Estimates for the Constant Mean Curvature Dirichlet Problem on Catenoids

TL;DR

This paper provides estimates for solutions to the constant mean curvature Dirichlet problem on small catenoidal necks in three-dimensional space, improving bounds and demonstrating differentiability, with applications to geometric gluing methods.

Contribution

It introduces new bounds for solutions on catenoids, enhances regularity estimates, and establishes differentiability at the limit, aiding geometric gluing constructions.

Findings

Solutions are bounded by a constant times r^{1+γ} with γ in (0,1).

Improved estimate to γ=1 for solutions.

Proved differentiability of solutions down to τ=0.

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 H\"older spaces with non-integer weight and are a-priori bounded by a uniform constant times $r^{1 + \gamma}$, where $r$ denotes the distance to the axis of the neck and where $\gamma$ belongs to the interval $(0, 1)$. By comparing the solutions with their limits on the disk, we improve the estimate to $\gamma =1$. As a corollary, we prove differentiability of solutions in $\tau$ down to $\tau = 0$. The surfaces we construct have applications to gluing constructions.