Prescribed Mean Curvature on the ball via a sign change in the derivative

2013/01/05 by Alvaro Ortiz, Gonzalo Garcia, Ortiz, Alvaro +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG

paper · pdf · doi:10.48550/arxiv.1301.0945

arxiv created 2026/07/29 · arxiv updated 2026/07/31

Abstract

We consider the problem of finding a conformal metric on the n-dimensional Euclidean ball Bⁿ, n≥ 3, with zero scalar curvature in the interior and prescribed mean curvature H on the boundary ∂ Bⁿ = S^n-1. For rotationally symmetric H=H(r) satisfying a flatness condition of order α∈(n-2, n-1) at each of its finitely many critical points in the region where H>0, we prove that the sign-change of H'(r) in that region is sufficient for existence. Combined with the necessary condition of Liu and Wang, this yields a characterization: in this class, the problem is solvable if and only if H is not monotone where it is positive. Along the way we prove that the single blow-up point of the subcritical approximation must be a local maximum of H, the boundary-trace analogue of a lemma used by Chen and Li on the sphere, and, under a separation condition on the values of the local maxima, we obtain multiple solutions. Our approach adapts the variational scheme of Chen and Li for the sphere to the boundary-trace setting, using subcritical approximation and the blow-up analysis of Escobar and García.

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