Comparison Geometry of Manifolds with Boundary under a Lower Weighted Ricci Curvature Bound

Abstract

We study Riemannian manifolds with boundary under a lower weighted Ricci curvature bound. We consider a curvature condition in which the weighted Ricci curvature is bounded from below by the density function. Under the curvature condition, and a suitable condition for the weighted mean curvature for the boundary, we obtain various comparison geometric results.

Geometric and spectral properties of directed graphs under a lower Ricci curvature bound

“…Let ρ ∂M : M → R denote the distance function from ∂M defined as ρ ∂M := d g (∂M, •), which is smooth on Int M \ Cut ∂M. Here Cut ∂M is the cut locus for the boundary (for its precise definition, see e.g., Subsection 2.3 in [33]). For z ∈ ∂M, the weighted mean curvature of ∂M at z is defined as…”

“…✷ Remark 9.7. On Riemannian manifolds with boundary with a lower Ricci curvature bound and a lower mean curvature bound for the boundary, it is well-known that one can derive a lower bound of its Dirichlet isoperimetric constant from a Laplacian comparison theorem for the distance function from the boundary, and integration parts formula (see Proposition 4.1 in [21], Lemma 8.9 in [33], and cf. Theorem 15.3.5 in [35]).…”

One dimensional weighted Ricci curvature and displacement convexity of entropies

“…Li and Xia [10] have produced formulas of Bochner type and Reilly type with respect to Ric ∇ 𝛼 . The author [24] has studied comparison geometry of manifolds with boundary under the curvature condition (1.2).…”

Lower $N$-weighted Ricci curvature bound with $\varepsilon$-range and displacement convexity of entropies

“…On Corollaries 5.3 and 5.4, the authors do not know whether the assumption that µ is m-constant can be dropped. Under the curvature condition (1.1), similar functional inequalities are known to be useful to analyze the gradient flow of entropy functionals (see e.g., [29,Chapters 23,24,25]). There might be some applications of our inequalities to the analysis of such gradient flow under the curvature condition (1.5).…”

“…In recent years, the validity of the N-weighted Ricci curvature with N ∈] − ∞, n[ has begun to be pointed out (see e.g., [5], [6], [7], [8], [9], [11], [14], [15], [16], [18], [20], [21], [22], [23], [25], [26], [31], [32]). Wylie-Yeroshkin [32] have proposed a curvature condition…”