Rigidity for Einstein Manifolds under Bounded Covering Geometry (Journal of Mathematical Study, 2025)
In this note, we prove three rigidity results for Einstein manifolds withbounded covering geometry. (1) An almost flat manifold $(M,g)$ must be flat if it isEinstein, i.e. ${\rm Ric}_g =λg$ for some real number $λ.$ (2) A compact Einstein manifold witha non-vanishing and almost maximal volume entropy is hyperbolic. (3) A compactEinstein 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.