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arXiv · 2308.04257

Estimates for the Constant Mean Curvature Dirichlet Problem on Catenoids

Stephen J. Kleene
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Abstract

In this article, we solve the constant mean curvature dirichlet problem on catenoidal necks with small scale in $\mb{R}^3$. 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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Stephen J. Kleene. 2023-09-15. Estimates for the Constant Mean Curvature Dirichlet Problem on Catenoids. https://arxiv.org/abs/2308.04257

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