Quantitative rigidity results for conformal immersions

TL;DR

This paper establishes quantitative rigidity results for conformal surface immersions in Euclidean space, demonstrating that surfaces with energy close to specific minimal or canonical shapes are geometrically close in the $W^{2,2}$-norm.

Contribution

It provides new quantitative estimates linking energy proximity to geometric closeness for conformal immersions near key minimal and canonical surfaces.

Findings

Surfaces close in energy to canonical shapes are close in the $W^{2,2}$-norm.

Rigidity results extend to complete, non-compact surfaces.

Quantitative bounds are established for various classical minimal surfaces.

Abstract

In this paper we prove several quantitative rigidity results for conformal immersions of surfaces in $\mathbb{R}^n$ with bounded total curvature. We show that (branched) conformal immersions which are close in energy to either a round sphere, a conformal Clifford torus, an inverted catenoid, an inverted Enneper's minimal surface or an inverted Chen's minimal graph must be close to these surfaces in the $W^{2,2}$-norm. Moreover, we apply these results to prove a corresponding rigidity result for complete, connected and non-compact surfaces.