Comparison Geometry for Integral Bakry–Émery Ricci Tensor Bounds

Abstract

We prove mean curvature and volume comparison estimates on smooth metric measure spaces when their integral Bakry-Émery Ricci tensor bounds, extending Wei-Wylie's comparison results to the integral case. We also apply comparison results to get diameter estimates, eigenvalue estimates and volume growth estimates on smooth metric measure spaces with their normalized integral smallness for Bakry-Émery Ricci tensor. These give generalizations of some work of Petersen-Wei, Aubry, Petersen-Sprouse, Yau and more.

Local Sobolev constant estimate for integral Bakry–Émery Ricci curvature

“…Wei-Wylie [21] has proved Ric f − ≥ 0 yields ∆ f r ≤ n−1 r + a, that is Ric f − p,f,a (r) ≡ 0 implies that ψ ≡ 0. In [22], this has been extended to integral Bakry-Émery Ricci curvature bound following the work of Petersen-Wei [16]…”

“…Remark 2.2. In fact, the proof of [22,Theorem 1.1] gives the following normalized form of Laplacian comparison…”

“…Remark 2.3. Here we choose the power 2p rather than 2p − 1, so the explicit expression of C(n, p, a, r 2 ) similar to the one in [16, Lemma 2.1] rather than the one in [22]. If we denote the volume of (n − 1)-dimensional unit ball in R n and the weighted volume of geodesic sphere ∂B 0 (t) by ω n and A(n, a, t), respectively.…”

“…Recently, Wu [22] extend the volume comparison in [16] to integral Bakry-Émery Ricci curvature case. In this paper we extend the local isoperimetric constant estimate in [10] to integral Bakry-Émery Ricci curvature and give applications.…”

Comparison geometry for integral generalized quasi-Einstein tensor bounds

“…On the other hand, Wu generalized Petersen, Sprouse, and Wei results [11,12,13] using Bakry-Emery Ricci tensor. In addition, Wu proved generalized Myers theorem, relative volume comparison for annulus, eigenvalue estimate, and volume growth estimate in [17].…”