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Gluing methods for the Yamabe problem with isolated singularities

Banff International Research Station for Mathematical Innovation and Discovery

We construct some solutions for the fractional Yamabe problem with isolated singularities, problem which arises in conformal geometry, $$ (-\Delta)^\gamma u= c_{n, {\gamma}}u^{\frac{n+2\gamma}{n-2\gamma}}, u>0 \ \mbox{in}\ {\mathbb R}^n \backslash \Sigma.$$ The fractional curvature, a generalization of the usual scalar curvature, is defined from the conformal fractional Laplacian, which is a non-local operator constructed on the conformal infinity of a conformally compact Einstein manifold. When the singular set $\Sigma$ is composed by one point, some new tools for fractional order ODE can be applied to show that a generalization of the usual Delaunay solves the fractional Yamabe problem with an isolated singularity at $\Sigma$. When the set $\Sigma$ is a finite number of points, using gluing methods, we will provide a solution for the fractional Yamabe problem with singularities at $\Sigma$. In order to preserve the non-locality of the problem, we need to glue infinitely many bubbles per point removed. This seems to be the first time that a gluing method is successfully applied to a non-local problem. This is a joint work with Weiwei Ao, Mar\'ia del Mar Gonz\'alez and Juncheng Wei.

Mathematics; Partial differential equations; Differential geometry

video/mp4

Author affiliation: University of Freiburg

2019-03-09

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/68555

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/