Regularity of Minimal Surfaces

Focusing on minimal surfaces with free boundaries, the book explores their boundary behavior and presents key results, including asymptotic expansions and Gauss-Bonnet formulas. It tackles the challenges of deriving regularity proofs for non-minimizers through indirect reasoning and monotonicity formulas. Geometric properties, enclosure theorems, and isoperimetric inequalities are examined, alongside discussions of obstacle problems and Plateau’s problem in Riemannian manifolds. The final chapter introduces a novel approach to the absence of interior branch points in area-minimizing solutions.