Gap theorems for ends of smooth metric measure spaces

Author:

Hua Bobo,Wu Jia-Yong

Abstract

In this paper, we establish two gap theorems for ends of smooth metric measure space ( M n , g , e f d v ) (M^n, g,e^{-f}dv) with the Bakry-Émery Ricci tensor Ric f ( n 1 ) \operatorname {Ric}_{f}\!\ge -(n-1) in a geodesic ball B o ( R ) B_{o}(R) with radius R R and center o M n o\in M^n . When Ric f 0 \operatorname {Ric}_{f}\ge 0 and f f has some degeneration outside B o ( R ) B_{o}(R) , we show that there exists an ϵ = ϵ ( n , sup B o ( 1 ) | f | ) \epsilon =\epsilon (n,\sup _{B_{o}(1)}|f|) such that such a space has at most two ends if R ϵ R\le \epsilon . When Ric f 1 2 \operatorname {Ric}_{f}\ge \frac 12 and f ( x ) 1 4 d 2 ( x , B o ( R ) ) + c f(x)\le \frac 14d^2(x,B_{o}(R))+c for some constant c > 0 c>0 outside B o ( R ) B_{o}(R) , we can also get the same gap conclusion.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference22 articles.

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