HARMONIC FUNCTIONS AND END NUMBERS ON SMOOTH METRIC MEASURE SPACES

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^-f dv). We prove that any simply connected, negatively curved smooth metric measure space with a small bound of |∇f| admits a unique f-harmonic function for a given boundary value at infinity. We also prove a sharp L²_f-decay estimate for a Schrödinger 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-Émery Ricci tensor is bounded below by certain positive spectrum. Finally we study the dimension of the first L²_f-cohomology of the smooth metric measure space.