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- Emery Ricci tensor Ric(f) >= -(n - 1) in a geodesic ball B-o(R) with radius R and center o is an element of M-n. When Ric(f) >= 0 and f has some degeneration outside B-o(R), we show that there exists an epsilon = epsilon(n, sup(Bo(1)) vertical bar f vertical bar) such that such a space has at most two ends if R <= epsilon. When Ric(f) >= 1/2 and f(x) <= 1/4 d(2)(x, B-o(R))+ c for some constant c > 0 outside B-o(R), we can also get the same gap conclusion.

Automated summary

In this paper, two gap theorems are established for the ends of smooth metric measure space with specific conditions, providing insights into the boundary properties of these spaces.