Rigidity for Einstein Manifolds under Bounded Covering Geometry

Abstract

In this note, we prove three rigidity results for Einstein manifolds with bounded covering geometry. (1) An almost flat manifold (M,g) must be flat if it is Einstein, i.e. Ric_g=λg for some real number λ. (2) A compact Einstein manifold with a non-vanishing and almost maximal volume entropy is hyperbolic. (3) A compact Einstein manifold admitting a uniform local rewinding almost maximal volume is isometric to a space form.

Keywords

Einstein / rigidity / almost nonnegative Ricci curvature / bounded covering geometry / space forms