Harmonic Functions, Entropy, and a Characterization of the Hyperbolic Space

Harmonic Functions, Entropy, and a Characterization of the Hyperbolic Space

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Xiaodong Wang, Michigan State University
Fine Hall 314

Complete Riemannian manifolds with nonnegative Ricci curvature have been well studied. Riemannian manifolds with a negative lower bound for Ricci curvature are considerably more complicated and less understood. I will first survey some recent results on such manifolds with positive bottom of spectrum. Then I will discuss a rigidity theorem which characterizes hyperbolic manifolds. The proof uses idea from potential theory and Brownian motion on Riemannian manifolds