Estimates for the Constant Mean Curvature Dirichlet Problem on Catenoids

2023/08/08 by Stephen J. Kleene, Kleene, Stephen J.
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Equations and Boundary Problems #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2308.04257

openalex publication_date 2023/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, we solve the constant mean curvature dirichlet problem on catenoidal necks with small scale in \mbR³. The solutions are found in exponentially weighted Hölder spaces with non-integer weight and are a-priori bounded by a uniform constant times r^{1 + γ}, where r denotes the distance to the axis of the neck and where γ belongs to the interval (0, 1). By comparing the solutions with their limits on the disk, we improve the estimate to γ=1. As a corollary, we prove differentiability of solutions in τ down to τ= 0. The surfaces we construct have applications to gluing constructions.

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