Minimal Hypersurfaces: Stability, Regularity, and Applications

These notes are based on a course on minimal hypersurfaces given at the Beijing Institute of Mathematical Sciences and Applications (BIMSA). They cover stability, curvature estimates, regularity, compactness, stable Bernstein-type problems, and applications to scalar curvature and mathematical general relativity.

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Contents

• 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

• References