Let 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.
Cite this article
Simon Brendle, Szu-Yu Sophie Chen, An existence theorem for the Yamabe problem on manifolds with boundary. J. Eur. Math. Soc. 16 (2014), no. 5, pp. 991–1016DOI 10.4171/JEMS/453