Optimal partitions for the Yamabe equation

Abstract: The Yamabe equation on a Riemannian manifold $(M, g)$ is of relevance in differential geometry. A positive solution to it gives rise to a metric on M which has constant scalar curvature and is conformally equivalent to the given metric $g$. An optimal $\ell$-partition for the Yamabe equation is a cover of M by $\ell$-pairwise disjoint open subsets such that the Yamabe equation with Dirichlet boundary condition has a least energy solution on each one of these sets, and the sum of the energies of these solutions is minimal. Such a partition induces a generalized metric that vanishes on a set of measure zero and is conformally equivalent to $g$ in the complement. I will present some results obtained in collaboration with Angela Pistoia (La Sapienza Universit`a di Roma) and Hugo Tavares (Universidade de Lisboa) that ensure the existence and establish qualitative properties of this type of partitions. To do this, we use some ideas from physics.

geometric topology

Audience: researchers in the topic

GEOTOP-A seminar

Series comments: Web-seminar series on Applications of Geometry and Topology