Myers’ type theorem with the Bakry–Émery Ricci tensor

Abstract

In this paper we prove a new Myers' type diameter estimate on a complete connected Reimannian manifold which admits a bounded vector field such that the Bakry-Émery Ricci tensor has a positive lower bound. The result is sharper than previous Myers' type results. The proof uses the generalized mean curvature comparison applied to the excess function instead of the classical second variation of geodesics.Ric + Hess f = λ g, where Hess is the Hessian of the metric g. By Perelman [19], any compact Ricci soliton is necessarily a gradient soliton; by Hamilton [9] and Ivey [10], any compact gradient non-shrinking Ricci soliton is Einstein. Hence, any compact non-shrinking Ricci soliton must be Einstein. A compact shrinking Ricci soliton, when the dimension of manifold is two or three, is also Einstein

Integral curvature bounds and bounded diameter with Bakry–Emery Ricci tensor

“…Several works have attempted to generalize this result (for example, see [4], [5], [9], and [10]), including that of Sprouse which can be summarized in the following three theorems.…”

Some Myers-Type Theorems and Comparison Theorems for Manifolds with Modified Ricci Curvature

“…Using the same arguments as in the prove of theorem 2.2 in [9], we get e ar A k (r,θ) is nonincreasing in r. Using the lemma 3 in [1] and integrating along the unit sphere, we get, for 0 < r ≤ R…”

Comparison Geometry for Integral Bakry–Émery Ricci Tensor Bounds

“…First, we will apply mean curvature comparisons of Section 2 to prove Theorem 1.3. The proof uses the excess function which is similar to the Wei-Wylie's argument [20]; see also [21]. and let e(x) be the excess function for the points p 1 and p 2 , that is,…”