| In this dissertation we explore smooth metric measure spaces with nonnegative Bakry-Emery Ricci tensor. In particular, we prove a Liouville-type theorem for smooth metric measure spaces (M, g, e−f dvol ) with nonnegative Bakry-Emery Ricci tensor. This generalizes a result of Yau, which is recovered in the case f is constant. This result follows from a gradient estimate for f-harmonic functions on smooth metric measure spaces with Bakry-Emery Ricci tensor bounded from below. We also extend Zhong-Yang's first eigenvalue estimate to smooth metric measure spaces with nonnegative Bakry-Emery Ricci tensor and explore some aspects of a first eigenvalue estimate combining the Zhong-Yang and Lichnerowicz estimates. |
