Quantitative rigidity results for conformal immersions

2014/05/28 by Tobias Lamm, Lamm, Tobias, Nguyen, Huy The
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1405.7335

openalex publication_date 2014/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we prove several quantitative rigidity results for conformal immersions of surfaces in ℝⁿ 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.

Citations

• A theorem on geometric rigidity and the derivation of nonlinear plate theory from three‐dimensional elasticity

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