Rigidity of the 1-Bakry-Émery inequality and sets of finite Perimeter in RCD spaces

DOI10.1007/s00039-019-00504-5zbMath1431.53046arXiv1812.07890OpenAlexW2963771020MaRDI QIDQ2317433

Elia Bruè, Daniele Semola, Luigi Ambrosio

Publication date: 9 August 2019

Published in: Geometric and Functional Analysis. GAFA (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1812.07890

zbMATH Keywords

Ricci curvaturegeometric measure theoryfinite perimeterRCD spaces

Mathematics Subject Classification ID

Geometric measure and integration theory, integral and normal currents in optimization (49Q15) Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces (53C23) Potential theory on fractals and metric spaces (31E05)

Related Items (24)

Boundary regularity and stability for spaces with Ricci bounded below ⋮ The isoperimetric problemviadirect method in noncompact metric measure spaces with lower Ricci bounds ⋮ Besov class via heat semigroup on Dirichlet spaces. II: BV functions and Gaussian heat kernel estimates ⋮ Constancy of the dimension in codimension one and locality of the unit normal on $\RCD(K,N)$ spaces ⋮ Calculus and fine properties of functions of bounded variation on RCD spaces ⋮ Monotonicity formulas for harmonic functions in \(\mathrm{RCD}(0,N)\) spaces ⋮ Subgraphs of BV functions on RCD spaces ⋮ About the general chain rule for functions of bounded variation ⋮ The metric measure boundary of spaces with Ricci curvature bounded below ⋮ On clusters and the multi-isoperimetric profile in Riemannian manifolds with bounded geometry ⋮ Obstacle problems on \(\mathrm{RCD}(K, N)\) metric measure spaces ⋮ The isoperimetric problem on Riemannian manifolds via Gromov–Hausdorff asymptotic analysis ⋮ Rectifiability of the reduced boundary for sets of finite perimeter over \(\operatorname{RCD} ( K , N )\) spaces ⋮ Rigidity of some functional inequalities on RCD spaces ⋮ Indecomposable sets of finite perimeter in doubling metric measure spaces ⋮ Splitting theorems on complete Riemannian manifolds with nonnegative Ricci curvature ⋮ The equality case in Cheeger's and Buser's inequalities on \(\mathsf{RCD}\) spaces ⋮ Rough traces of BV functions in metric measure spaces ⋮ Indeterminacy estimates and the size of nodal sets in singular spaces ⋮ Equality in the logarithmic Sobolev inequality ⋮ Quantitative estimates for the Bakry-Ledoux isoperimetric inequality ⋮ Volume bounds for the quantitative singular strata of non collapsed RCD metric measure spaces ⋮ Characterising rectifiable metric spaces using tangent spaces ⋮ On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth

Cites Work

• Unnamed Item
• Unnamed Item
• Independence on \(p\) of weak upper gradients on \(\mathsf{RCD}\) spaces
• Heat kernel bounds on metric measure spaces and some applications
• Well-posedness of Lagrangian flows and continuity equations in metric measure spaces
• Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds
• An overview of the proof of the splitting theorem in spaces with non-negative Ricci curvature
• Euclidean spaces as weak tangents of infinitesimally Hilbertian metric measure spaces with Ricci curvature bounded below
• On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces
• Geometry of measures in \(R^ n:\) Distribution, rectifiability, and densities
• From volume cone to metric cone in the nonsmooth setting
• Nuovi teoremi relativi alle misure \((r - 1)\)-dimensionali in uno spazio ad \(r\) dimensioni
• Functions of bounded variation on ``good metric spaces
• Characterization of \(n\)-rectifiability in terms of Jones' square function. II
• A new proof of the compactness theorem for integral currents
• Differentiability of Lipschitz functions on metric measure spaces
• Fine properties of sets of finite perimeter in doubling metric measure spaces
• Short-time behavior of the heat kernel and Weyl's law on \(\mathrm{RCD}^*(K,N)\) spaces
• Lipschitz algebras and derivations. II: Exterior differentiation
• Analysis on local Dirichlet spaces. III: The parabolic Harnack inequality
• Behaviour of the reference measure on \(\mathsf{RCD}\) spaces under charts
• Equivalence of two different notions of tangent bundle on rectifiable metric measure spaces
• The globalization theorem for the curvature-dimension condition
• Equivalent definitions of \(BV\) space and of total variation on metric measure spaces
• Rectifiability of sets of finite perimeter in Carnot groups: existence of a tangent hyperplane
• Ricci curvature for metric-measure spaces via optimal transport
• Structure theory of metric measure spaces with lower Ricci curvature bounds
• On a class of first order Hamilton-Jacobi equations in metric spaces
• Metric measure spaces with Riemannian Ricci curvature bounded from below
• On local Poincaré via transportation
• On the geometry of metric measure spaces. I
• On the geometry of metric measure spaces. II
• Normal and integral currents
• Su una teoria generale della misura \((r-1)\)-dimensionale in uno spazio ad \(r\) dimensioni
• Alexandrov meets Lott--Villani--Sturm
• On metric viscosity solutions for Hamilton-Jacobi equations of evolution type
• On the differential structure of metric measure spaces and applications
• Metric spaces with unique tangents
• On a measure-theoretic area formula
• Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows
• Non-collapsed spaces with Ricci curvature bounded from below
• On the volume measure of non-smooth spaces with Ricci curvature bounded below
• Nonsmooth differential geometry– An approach tailored for spaces with Ricci curvature bounded from below
• New stability results for sequences of metric measure spaces with uniform Ricci bounds from below
• On a conjecture of Cheeger
• Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows
• CALCULUS, HEAT FLOW AND CURVATURE-DIMENSION BOUNDS IN METRIC MEASURE SPACES
• Asymptotic behavior of $BV$ functions and sets of finite perimeter in metric measure spaces
• Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces
• Riemannian Ricci curvature lower bounds in metric measure spaces with 𝜎-finite measure
• Flat Chains Over a Finite Coefficient Group
• Optimal Transport
• Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces
• Examples of strictly weakly regular points
• Characterization of \(n\)-rectifiability in terms of Jones' square function. I

This page was built for publication: Rigidity of the 1-Bakry-Émery inequality and sets of finite Perimeter in RCD spaces