A Variational Characterization of the Catenoid

Weierstrass representation for semi-discrete minimal surfaces, and comparison of various discretized catenoids

We give a Weierstrass type representation for semi-discrete minimal surfaces in Euclidean 3-space. We then give explicit parametrizations of various smooth, semi-discrete and fully-discrete catenoids, determined from either variational or integrable systems principles. Finally, we state the shared properties that those various catenoids have.

亏格一临界悬链面

Genus one critical catenoid

本文用变分方法在三维单位球中构造了亏格一、双边界、棱柱对称的自由边界极小曲面，扩展了等变极小极大理论并发现双参数扫描。

The catenoid estimate and its geometric applications

We prove a sharp area estimate for catenoids that allows us to rule out the phenomenon of multiplicity in min-max theory in several settings. We apply it to prove that i) the width of a three-manifold with positive Ricci curvature is realized by an orientable minimal surface ii) minimal genus Heegaard surfaces in such manifolds can be isotoped to be minimal and iii) the "doublings" of the Clifford torus by Kapouleas-Yang can be constructed variationally by an equivariant min-max procedure. In higher dimensions we also prove that the width of manifolds with positive Ricci curvature is achieved …

Harmonic maps with prescribed degrees on the boundary of an annulus and bifurcation of catenoids

Let $A \subset \mathbb{R} ^2 $ be a smooth doubly connected domain. We consider the Dirichlet energy $E(u)=\int_{A} |\nabla u|^2$, where $u:A \rightarrow \mathbb{C}$, and look for critical points of this energy with prescribed modulus $|u|=1$ on $\partial A$ and with prescribed degrees on the two connected components of $\partial A$. This variational problem is a problem with lack of compactness hence we can not use the direct methods of calculus of variations. Our analysis relies on the so-called Hopf differential and on a strong link between this problem and the problem of finding all minima…

On a free boundary problem and minimal surfaces

From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov-Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension $8$, using variational arguments, we also obtain solutions which are global minimizers of the corresponding energy functional. This shows that Savin's theorem is optimal.