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Gaoming Wang

Gaoming Wang

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Notes Minimal Hypersurfaces Preface

Background on Riemannian Manifolds and Minimal Immersions

Riemannian Manifolds
Submanifolds
First and Second Variation Formulas
Properties of Minimal Submanifolds in R^N
Monotonicity Formula
Area-Minimizing Hypersurfaces and Calibration
Minimal Graphs
The Dirichlet Problem for the Minimal Surface Equation
Bernstein’s Theorem and Generalizations

Simons’ Inequality and Generalized Bernstein Theorems

Simons’ Identity and Inequality
Classification of Stable Minimal Cones for n<= 6
Sub-minimal Sets and Sub-calibrations

Background on Geometric Measure Theory

Varifolds
Sets of Finite Perimeter
Dimension Reduction and Regularity of Perimeter Minimizers
Proof of Bernstein Theorem

Regularity and Compactness Theorems

Regularity and Compactness Results for Stable Minimal Hypersurfaces
Key Steps in the Regularity Proofs
Regularity in the Immersed Setting
Bellettini’s Sheeting Theorem and Schoen–Simon Regularity
Sketch of Wickramasekera’s Regularity Theorem

Stable Bernstein Theorems

Stable Bernstein in Dimension Two
Higher-Dimensional Stable Bernstein Theorems
Green-kernel proof of the stable Bernstein theorem in R^4

Applications to Positive Scalar Curvature and General Relativity

The conformal Laplacian
Conformal descent on a stable minimal hypersurface
Geroch’s Conjecture by Dimension Reduction
The Positive Mass Theorem and the Reduction to Geroch’s Conjecture
The Horowitz–Myers Conjecture

Simons’ Inequality and Generalized Bernstein Theorems

We now discuss several generalizations of Bernstein’s theorem and curvature estimates for stable minimal hypersurfaces.

Sections

Simons’ Identity and Inequality
Classification of Stable Minimal Cones for n<= 6
Sub-minimal Sets and Sub-calibrations

Previous Bernstein’s Theorem and Generalizations Next Simons’ Identity and Inequality

Gaoming Wang

Assistant Professor

My research interests include Geometric Analysis and Partial Differential Equations.

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