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An existence theorem for the Yamabe problem on manifolds with boundary

Simon Brendle; Szu-Yu Sophie Chen — 2014

Journal of the European Mathematical Society

Let $(M, g)$ be a compact Riemannian manifold with boundary. We consider the problem (first studied by Escobar in 1992) of finding a conformal metric with constant scalar curvature in the interior and zero mean curvature on the boundary. Using a local test function construction, we are able to settle most cases left open by Escobar’s work. Moreover, we reduce the remaining cases to the positive mass theorem.

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