Minimal Varieties in Riemannian Manifolds

1968/07/01 by James Simons · 101 citations
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals

paper · doi:10.2307/1970556

Cited by

• Local Rigidity Theorems for Minimal Hypersurfaces
• On Chernʼs problem for rigidity of minimal hypersurfaces in the spheres
• First stability eigenvalue characterization of Clifford hypersurfaces
• A spectral characterization of the H(r)-torus by the first stability eigenvalue
• First eigenvalue of a Jacobi operator of hypersurfaces with a constant scalar curvature
• A lower bound for 𝐿₂ length of second fundamental form on minimal hypersurfaces
• First stability eigenvalue of singular hypersurfaces with constant mean curvature in the unit sphere
• The scalar curvature of minimal hypersurfaces in spheres
• Rigidity of compact minimal submanifolds in a unit sphere
• Scalar Curvature and Product Manifolds
• Anisotropic minimal surface equation with Dirichlet boundary condition
• Volume gap for minimal submanifolds in spheres
• On the Morse index of harmonic maps and minimal immersions
• Willmore submanifolds in a sphere
• A survey on Geometry of Slant Submanifolds
• Biharmonic submanifolds with parallel mean curvature in\n mathbbSⁿ\×\ℝ
• The geometry of Grassmannian manifolds and Bernstein type theorems for higher codimension
• Simons' equation and minimal hypersurfaces in space forms
• Stability properties for the higher dimensional catenoid in \rrⁿ⁺¹
• Chern conjecture and isoparametric hypersurfaces
• Regularity of area-minimizing surfaces in 3D polytopes and of invariant hypersurfaces in Rⁿ
• Pinching rigidity of surfaces with parallel mean curvature vector in spheres
• On the first stability eigenvalue of constant mean curvature surfaces\n into homogeneous 3-manifolds
• Regularity of Cohomogeneity two equivariant isotopy minimization problems and minimal hypersurfaces with large first Betti number on spheres
• Chern Conjecture on Minimal Willmore Hypersurfaces with Constant Scalar Curvature
• Detect duality obstruction of calibrations in smooth category
• Note on Regularity theory for Nonlinear elliptic equations
• The second boundary value problem of the prescribed affine mean\n curvature equation and related linearized Monge-Amp `ere equation
• Higher order geometric flow of hypersurfaces in a Riemannian manifold
• A Bernstein type result for graphical self-shrinkers in ℝ⁴
• Phase transitions in two-component Bose-Einstein condensates with Rabi frequency (II): The De Giorgi conjecture for the nonlocal problem in ℝ² or ℝ³
• Harmonic maps from S³ to S² and the rigidity of the Hopf fibration
• Curvature estimates for submanifolds with prescribed Gauss image and mean curvature
• Free boundary minimal surfaces in the unit ball : recent advances and open questions
• A new characterization of the Clifford torus via scalar curvature pinching
• Topological obstructions for submanifolds in low codimension
• Stability of Homogeneous minimal hypersurfaces in the Page space and Y^p,q Sasaki-Einstein manifolds
• Spacetime Foliations and p-Brane Dynamics
• On stable compact minimal submanifolds
• Some results on Chern's problem
• An optimal pinching theorem of minimal Legendrian submanifolds in the unit sphere
• A lower bound for L₂ length of second fundamental form on minimal hypersurfaces
• A proof by foliation that Lawson's cones are A_Φ-minimizing
• Stable solutions of equations with a quadratic gradient term
• Curvature estimates for stable marginally trapped surfaces
• A gap theorem for free boundary minimal surfaces in geodesic balls of hyperbolic space and hemisphere
• Bifurcation of constant mean curvature tori in Euclidean spheres
• Saddle-shaped solutions of bistable diffusion equations in all of ℝ^2m
• Scale-Separated Curvature Estimates for Strongly Stable Hypersurfaces
• Topological and differentiable rigidity of submanifolds in space forms
• Rectifiability; a survey
• A Bernstein theorem for two-valued minimal graphs in dimension four
• Surfaces of Prescribed Mean Curvature in Quasi-Fuchsian Manifolds
• Boundary Regularity of Minimal Oriented Hypersurfaces on a Manifold
• A new pinching theorem for complete self-shrinkers and its generalization
• Ancient solutions to mean curvature flow for isoparametric submanifolds
• Mean curvature flow into an ambient Riemannian manifold evolving by Ricci flow coupled with harmonic map heat flow
• Linearized equation and generic regularity in the Alt-Caffarelli problem
• Some configuration results for area-minimizing cones
• Mean curvature flow in Fuchsian manifolds
• On FKM isoparametric hypersurfaces in \mathbbSⁿ × \mathbbSⁿ and new area-minimizing cones
• Note on energy index and first eigenvalue of minimal surfaces in spheres
• Uniqueness in the Plateau problem for calibrated currents
• Minimal surfaces and comparison geometry
• Minimally embedded Riemann surfaces in \mbS³ and the conformal deformation of their metrics
• The geometry of closed conformal vector fields on Riemannian spaces
• Local well-posedness of the Skew mean curvature flow for small data in d≥ 2 dimensions
• Special Lagrangian Cones
• Minimal Hypersurfaces with constant scalar curvature in S⁶
• The second pinching theorem for hypersurfaces with constant mean curvature in a sphere
• On the Morse Index with Constraints I: An Abstract Formulation
• Phase transitions in two-component Bose-Einstein condensates with Rabi frequency (I): The De Giorgi conjecture for the local problem in ℝ³
• Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery
• Uniqueness and non-existence of minimal submanifolds
• New Homogeneous Solutions for the One-Phase Free Boundary Problem
• The non-parametric problem of Plateau in arbitrary codimension
• Liouville Type Theorems for Minimal Surface Equations in Half Space
• The topological rigidity theorem for submanifolds in space forms
• Spherical Bernstein theorems for codimension 1 and 2
• Calibrated submanifolds with flat normal bundles
• Scalar Curvature on some Open 3-Manifolds
• A proof of the reverse isoperimetric inequality using a geometric-analytic approach
• Four manifolds with two-positive Ricci curvature
• Compact special Legendrian surfaces in S⁵
• Convex functions on Grassmannian manifolds and Lawson-Osserman problem
• On Yau rigidity theorem for minimal submanifolds in spheres
• Free boundary minimal Möbius band in spherical caps
• Second eigenvalue of a Jacobi operator of hypersurfaces with constant scalar curvature
• The regularity of harmonic maps into spheres and applications to Bernstein problems
• Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones
• Normal scalar curvature inequality on the focal submanifolds of isoparametric hypersurfaces
• Spectral properties of a family of minimal tori of revolution in\n five-dimensional sphere
• Stable prescribed-mean-curvature integral varifolds of codimension 1: regularity and compactness
• Uniqueness of certain cylindrical tangent cones
• Singular behavior and generic regularity of min-max minimal hypersurfaces
• Simons' formula [wikipedia]
• A Björling representation for Jacobi fields on minimal surfaces and soap film instabilities. [europepmc]
• Local Well-Posedness of Skew Mean Curvature Flow for Small Data in d ≥ 4 Dimensions. [europepmc]
• Marginally outer trapped tubes in de Sitter spacetime. [europepmc]
• Smoothness and stability in the Alt-Phillips problem. [europepmc]
• On the Dimension of the Singular Set of Perimeter Minimizers in Spaces with a Two-Sided Bound on the Ricci Curvature [europepmc]