Bounds on the Volume Entropy and Simplicial Volume in Ricci Curvature Lp-Bounded from Below

Comparison Geometry for Integral Bakry–Émery Ricci Tensor Bounds

“…It is easy to see that Ric H − p (R) ≡ 0 if and only if Ric ≥ (n − 1)H. Under certain assumption of Ric H − p (R), Petersen and Wei [15,16] generalized classical comparison theorems to the integral case. For more related results, the reader can refer to [1,2,5,6,7,8,13,14,17,23] and references therein.…”

“…From the work of [1,2,13] we expect Aubry's type diameter estimate, finiteness fundamental group theorem, first Betti number estimate and Gromov's bounds on the volume entropy in the integral sense can be generalized to smooth metric measure spaces. These will be treated in separate paper.…”

“…He also got finite fundamental group theorem in the integral Ricci tensor sense. For more results, see for example [1,2,8,9,10,11,19,23].…”

Almost maximal volume entropy rigidity for integral Ricci curvature in the non-collapsing case

“…If M has integral Ricci curvature bound, then by [2], the universal cover M also has integral Ricci curvature bound:…”

“…Theorem 2.3 ([16, 2] Precompactness). For n ≥ 2, p > n 2 , H, there exists c(n, p, H) such that if a sequence of compact Riemannian n-manifold M i satisfies that diam(M ) 2 ki (H, p) ≤ c(p, n, H), then there are subsequences of {(M i , x i )} and {( Mi , xi )} that converge in the pointed Gromov-Hausdorff topology where Mi is the universal cover of M i .…”