- 6635 South Hall
Abstract: Bishop-Gromov's volume comparison has been a very useful tool for studying manifolds with Ricci curvature lower bound. Its essential information is contained in the monotonicity of a volume ratio. Perelman obtained several monotonicity formulas along the Ricci flow which are key tools in his work on the geometrization conjecture. Recently Colding and Minicozzi-Colding introduced some monotonicity formulas associated with harmonic functions. Joint with B. Song and G. Wu we extend and generalize these to Bakry-Emery Ricci curvature.