Rigidity of min-max minimal spheres in three-manifolds

DOI10.1215/00127094-1813410zbMath1260.53079arXiv1105.4632OpenAlexW3105794262MaRDI QIDQ691565

André Neves, Fernando Codá Marques

Publication date: 3 December 2012

Published in: Duke Mathematical Journal (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1105.4632

zbMATH Keywords

minimal surface3-sphererigiditywidthRicci flowindexmini-max method

Mathematics Subject Classification ID

Second-order elliptic equations (35J15) Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) (53C42) Rigidity results (53C24) Variational methods for second-order elliptic equations (35J20)

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Cites Work

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• Rigidity of area-minimizing hyperbolic surfaces in three-manifolds
• Deformations of the hemisphere that increase scalar curvature
• Rigidity of area-minimizing two-spheres in three-manifolds
• Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature
• Nonlinear analysis in geometry
• Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with non-negative scalar curvature
• Stable constant mean curvature tori and the isoperimetric problem in three space forms
• On the proof of the positive mass conjecture in general relativity
• Scalar curvature rigidity of asymptotically hyperbolic spin manifolds
• Sharp estimates and the Dirac operator
• Scalar curvature rigidity for asymptotically locally hyperbolic manifolds
• A new proof of the positive energy theorem.
• The mass of asymptotically hyperbolic manifolds.
• Curvature estimates and compactness theorems in 3-manifolds for surfaces that are stationary for parametric elliptic functionals
• Embedded minimal surfaces in manifolds diffeomorphic to the three-dimensional ball or sphere
• Three-manifolds with positive Ricci curvature
• The mass of asymptotically hyperbolic Riemannian manifolds.
• Rigidity theorems for compact manifolds with boundary and positive Ricci curvature
• Embedded Weingarten tori in \(S^3\)
• Non-orientable surfaces in orientable 3-manifolds
• On the fundamental group of a compact minimal submanifolds
• The unknottedness of minimal embeddings
• The stabilization problem for Heegaard splittings of Seifert fibered spaces
• The Existence of Least Area Surfaces in 3-Manifolds
• Genus bounds for minimal surfaces arising from min-max constructions
• Area-minimizing projective planes in 3-manifolds
• Curvature estimates for minimal surfaces in $3$-manifolds
• Equivariant minimax and minimal surfaces in geometric three-manifolds
• The second variation of nonorientable minimal submanifolds
• Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman
• The size of isoperimetric surfaces in $3$-manifolds and a rigidity result for the upper hemisphere
• Lectures on the topology of 3-manifolds. An introduction to the Casson invariant
• Rigidity of area minimizing tori in 3-manifolds of nonnegative scalar curvature

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