Convergent sequences of closed minimal surfaces embedded in $\S3$

TL;DR

This paper proves the existence of convergent sequences of embedded minimal surfaces in the 3-sphere, showing conditions under which they approach a given surface and relate eigenvalue embeddings.

Contribution

It establishes conditions for the convergence of minimal surfaces in -sphere and links eigenvalue embedding properties between surfaces.

Findings

Existence of convergent sequences of minimal surfaces in -sphere.

Conditions under which convergence occurs.

Relation between eigenvalue embedding and surface properties.

Abstract

given two minimal surfaces embedded in $\S3$ of genus $g$ we prove the existence of a sequence of non-congruent compact minimal surfaces embedded in $\S3$ of genus $g$ that converges in $C^{2,\alpha}$ to a compact embedded minimal surface provided some conditions are satisfied. These conditions also imply that, if any of these two surfaces is embedded by the first eigenvalue, so is the other.