Markov convexity and local rigidity of distorted metrics

Cartan-Hadamard流形中超曲面的凸性与刚性

Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds

本文证明Cartan-Hadamard流形中曲率在切平面消失的闭无穷小凸超曲面围成凸平坦区域，推广了经典刻画与刚性结果，并应用Gauss-Codazzi方程和Alexandrov几何技巧。

Circumcenters in Finsler unitary groups

We study convexity properties of distance functions in Finsler unitary groups, where the Finsler structure is defined by translation of the $p$-Schatten norm on the Lie algebra. As a result we prove the existence of circumcenters for sets with radius less than $π/2$ in several metrics. This result is applied to a fixed point property and to quantitative metric bounds in certain rigidity problems. Bounds for convexity, existence of circumcenters and rigidity are shown to be optimal.

Characterizations of monotonicity of vector fields on metric measure spaces

We characterize the convexity of functions and the monotonicity of vector fields on metric measure spaces with Riemannian Ricci curvature bounded from below. Our result offers a new approach to deal with some rigidity theorems such as `splitting theorem' and `volume cone implies metric cone theorem' in non-smooth context.

On the Unique Determination of Domains by the Condition of the Local Isometry of the Boundaries in the Relative Metrics

The article contains the results of the author's recent investigations of rigidity problems of domains in Euclidean spaces carried out for developing a new approach to the classical problem of the unique determination of bounded closed convex surfaces [A.V.Pogorelov, Extrinsic Geometry of Convex Surfaces, AMS, Providence (1973)], rather completely presented in [A.P.Kopylov, On the unique determination of domains in Euclidean spaces, J. of Math. Sciences, 153, no.6, 869-898 (2008)]. We give a complete characterization of a plane domain $U$ with smooth boundary (i.e., the Euclidean boundary $\ma…