A classical isoperimetric inequality by A. D. Alexandrov says that for any simply-connected domain Ω on a surface, L^2>=4π*A-K*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, no such isoperimetric-typed rigidity with respect to the upper sectional curvature bound is known. In this talk, we consider Heintze-Reilly'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)(1-ε), then M is embedded, and the enclosed domain Ω is C^{1,α}-close to a geodesic ball of constant curvature K.
Such quantitative rigidity is known before only in simply connected space forms or the infinitesimal Euclidean case. By counterexamples, bound of 2nd fundamental form's L^q-norm 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, Cheeger-Gromov's convergence theorem, and convergence of eigenvalues under L^p integral Ricci curvature bound in Cheeger-Colding's theory.
