A Bishop-type inequality on metric measure spaces with Ricci curvature bounded below

2016/11/03 by Yu Kitabeppu · 2 citations
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Advanced Differential Geometry Research

paper · pdf · doi:10.1090/proc/13517

Abstract

We define a Bishop-type inequality on metric measure spaces with Riemannian curvature-dimension condition. The main result in this short article is that any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R upper C upper D"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mi>C</mml:mi> <mml:mi>D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">RCD</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spaces with the Bishop-type inequalities possess only one regular set in not only the measure theoretical sense but also the set theoretical one. As a corollary, the Hausdorff dimension of such <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R upper C upper D Superscript asterisk Baseline left-parenthesis upper K comma upper N right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mi>C</mml:mi> <mml:msup> <mml:mi>D</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">RCD^*(K,N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spaces is exactly <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We also prove that every tangent cone at any point on such <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R upper C upper D"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mi>C</mml:mi> <mml:mi>D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">RCD</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spaces is a metric cone.

Citations

• Ricci curvature for metric-measure spaces via optimal transport
• On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces
• Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications

Cited by

• On the structure of \mathrm RCD spaces with upper curvature bounds
• A sufficient condition to a regular set of positive measure on RCD spaces

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