Gap theorems for ends of smooth metric measure spaces

TL;DR

This paper proves new gap theorems for the number of ends of smooth metric measure spaces under certain Ricci curvature and potential function conditions, revealing geometric constraints based on curvature bounds and function behavior.

Contribution

It establishes two novel gap theorems for ends of smooth metric measure spaces with specific Ricci curvature bounds and potential function conditions, extending understanding of their geometric structure.

Findings

Spaces with Ric_f ≥ 0 and degenerate f outside a ball have at most two ends if the radius is sufficiently small.

Spaces with Ric_f ≥ 1/2 and controlled quadratic growth of f outside a ball also have at most two ends under certain conditions.

The results connect curvature bounds, potential function behavior, and the topology of the space's ends.

Abstract

In this paper, we establish two gap theorems for ends of smooth metric measure space $(M^n, g,e^{-f}dv)$ with the Bakry-\'Emery Ricci tensor $\mathrm{Ric}_f\ge-(n-1)$ in a geodesic ball $B_o(R)$ with radius $R$ and center $o\in M^n$. When $\mathrm{Ric}_f\ge 0$ 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 $\mathrm{Ric}_f\ge\frac 12$ 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.