguments as Theorem 1.1 to derive is isometric to a ball in space form.
A crucial difficulty is that, if is far away from an embedded diffeomorphic sphere, then generally fails, and thus the relation between pinching condition (1.8) and limit geometry is lost.
Here is our approach. Instead of looking at the limit, we will translate pinching condition (1.8) along to its position vector at such that is close to (up to a rescaling) and perpendicular to in the sense. By refining the relation between divergence of on and mean curvature (see Lemma 3.5, cf. Lemma 2.5), and are close in the sense, where . Since by (M1), is close to , we see that is also close to along in the sense. Via the monotonicity of ’s volume in extrinsic balls (cf. ), we transmit the estimate on over to points sufficient dense in , and finally prove (M2). This is done in §3.
The remaining of the paper is organized as follows. We recall some necessary facts and tools in §2, and give a proof of Theorem 1.1 as a preliminary knowledge. §3 and §4 are devoted to the proof of Main Theorem. In §5 we prove Theorem 1.4. Further discussion via examples about Conjecture 1.5 is given in §6. Appendix is for proofs of some technical lemmas.
Acknowledgements. The first author was supported by China Postdoctoral Science Foundation (No.2018M641317). The second author was supported partially by National Natural Science Foundation of China , , by research funds of Beijing Municipal Education Commission and Youth Innovative Research Team of Capital Normal University.
In this section we provide notations and facts used later. The -sine function is defined by
and -cosine function is defined by . Clearly, the following identities hold:
2.1. Convexity radius
Let be a (maybe non-complete) Riemannian manifold. The exponential map, , from tangent space at to is well-defined locally. The injectivity radius of a point , , is defined to be the supremum of radii of open balls centered at origin of where the restriction is a well-defined diffeomorphism onto its image. The conjugate radius of , , is defined to be the supremum of radii of open balls in which contains no critical point of . The convexity radius of , , is defined to be the supremum of radii of open balls centered at that is strongly convex (cf. ). We call an open set convex, if any two points of are joined by a unique minimal geodesic in and its image lies in . An open ball is called strongly convex, if any is convex.
By definition, , and . The following pointwise estimates of and will be used later.
Assume that has a compact closure in . For any point , the followings hold.
where is the upper bound of sectional curvature in
2.2. Sobolev inequality
The well-known Sobolev inequality for Riemannian submanifolds due to Hoffman and Spruck  is a fundamental tool applied in the proof of Main Theorem.
Theorem 2.2 ().
Let be a complete Riemannian manifold with . Let be a compact immersed submanifold in . Let be nonnegative. For , in addition we assume that the volume of has an upper bound,
Then there exists a positive constant such that
We now verify that Theorem 2.2 is applicable for hypersurface in Main Theorem by localizing injectivity radius along whole to one point.
Let be a compact hypersurface immersed into a geodesic contractible ball , where sectional curvature of ambient space and . In order to apply Theorem 2.2 for , we need to justify (2.4) under the condition of Main Theorem.
Let us consider the ball in the tangent space with the pullback metric by . Since for any , it is easy to see by Lemma 2.1 that, for any , .
On the other hand, since is geodesic contractible, can be lifted by the inverse of into . Then by (2.2), for any point of the image of in , .
2.3. Convergence theorems in Gromov-Hausdorff topology
We recall convergence results for Riemannian manifolds under Gromov-Hausdorff topology.
We say that a sequence of metric spaces -converges to in Gromov-Hausdorff topology, denoted by , if there is -isometries with , i.e., for any , , and -neighborhood of covers .
Gromov’s compactness theorem (cf. ) says that for any collection of compact metric spaces of bounded diameter, if they are uniformly and totally bounded (i.e., there is a nonnegative function such that for each , any maximal -discrete net of contains points at most ), then is precompact (i.e,. has a compact closure) in the Gromov-Hausdorff topology. Such precompactness can be guaranteed by the relative volume comparison theorem under lower bounded Ricci curvature.
