On the differential structure of metric measure spaces and applications

DOI10.1090/memo/1113zbMath1325.53054arXiv1205.6622OpenAlexW2963600169MaRDI QIDQ2944995

Nicola Gigli

Publication date: 9 September 2015

Published in: Memoirs of the American Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1205.6622

zbMATH Keywords

differential geometrymetric geometrylower Ricci bounds

Mathematics Subject Classification ID

Geometric measure and integration theory, integral and normal currents in optimization (49Q15) Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) (58J60) Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces (53C23) Analysis on metric spaces (30L99)

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Cites Work

• Unnamed Item
• Unnamed Item
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• Unnamed Item
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• Unnamed Item
• Unnamed Item
• Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces
• Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in \(\text{RCD}(K, \infty)\) metric measure spaces
• A PDE approach to nonlinear potential theory in metric measure spaces
• Local Poincaré inequalities from stable curvature conditions on metric spaces
• Local curvature-dimension condition implies measure-contraction property
• Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm
• Nonlinear potential theory on metric spaces
• Optimal maps in non branching spaces with Ricci curvature bounded from below
• Characterization of absolutely continuous curves in Wasserstein spaces
• Weak curvature conditions and functional inequalities
• Duality on gradient estimates and Wasserstein controls
• Localization and tensorization properties of the curvature-dimension condition for metric measure spaces
• Finsler interpolation inequalities
• Differentiability of Lipschitz functions on metric measure spaces
• The Riemannian structure of Alexandrov spaces
• On the structure of spaces with Ricci curvature bounded below. I
• On the structure of spaces with Ricci curvature bounded below. II
• On the structure of spaces with Ricci curvature bounded below. III
• Newtonian spaces: An extension of Sobolev spaces to metric measure spaces
• Lipschitz algebras and derivations. II: Exterior differentiation
• Equivalent definitions of \(BV\) space and of total variation on metric measure spaces
• Splitting theorems for Finsler manifolds of nonnegative Ricci curvature
• Non-contraction of heat flow on Minkowski spaces
• Ricci curvature for metric-measure spaces via optimal transport
• Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
• Bochner-Weitzenböck formula and Li-Yau estimates on Finsler manifolds
• Metric measure spaces with Riemannian Ricci curvature bounded from below
• On the geometry of metric measure spaces. I
• On the geometry of metric measure spaces. II
• Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds
• Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon-Nikodym property
• Local semiconvexity of Kantorovich potentials on non-compact manifolds
• Heat flow on Finsler manifolds
• Nonsmooth calculus
• Harmonic functions on Alexandrov spaces and their applications
• Heat Flow on Alexandrov Spaces
• Riemannian Ricci curvature lower bounds in metric measure spaces with 𝜎-finite measure
• Products, cones, and suspensions of spaces with the measure contraction property
• Optimal Transport
• On the heat flow on metric measure spaces: existence, uniqueness and stability

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