Harmonic functions and end numbers on smooth metric measure spaces

Abstract

In this paper, we study properties of functions on smooth metric measure space $(M,g,e^{-f}dv)$. We prove that any simply connected, negatively curved smooth metric measure space with a small bound of $|\nabla f|$ admits a unique $f$-harmonic function for a given boundary value at infinity. We also prove a sharp $L_f^2$-decay estimate for a Schr\"odinger equation under certain positive spectrum. As applications, we discuss the number of ends on smooth metric measure spaces. We show that the space with finite $f$-volume has a finite number of ends when the Bakry-\'Emery Ricci tensor and the bottom of Neumann spectrum satisfy some lower bounds. We also show that the number of ends with infinite $f$-volume is finite when the Bakry-\'Emery Ricci tensor is bounded below by certain positive spectrum. Finally we study the dimension of the first $L^2_f$-cohomology of the smooth metric measure space.