Constant mean curvature surfaces of annular type
Let \(\gamma_1\) and \(\gamma_2\) be two disjoint oriented Jordan curves in \(\mathbb{R}^3\), denote by \(\mathcal{S}(\gamma_1\vee\gamma_2,K)\) the function space of maps from annular domains in \(\mathbb{R}^2\) to \(\mathbb{R}^3\) which map the boundary of the domain to \(\gamma_1\) and \(\gamma_2\) monotonically, and the volume of the maps equal to a constant \(K\). In a previous paper [Math. Z. 120, 277--288 (1971; Zbl 0214.11101)] the author proved that if the Douglas condition is satisfied then there exists an immersion \(x_K\in \mathcal{S}(\gamma_1\vee\gamma_2,K)\) which minimizes the Dirichlet integral \(D(x)\) and \(x_K\) is a constant mean curvature surfaces. In this paper, the author studies the behavior of solutions \(x_{K_n}\in \mathcal{S}(\gamma_1\vee\gamma_2,K_n)\) as \(K_n\to\infty\). By rescaling, he makes the boundary curves shrink to the origin with the constant volume \(V(x)=1\), he then proves that the mean curvature \(H_n\) of \(x_n\) converges to \(H_0=(-4\pi/3)^{\frac13}\) and \(D(x_n)\) converges to \(D_0\), where \(H_0\) is the mean curvature of the sphere with volume 1. Moreover, there exists a subsequence converging to a limit map \(x_0: \mathbb{R}^2\to \mathbb{R}^3\), where \(x_0\) is a stereographic representation of the unit volume sphere.
scientific article; zbMATH DE number 1756809