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The Yamabe problem for manifolds with boundary and its quantitative stability
Abstract
The classical Yamabe problem seeks for a metric conformal to a given metric with constant scalar curvature. By rewriting the conformal factor in a suitable way, it is equivalent to minimizing the L² norm of the gradient under a constraint on the L^{2^*} - norm. This talk concerns an analogous variational problem for manifolds with boundary introduced by Escobar, which seeks for metrics with zero scalar curvature in the interior and constant mean curvature on the boundary. The variational formulation for functions is very similar, but the constraint is now on the boundary values of the function. I will introduce the problem and show how a compactness argument of Engelstein-Neumayer-Spolaor can be adapted to give quantitative stability for minimizers of this problem.
