Home >> Seminar 

Quantitative rigidity for domains and immersed hypersurfaces in a Riemannian manifold
胥世成副教授(首都师范大学)
Thursday, October 18th, 2018, 1:00 PM 闵行4教414 

Abstract:
A classical isoperimetric inequality by A. D. Alexandrov says that for any simplyconnected domain Ω on a surface, L^2>=4π*AK*A^2, where L is the length of boundary, A the area of Ω, and K the upper bound of Ω's Gaussian curvature. Moreover, "=" holds if and only if Ω is a geodesic ball of constant curvature K. For domains in higher dimensional Riemannian manifolds, however, such isoperimetrictyped rigidity with respect to the upper sectional curvature bound is rarely known.
In this talk, we consider a similar rigidity via HeintzeReilly's inequality for immersed hypersurface M^n in a convex ball B(p,R) of a (n+1)manifold N: λ_1(M)<= n(K+max H), where λ_1 is 1st eigenvalue of Laplacian on M, H the mean curvature of immersion, and K=max K_N the upper sectional curvature bound of N.
We prove its quantitative rigidity: under some natural restrictions on R, vol(M), mean curvature H and L^q norm (q>n) of 2nd fundamental form of M, if λ_1(M)>= n(K+max H)(1ε), then not only M is embedded, diffeomorphic and C^αclose to a round sphere, but also the whole enclosed domain Ω is C^{1,α}close to a geodesic ball of constant curvature K.
Such quantitative rigidity is known before only in simplyconnected space forms or the infinitesimal case that diam M goes to 0. We construct counterexamples to show that both the bound of 2nd fundamental form's L^qnorm (q>n) and the convexity of B(p,R) are necessary. Our proof is based on tools from comparison Riemannian geometric, geometric analysis and metric geometry, such as, Moser iteration, CheegerGromov's convergence theorem, and C^α convergence of pointwise noncollapsing manifolds with a L^p integral Ricci curvature bound in CheegerColding's theory. This is a joint work with Yingxiang Hu.



