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Gaoming Wang
Gaoming Wang
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Notes Minimal Hypersurfaces Preface
Background on Riemannian Manifolds and Minimal Immersions
  1. Riemannian Manifolds
  2. Submanifolds
  3. First and Second Variation Formulas
  4. Properties of Minimal Submanifolds in R^N
  5. Monotonicity Formula
  6. Area-Minimizing Hypersurfaces and Calibration
  7. Minimal Graphs
  8. The Dirichlet Problem for the Minimal Surface Equation
  9. Bernstein’s Theorem and Generalizations
Simons’ Inequality and Generalized Bernstein Theorems
  1. Simons’ Identity and Inequality
  2. Classification of Stable Minimal Cones for n<= 6
  3. Sub-minimal Sets and Sub-calibrations
Background on Geometric Measure Theory
  1. Varifolds
  2. Sets of Finite Perimeter
  3. Dimension Reduction and Regularity of Perimeter Minimizers
  4. Proof of Bernstein Theorem
Regularity and Compactness Theorems
  1. Regularity and Compactness Results for Stable Minimal Hypersurfaces
  2. Key Steps in the Regularity Proofs
  3. Regularity in the Immersed Setting
  4. Bellettini’s Sheeting Theorem and Schoen–Simon Regularity
  5. Sketch of Wickramasekera’s Regularity Theorem
Stable Bernstein Theorems
  1. Stable Bernstein in Dimension Two
  2. Higher-Dimensional Stable Bernstein Theorems
  3. Green-kernel proof of the stable Bernstein theorem in R^4
Applications to Positive Scalar Curvature and General Relativity
  1. The conformal Laplacian
  2. Conformal descent on a stable minimal hypersurface
  3. Geroch’s Conjecture by Dimension Reduction
  4. The Positive Mass Theorem and the Reduction to Geroch’s Conjecture
  5. The Horowitz–Myers Conjecture
References

Background on Geometric Measure Theory

We collect some basic definitions and facts about the geometric measure theory. Readers can refer to [All72, Sim83] for more details.

Sections

  • Varifolds
  • Sets of Finite Perimeter
  • Dimension Reduction and Regularity of Perimeter Minimizers
  • Proof of Bernstein Theorem
Previous Sub-minimal Sets and Sub-calibrations Next Varifolds
Gaoming Wang
Gaoming Wang
Assistant Professor

My research interests include Geometric Analysis and Partial Differential Equations.

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