Improved geodesics for the reduced curvature-dimension condition in branching metric spaces

Tapio Rajala

doi:10.3934/dcds.2013.33.3043

Article Contents

2013, Volume 33, Issue 7: 3043-3056. Doi: 10.3934/dcds.2013.33.3043

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Improved geodesics for the reduced curvature-dimension condition in branching metric spaces

Tapio Rajala^1,

1.
Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56127 Pisa

Received: February 29, 2012

Revised: February 29, 2012

Published: June 2013

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Abstract

In this note we show that in metric measure spaces satisfying the reduced curvature-dimension condition $CD^*(K,N)$ we always have geodesics in the Wasserstein space of probability measures that satisfy the critical convexity inequality of $CD^*(K,N)$ also for intermediate times and in addition the measures along these geodesics have an upper-bound on their densities. This upper-bound depends on the bounds for the densities of the end-point measures, the lower-bound $K$ for the Ricci-curvature, the upper-bound $N$ for the dimension, and on the diameter of the union of the supports of the end-point measures.

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Mathematics Subject Classification: Primary: 53C23; Secondary: 28A33, 49Q20.

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