Slicing theorems and rigidity phenomena for self‐affine carpets

Two Rigidity Theorems for Gravity

In this paper we prove two theorems on initial data for general relativity. The results present “rigidity phenomena” for the extrinsic curvature, caused by the negativity of the scalar curvature. More precisely, Theorem 1 states that in the case of non-vacuum initial data if the metric has everywhere non-positive scalar curvature then the extrinsic curvature cannot be compactly supported. Theor...

On Family Rigidity Theorems

Some Rigidity Theorems for Finsler Manifolds

This is a survey article on global rigidity theorems for complete Finsler manifolds without boundary.

On Family Rigidity Theorems II

In [LM], we proved a family version of the famous Witten rigidity theorems and several family vanishing theorems for elliptic genera. In this paper, we gerenalize our theorems [LM] in two directions. First we establish a family rigidity theorem for the Dirac operator on loop space twisted by general positive energy loop group representations. Second we prove a family rigidity theorem for spin-m...

Rigidity Theorems in Riemannian geometry

The purpose of this chapter is to survey some recent results and state open questions concerning the rigidity of Riemannian manifolds. The starting point will be the boundary rigidity and conjugacy rigidity problems. These problems are connected to many other problems (Mostow-Margulis type rigidity, isopectral problems, isoperimetric inequalities etc.). We will restrict our attention to those r...