Journal of the Mathematical Society of Japan

Long-time existence of the edge Yamabe flow


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This article presents an analysis of the normalized Yamabe flow starting at and preserving a class of compact Riemannian manifolds with incomplete edge singularities and negative Yamabe invariant. Our main results include uniqueness, long-time existence and convergence of the edge Yamabe flow starting at a metric with everywhere negative scalar curvature. Our methods include novel maximum principle results on the singular edge space without using barrier functions. Moreover, our uniform bounds on solutions are established by a new ansatz without in any way using or redeveloping Krylov–Safonov estimates in the singular setting. As an application we obtain a solution to the Yamabe problem for incomplete edge metrics with negative Yamabe invariant using flow techniques. Our methods lay groundwork for studying other flows like the mean curvature flow as well as the porous medium equation in the singular setting.

Article information

J. Math. Soc. Japan, Volume 71, Number 2 (2019), 651-688.

Received: 1 June 2017
Revised: 28 December 2017
First available in Project Euclid: 8 March 2019

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Zentralblatt MATH identifier

Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 58J35: Heat and other parabolic equation methods 35K08: Heat kernel

incomplete edge metrics Yamabe flow long-time existence


BAHUAUD, Eric; VERTMAN, Boris. Long-time existence of the edge Yamabe flow. J. Math. Soc. Japan 71 (2019), no. 2, 651--688. doi:10.2969/jmsj/78147814.

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