The paper connects a uniqueness conjecture for a nonlinear Neumann problem to sharp boundary-area bounds under nonnegative Ricci curvature and convexity, and proves the dimension-three area and volume bounds.
Escobar, Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary
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On compact Riemannian manifolds with convex boundary and Ricci curvature bounded from below
The paper connects a uniqueness conjecture for a nonlinear Neumann problem to sharp boundary-area bounds under nonnegative Ricci curvature and convexity, and proves the dimension-three area and volume bounds.