Embedded tori with prescribed mean curvature

TL;DR

This paper constructs a sequence of embedded genus-one surfaces in Euclidean 3-space with prescribed mean curvature approaching a constant, using advanced analytical methods to understand their geometric properties.

Contribution

It introduces a novel construction of embedded tori with prescribed mean curvature that approximates unduloids, employing Jacobi operator analysis and Lyapunov-Schmidt reduction.

Findings

Surfaces have prescribed mean curvature of the form $H(X)=1+{A}{|X|^{-\gamma}}$ for large $|X|$

Constructed surfaces are close to unduloids with small necksize

Method involves deep analysis of Jacobi operators and variational techniques

Abstract

We construct a sequence of compact, oriented, embedded, two-dimensional surfaces of genus one into Euclidean 3-space with prescribed, almost constant, mean curvature of the form $H(X)=1+{A}{|X|^{-\gamma}}$ for $|X|$ large, when $A<0$ and $\gamma\in(0,2)$. Such surfaces are close to sections of unduloids with small necksize, folded along circumferences centered at the origin and with larger and larger radii. The construction involves a deep study of the corresponding Jacobi operators, an application of the Lyapunov-Schmidt reduction method and some variational argument.