On the structure of \mathrm RCD spaces with upper curvature bounds

2022/04/20 by Vitali Kapovitch, Martin Kell, Christian Ketterer +1
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Advanced Differential Geometry Research #Geometry and complex manifolds

paper · pdf · doi:10.1007/s00209-022-03015-6

Abstract

Abstract We develop a structure theory for \mathrm RCD <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>RCD</mml:mi> </mml:math> spaces with curvature bounded above in Alexandrov sense. In particular, we show that any such space is a topological manifold with boundary whose interior is equal to the set of regular points. Further the set of regular points is a smooth manifold and is geodesically convex. Around regular points there are \mathrm DC <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>DC</mml:mi> </mml:math> coordinates and the distance is induced by a continuous \mathrm BV <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>BV</mml:mi> </mml:math> Riemannian metric.

Citations

• Ricci curvature for metric-measure spaces via optimal transport
• Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications
• Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds
• A Bishop-type inequality on metric measure spaces with Ricci curvature bounded below
• Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows

Related

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