An intrinsic proof of Gromoll-Grove diameter rigidity theorem

A Hamiltonian version of a result of Gromoll and Grove

The theorem that if all geodesics of a Riemannian two-sphere are closed they are also simple closed is generalized to real Hamiltonian structures on $\mathbb{R}P^3$. For reversible Finsler $2$-spheres all of whose geodesics are closed this implies that the lengths of all geodesics coincide.

Rigidity of manifolds with boundary under a lower Ricci curvature bound

We study Riemannian manifolds with boundary under a lower Ricci curvature bound, and a lower mean curvature bound for the boundary. We prove a volume comparison theorem of Bishop-Gromov type concerning the volumes of the metric neighborhoods of the boundaries. We conclude several rigidity theorems. As one of them, we obtain a volume growth rigidity theorem. We also show a splitting theorem of Cheeger-Gromoll type under the assumption of the existence of a single ray.

Ricci Curvature on Alexandrov spaces and Rigidity Theorems

In this paper, we introduce a new notion for lower bounds of Ricci curvature on Alexandrov spaces, and extend Cheeger-Gromoll splitting theorem and Cheng's maximal diameter theorem to Alexandrov spaces under this Ricci curvature condition.

Rigidity for positively curved Alexandrov spaces with boundary

Inspired by a recent work of Grove-Petersen in [GP18], where the authors studied Alexandrov spaces with largest possible boundary. We study Alexandrov spaces with lower curvature bound 1 and with small boundary. When the radius of X is π/2, and the boundary has diameter π/2, we classify the total space X.

Isoperimetric rigidity and distributions of 1-Lipschitz functions

We prove that if a geodesic metric measure space satisfies a comparison condition for isoperimetric profile and if the observable variance is maximal, then the space is foliated by minimal geodesics, where the observable variance is defined to be the supremum of the variance of 1-Lipschitz functions on the space. Our result can be considered as a variant of Cheeger-Gromoll's splitting theorem and also of Cheng's maximal diameter theorem. As an application, we obtain a new isometric splitting theorem for a complete weighted Riemannian manifold with a positive Bakry-Émery Ricci curvature.

非光滑Riemannian与semi-Riemannian度量的Ricci曲率界限与刚性

Ricci curvature bounds and rigidity for non-smooth Riemannian and semi-Riemannian metrics

本文研究低正则性度量的Riemannian与semi-Riemannian流形刚性问题，证明了Cheeger-Gromoll分裂定理和C^1平坦性准则，建立了非光滑度量和向量场的Bochner-Weitzenböck恒等式，并给出了分布Ricci曲率与RCD型界限等价性的新证明。