Conformal Deformation to Scalar Flat Metrics with Constant Mean Curvature on the Boundary in Higher Dimensions

Abstract

In 1992, motivated by Riemann mapping theorem, Escobar considered a version of Yamabe problem on manifolds of dimension n greater than 2 with boundary. The problem consists in finding a conformal metric such that the scalar curvature is zero and the mean curvature is constant on the boundary. By using a local test function construction, we are able to seattle the most cases left by Escobar's and Marques's works. Moreover, we reduce the remaining case to the positive mass theorem. In this proof, we use the method developed in previous works by Brendle and by Brendle and the author.

27 pages, title changed, introduction expanded

References in corpus (1)

• An existence theorem for the Yamabe problem on manifolds with boundary

Cited by in corpus (10)

• Barycenter technique and the Riemann mapping problem of Escobar
• A compactness theorem for scalar-flat metrics on manifolds with boundary
• Blow-up phenomena for scalar-flat metrics on manifolds with boundary
• Convergence of scalar-flat metrics on manifolds with boundary under a Yamabe-type flow
• A compactness theorem for scalar-flat metrics on 3-manifolds with boundary
• Fractional Yamabe problem on locally flat conformal infinities of Poincare-Einstein manifolds
• Proof of the remaining cases of the Yamabe boundary problem
• A compactness result for scalar-flat metrics on manifolds with umbilic boundary
• Compactness of scalar-flat conformal metrics on low-dimensional manifolds with constant mean curvature on boundary
• Existence of conformal metrics with constant scalar curvature and constant boundary mean curvature on compact manifolds