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.
Highlights
Using Anti-de Sitter geometry, the result of Labourie and Schoen has been generalized in various directions: in [BS10, BS18] in the setting of universal Teichmüller space; in [Tou16] for closed hyperbolic surfaces with cone singularities of angles in (0, π), provided the diffeomorphism Σ1 → Σ2 maps cone points to cone points of the same angles
We remark that interesting results in a similar spirit have been obtained for minimal Lagrangian diffeomorphisms between bounded domains in the Euclidean plane ([Del91, Wol97]) and in a complete non-positively curved Riemannian surface ([Bre08])
We show that two spherical cone surfaces do not admit any minimal Lagrangian diffeomorphism unless they are isometric
Summary
Minimal Lagrangian maps have played an important role in the study of hyperbolic structures on surfaces. Using Anti-de Sitter geometry, the result of Labourie and Schoen has been generalized in various directions: in [BS10, BS18] in the setting of universal Teichmüller space; in [Tou16] for closed hyperbolic surfaces with cone singularities of angles in (0, π), provided the diffeomorphism Σ1 → Σ2 maps cone points to cone points of the same angles. — Given two closed spherical cone surfaces (Σ1, p1, g1) and (Σ2, p2, g2), any minimal Lagrangian diffeomorphism φ : (Σ1, p1, g1) → (Σ2, p2, g2) is an isometry. A priori, we do not assume that such a smooth map extends smoothly to the cone points
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