## Journal of the Mathematical Society of Japan

### Long-time existence of the edge Yamabe flow

#### Abstract

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

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

Dates
Revised: 28 December 2017
First available in Project Euclid: 8 March 2019

https://projecteuclid.org/euclid.jmsj/1552035633

Digital Object Identifier
doi:10.2969/jmsj/78147814

Mathematical Reviews number (MathSciNet)
MR3943455

Zentralblatt MATH identifier
07090060

#### Citation

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. https://projecteuclid.org/euclid.jmsj/1552035633

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