HARMONIC FUNCTIONS AND END NUMBERS ON SMOOTH METRIC MEASURE SPACES

. In this paper, we study properties of functions on smooth metric measure space (M, g, e-fdv). We prove that any simply connected, negatively curved smooth metric measure space with a small bound of |del f| admits a unique f-harmonic function for a given boundary value at infinity. We also prove a sharp L2f-decay estimate for a Schro<spacing diaeresis>dinger 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-E<acute accent>mery 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-E<acute accent>mery Ricci tensor is bounded below by certain positive spectrum. Finally we study the dimension of the first L2f-cohomology of the smooth metric measure space.