For -manifolds with uniformly bounded (sectional or Ricci) curvature and under certain appropriate non-collapsing assumptions, -convergence implies higher regularity of metric tensors.
Let be a sequence of Riemannian -manifolds whose sectional curvature , diameter and injectivity radius . Then there is a subsequence whose -limit is isometric to a -Riemannian manifold , and there are diffeomorphisms for all sufficient large such that the pullback metric converges to in the topology for any (i.e., there is a fixed coordinates system such that converges to on each chart in the -norm).
A compact Riemannian -manifold is said to be of -bounded Ricci curvature, if for real numbers and ,
A manifold is called to be -non-collapsing with at scale , if
where is an Euclidean ball of radius .
By , the relative volume comparison of balls holds on manifolds of -bounded Ricci curvature. Since in harmonic coordinates, the -bound of the Ricci curvature gives the -bound of the metric tensor (cf. ), by the -harmonic radius estimate [42, Theorem 2.35], the following -regularity convergence result holds.
For any , there is such that for any sequence of Riemannian -manifolds of -bounded Ricci curvature, -non-collapsing volume at scale and uniformly bounded diameter, there is a subsequence whose -limit is isometric to a -Riemannian manifold , and there are diffeomorphisms for all sufficient large such that the pullback metric converges to in the topology.
2.4. Center of mass
Let be an immersed submanifold in a geodesic contractible ball . If in addition with , then we assume . By lifting to as the same argument below Theorem 2.2 and by (2.3), we assume without loss of generality that is convex.
Let be an energy function defined by
where is the modified distance function defined by
We claim that there is a unique minimum point of in . We call the center of mass of with respect to modified distance.
First, by and (2.3), every is convex. Hence is a strictly convex function on , which admits a unique minimum point such that . Note that it is equivalent to , where is a vector field defined by
Secondly, because the vector field defined above, by the convexity of , points into interior of along the boundary. It follows that the minimum point of lies in .
By definition, it is clear that in the normal coordinates of , (2.8) becomes
2.5. Test functions for 1st eigenvalue of Laplace-Beltrami operator
Now let be an immersed oriented hypersurface with . If , we further assume that .
Let be the center of mass of and . We call the vector field the position vector about . By Rayleigh’s principle and (2.9), each component of the position vector provides a test function of , such that
In order to estimate , we need the following lemmas from . Let be ’s tangential projection over , then
For any vector field on , the divergence of along is defined by
where is an orthonormal basis of .
Lemma 2.5 ().
Let the assumptions be as above. The following inequalities hold.
(i) Let be the normal vector of from its orientation, then
(ii) The covariant derivatives of components of , satisfies
Since the proof of (i) is used in the proof of Main Theorem, we give a proof. For (ii) we refer to .
Let and be an orthonormal basis of . If at , then for all ; Otherwise, we take and for . Then we get
where is a unit vector such that . By Hessian comparison theorem for , we get
where we used the fact that and the standard Jacobi field estimates. Thus we have
By the identity
2.6. Proof of Theorem 1.1
Proof of Theorem 1.1.
Let be the center of mass of and , and let be the position vector with respect to and be its tangential projection over . We claim that
By the claim, the image of is a geodesic sphere centered at , and the Laplacian of . Since , it implies
By the rigidity of Hessian comparison, must be a geodesic ball of constant curvature .
If , then by the simply connectedness of , is embedded. For , by the fact that , where is ’s length, and the 1st eigenvalue of its image satisfies (1.2) or (1.3), it is clear that . Hence is also embedded.
The claim can be directly seen from Heintze’s proof . For completeness, we give its verification below by dividing into three cases: , and .
Case 1 for . Let us take , as test functions in Rayleigh quotient. Since is the center of mass of , we have . By Rayleigh quotient,
Therefore, , . By (2.18) again .
Case 2 for . Let us take , and as test functions, where . Since and , we derive