Gap theorems for ends of smooth metric measure spaces.

In this paper, we establish two gap theorems for ends of smooth metric measure space (M^n, g,e^{-f}dv) with the Bakry-Émery Ricci tensor \operatorname {Ric}_{f}\!\ge -(n-1) in a geodesic ball B_{o}(R) with radius R and center o\in M^n. When \operatorname {Ric}_{f}\ge 0 and f has some degeneration outside B_{o}(R), we show that there exists an \epsilon =\epsilon (n,\sup _{B_{o}(1)}|f|) such that such a space has at most two ends if R\le \epsilon. When \operatorname {Ric}_{f}\ge \frac 12 and f(x)\le \frac 14d^2(x,B_{o}(R))+c for some constant c>0 outside B_{o}(R), we can also get the same gap conclusion. [ABSTRACT FROM AUTHOR]