Regularity of minimal and CMC stable hypersurfaces: a brief overview

DOI10.2969/aspm/08510491zbMath1470.53004OpenAlexW3134490777MaRDI QIDQ2042488

Neshan Wickramasekera

Publication date: 20 July 2021

Full work available at URL: https://doi.org/10.2969/aspm/08510491

zbMATH Keywords

minimal hypersurfacesAllen-Cahn equationCMC hypersurfacesintegral varifoldsgeneralised mean curvature

Mathematics Subject Classification ID

Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) (53C42) Research exposition (monographs, survey articles) pertaining to differential geometry (53-02) Research exposition (monographs, survey articles) pertaining to partial differential equations (35-02)

Cites Work

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• The existence of embedded minimal hypersurfaces
• A sharp strong maximum principle and a sharp unique continuation theorem for singular minimal hypersurfaces
• Morse index and multiplicity of \(\min\text{-}\max\) minimal hypersurfaces
• Nuovi teoremi relativi alle misure \((r - 1)\)-dimensionali in uno spazio ad \(r\) dimensioni
• Solution of the Plateau problem for \(m\)-dimensional surfaces of varying topological type
• On the oriented Plateau problem
• On stable critical points for a singular perturbation problem
• The structure of weakly stable constant mean curvature hypersurfaces
• A regularity and compactness theory for immersed stable minimal hypersurfaces of multiplicity at most 2
• Stability of hypersurfaces with constant mean curvature
• Curvature estimates for minimal hypersurfaces
• Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature
• The Allen-Cahn equation on closed manifolds
• Min-max for phase transitions and the existence of embedded minimal hypersurfaces
• Density of minimal hypersurfaces for generic metrics
• Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory.
• Minimal surfaces and the Allen-Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates
• The second inner variation of energy and the Morse index of limit interfaces
• Curvature estimates and sheeting theorems for weakly stable CMC hypersurfaces
• On the second inner variations of Allen-Cahn type energies and applications to local minimizers
• Existence of infinitely many minimal hypersurfaces in positive Ricci curvature
• A general regularity theory for stable codimension 1 integral varifolds
• Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem
• Minimal varieties in Riemannian manifolds
• Normal and integral currents
• On the first variation of a varifold
• Su una teoria generale della misura \((r-1)\)-dimensionale in uno spazio ad \(r\) dimensioni
• Stable phase interfaces in the van der Waals–Cahn–Hilliard theory
• Regularity of stable minimal hypersurfaces
• The Motion of a Surface by Its Mean Curvature. (MN-20)
• Entire solutions of semilinear elliptic equations in ℝ³ and a conjecture of De Giorgi
• Spectrum and index of two-sided Allen–Cahn minimal hypersurfaces
• On the second inner variation of the Allen-Cahn functional and its applications
• Regularity and compactness for stable codimension $1$ CMC varifolds
• Finite Morse Index Implies Finite Ends
• The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension