Paper · Jul 14, 2022
The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant
As on arXiv
Abstract
We study the Yamabe flow on asymptotically flat manifolds with non-positive Yamabe constant . Previous work by the second and third named authors showed that while the Yamabe flow always converges in a global weighted sense when , the flow must diverge when . We show here in the case however that after suitable rescalings, the Yamabe flow starting from any asymptotically flat manifold must converge to the unique positive function which solves the Yamabe problem on a compactification of the original manifold.
Comments: 24 pages
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Background reading
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In the guidebook
- 6C.4The Yamabe Problem and Critical Exponents
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.