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Rigidity of minimal Lagrangian diffeomorphisms between spherical cone surfaces

Christian El Emam, Andrea Seppi

Abstract

We prove that any minimal Lagrangian diffeomorphism between two closed spherical surfaces with cone singularities is an isometry, without any assumption on the multiangles of the two surfaces. As an application, we show that every branched immersion of a closed surface of constant positive Gaussian curvature in Euclidean three-space is a branched covering onto a round sphere, thus generalizing the classical rigidity theorem of Liebmann to branched immersions.

Rigidity of minimal Lagrangian diffeomorphisms between spherical cone surfaces

Abstract

We prove that any minimal Lagrangian diffeomorphism between two closed spherical surfaces with cone singularities is an isometry, without any assumption on the multiangles of the two surfaces. As an application, we show that every branched immersion of a closed surface of constant positive Gaussian curvature in Euclidean three-space is a branched covering onto a round sphere, thus generalizing the classical rigidity theorem of Liebmann to branched immersions.

Paper Structure

This paper contains 17 sections, 11 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Main statement
  3. Surfaces of constant Gauss curvature
  4. Outline of the proof of Theorem \ref{['thm:main1']}.
  5. Definitions and setup
  6. Conical metrics
  7. Minimal Lagrangian maps
  8. Defining the pair $(G,B)$
  9. A maximum principle
  10. A bounded subharmonic function
  11. Proof of Theorem \ref{['thm:main1']}
  12. Immersions in Euclidean space
  13. Proof of Proposition \ref{['prop:conf type']}
  14. Proof of Corollary \ref{['cor:rigidity']}
  15. Final remarks
  16. ...and 2 more sections

Key Result

Theorem 1.1

Given two closed spherical cone surfaces $(\Sigma_1,\mathfrak p_1,g_1)$ and $(\Sigma_2,\mathfrak p_2,g_2)$, any minimal Lagrangian diffeomorphism $\varphi:(\Sigma_1,\mathfrak p_1,g_1)\to(\Sigma_2,\mathfrak p_2,g_2)$ is an isometry.

Theorems & Definitions (31)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4: zbMATH05200423
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Lemma 2.8
  • ...and 21 more