scholarly journals Lower bound of coarse Ricci curvature on metric measure spaces and eigenvalues of Laplacian

2013 ◽  
Vol 169 (1) ◽  
pp. 99-107 ◽  
Author(s):  
Yu Kitabeppu
Keyword(s):  
Lower Bound ◽  
Ricci Curvature ◽  
Metric Measure Spaces ◽  
Measure Spaces ◽  
Coarse Ricci Curvature

scholarly journals Eigenvalues of the drifted Laplacian on complete metric measure spaces

2016 ◽  
Vol 19 (01) ◽  
pp. 1650001 ◽  
Author(s):  
Xu Cheng ◽  
Detang Zhou
Keyword(s):  
Lower Bound ◽  
Measure Space ◽  
Logarithmic Sobolev Inequality ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Dimensional Manifold ◽  
Metric Measure Spaces ◽  
Smooth Metric Measure Space ◽  
Measure Spaces ◽  
Multiplicity Formula

In this paper, first we study a complete smooth metric measure space [Formula: see text] with the ([Formula: see text])-Bakry–Émery Ricci curvature [Formula: see text] for some positive constant [Formula: see text]. It is known that the spectrum of the drifted Laplacian [Formula: see text] for [Formula: see text] is discrete and the first nonzero eigenvalue of [Formula: see text] has lower bound [Formula: see text]. We prove that if the lower bound [Formula: see text] is achieved with multiplicity [Formula: see text], then [Formula: see text], [Formula: see text] is isometric to [Formula: see text] for some complete [Formula: see text]-dimensional manifold [Formula: see text] and by passing an isometry, [Formula: see text] must split off a gradient shrinking Ricci soliton [Formula: see text], [Formula: see text]. This result can be applied to gradient shrinking Ricci solitons. Secondly, we study the drifted Laplacian [Formula: see text] for properly immersed self-shrinkers in the Euclidean space [Formula: see text], [Formula: see text] and show the discreteness of the spectrum of [Formula: see text] and a logarithmic Sobolev inequality.

scholarly journals Distribution-Valued Ricci Bounds for Metric Measure Spaces, Singular Time Changes, and Gradient Estimates for Neumann Heat Flows

2020 ◽  
Vol 30 (6) ◽  
pp. 1648-1711
Author(s):  
Karl-Theodor Sturm
Keyword(s):  
Lower Bound ◽  
Second Fundamental Form ◽  
Gradient Estimates ◽  
List Type ◽  
Metric Measure Spaces ◽  
Heat Flows ◽  
Measure Spaces ◽  
Neumann Laplacian ◽  
Boundary Curvature ◽  
Time Changes

AbstractWe will study metric measure spaces $$(X,\mathsf{d},{\mathfrak {m}})$$ ( X , d , m ) beyond the scope of spaces with synthetic lower Ricci bounds. In particular, we introduce distribution-valued lower Ricci bounds $$\mathsf{BE}_1(\kappa ,\infty )$$ BE 1 ( κ , ∞ ) for which we prove the equivalence with sharp gradient estimates, the class of which will be preserved under time changes with arbitrary $$\psi \in \mathrm {Lip}_b(X)$$ ψ ∈ Lip b ( X ) , and which are satisfied for the Neumann Laplacian on arbitrary semi-convex subsets $$Y\subset X$$ Y ⊂ X . In the latter case, the distribution-valued Ricci bound will be given by the signed measure $$\kappa = k\,{\mathfrak {m}}_Y + \ell \,\sigma _{\partial Y}$$ κ = k m Y + ℓ σ ∂ Y where k denotes a variable synthetic lower bound for the Ricci curvature of X and $$\ell $$ ℓ denotes a lower bound for the “curvature of the boundary” of Y, defined in purely metric terms. We also present a new localization argument which allows us to pass on the RCD property to arbitrary open subsets of RCD spaces. And we introduce new synthetic notions for boundary curvature, second fundamental form, and boundary measure for subsets of RCD spaces.

scholarly journals Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma $-finite measure

2015 ◽  
Vol 367 (7) ◽  
pp. 4661-4701 ◽  
Author(s):  
Luigi Ambrosio ◽  
Nicola Gigli ◽  
Andrea Mondino ◽  
Tapio Rajala
Keyword(s):  
Ricci Curvature ◽  
Lower Bounds ◽  
Finite Measure ◽  
Metric Measure Spaces ◽  
Measure Spaces

scholarly journals On interpolation and curvature via Wasserstein geodesics

2017 ◽  
Vol 10 (2) ◽  
pp. 125-167 ◽  
Author(s):  
Martin Kell
Keyword(s):  
Ricci Curvature ◽  
Comparison Theorem ◽  
Similar Condition ◽  
Interpolation Inequality ◽  
Metric Measure Spaces ◽  
Main Ingredient ◽  
Curvature Condition ◽  
Finsler Manifolds ◽  
Measure Spaces ◽  
Positively Curved

AbstractIn this article, a proof of the interpolation inequality along geodesics in p-Wasserstein spaces is given. This interpolation inequality was the main ingredient to prove the Borel–Brascamp–Lieb inequality for general Riemannian and Finsler manifolds and led Lott–Villani and Sturm to define an abstract Ricci curvature condition. Following their ideas, a similar condition can be defined and for positively curved spaces one can prove a Poincaré inequality. Using Gigli’s recently developed calculus on metric measure spaces, even a q-Laplacian comparison theorem holds on q-infinitesimal convex spaces. In the appendix, the theory of Orlicz–Wasserstein spaces is developed and necessary adjustments to prove the interpolation inequality along geodesics in those spaces are given.

scholarly journals Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds

2016 ◽  
Vol 208 (3) ◽  
pp. 803-849 ◽  
Author(s):  
Fabio Cavalletti ◽  
Andrea Mondino
Keyword(s):  
Ricci Curvature ◽  
Isoperimetric Inequalities ◽  
Metric Measure Spaces ◽  
Curvature Bounds ◽  
Measure Spaces

scholarly journals Inscribed Radius Bounds for Lower Ricci Bounded Metric Measure Spaces with Mean Convex Boundary

2020 ◽  
Author(s):  
Annegret Burtscher ◽  
◽  
Christian Ketterer ◽  
Robert J. McCann ◽  
Eric Woolgar ◽  
...  
Keyword(s):  
Lower Bound ◽  
Mean Curvature ◽  
Measure Space ◽  
Riemannian Curvature ◽  
Metric Measure Spaces ◽  
Contraction Property ◽  
Measure Spaces ◽  
Generalized Mean Curvature ◽  
Measure Contraction Property ◽  
Mean Convex

Consider an essentially nonbranching metric measure space with the measure contraction property of Ohta and Sturm, or with a Ricci curvature lower bound in the sense of Lott, Sturm and Villani. We prove a sharp upper bound on the inscribed radius of any subset whose boundary has a suitably signed lower bound on its generalized mean curvature. This provides a nonsmooth analog to a result of Kasue (1983) and Li (2014). We prove a stability statement concerning such bounds and - in the Riemannian curvature-dimension (RCD) setting - characterize the cases of equality.

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