Geometric Analysis : Partial Differential Equations and Surfaces.

Intro -- Preface -- Geometric PDEs in the presence of isolated singularities -- 1. Introduction -- 2. The Hessian one equation: preliminaries -- 3. Isolated singularities of ₓₓ _{ }- _{ }²=1: local study -- 4. Isolated singularities of ₓₓ _{ }- _{ }²=1: global study -- 5. Surfaces of constant curvature in ℝ³ -- 6. Isolated singularities of the constant curvature equation -- 7. Immersed isolated singularities -- 8. Peaked spheres in ℝ³ -- 9. Further results and open problems -- References -- Constant mean curvature surfaces in metric Lie groups -- 1. Introduction -- 2. Lie groups and homogeneous three-manifolds -- 3. Surface theory in three-dimensional metric Lie groups -- 4. Open problems and unsolved conjectures for -surfaces in three-dimensional metric Lie groups -- References -- Stochastic methods for minimal surfaces -- 1. Introduction -- 2. Basic notions -- 3. Parabolicity and area growth -- 4. Coupling -- References -- The role of minimal surfaces in the study of the Allen-Cahn equation -- 1. The role of minimal hypersurfaces -- 2. Entire solutions of the Allen-Cahn equation in Euclidean space -- 3. Proof of Theorem 1.1 -- References -- On curvature estimates for constant mean curvature surfaces -- Introduction -- 1. Bounding | | -- 2. Schoen Curvature Estimate -- 3. Choi-Schoen Curvature Estimate -- 4. Colding-Minicozzi One-Sided Curvature Estimate -- 5. Curvature Estimate For CMC Disks -- References.

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