The Ricci flow on manifolds with boundary

2012/10/02 by Panagiotis Gianniotis, Gianniotis, Panagiotis
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG

paper · pdf · doi:10.48550/arxiv.1210.0813

Improved exposition and statement on the control of the existence time of the Ricci flow, and updated/added some references. Also, a new section is added, demonstrating a situation in which the existence time of the Ricci flow (with boundary) with flat initial data and well behaved boundary data may become arbitrarily small

arxiv created 2015/04/11 · arxiv updated 2015/04/14

Abstract

We study the short-time existence and regularity of solutions to a boundary value problem for the Ricci-DeTurck equation on a manifold with boundary. Using this, we prove the short-time existence and uniqueness of the Ricci flow prescribing the mean curvature and conformal class of the boundary, with arbitrary initial data. Finally, we establish that under suitable control of the boundary data the flow exists as long as the ambient curvature and the second fundamental form of the boundary remain bounded.

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