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Metric Perspectives of the Ricci Flow Applied to Disjoint Unions

100%

Lakzian S., Munn M.

Analysis and Geometry in Metric Spaces

2014

tom 2

nr 1

In this paper we consider compact, Riemannian manifolds M1, M2 each equipped with a oneparameter family of metrics g1(t), g2(t) satisfying the Ricci flow equation. Adopting the characterization of super-solutions to the Ricci flow developed by McCann-Topping, we define a super Ricci flow for a family of distance metrics defined on the disjoint union M1 ⊔ M2. In particular, we show such a super Ricci flow property holds provided the distance function between points in M1 and M2 is itself a super solution of the heat equation on M1 × M2. We also discuss possible applications and examples.

Characterization of Low Dimensional RCD*(K, N) Spaces

100%

Kitabeppu Y., Lakzian S.

Analysis and Geometry in Metric Spaces

2016

tom 4

nr 1

In this paper,we give the characterization of metric measure spaces that satisfy synthetic lower Riemannian Ricci curvature bounds (so called RCD*(K, N) spaces) with non-empty one dimensional regular sets. In particular, we prove that the class of Ricci limit spaces with Ric ≥ K and Hausdorff dimension N and the class of RCD*(K, N) spaces coincide for N < 2 (They can be either complete intervals or circles). We will also prove a Bishop-Gromov type inequality (that is ,roughly speaking, a converse to the Lévy-Gromov’s isoperimetric inequality and was previously only known for Ricci limit spaces) which might be also of independent interest.

Ograniczanie wyników

2 Analysis and Geometry in Metric Spaces

2 Lakzian S.

1 Kitabeppu Y.

1 Munn M.

1 2016

1 2014

Wyniki wyszukiwania

Metric Perspectives of the Ricci Flow Applied to Disjoint Unions

Characterization of Low Dimensional RCD*(K, N) Spaces