# Geometric Analysis: Partial Differential Equations and Surfaces

Share this page *Edited by *
*Joaquín Pérez; José A. Gálvez*

This volume contains research and expository articles
from the courses and talks given at the RSME Lluis A. Santaló Summer
School, “Geometric Analysis”, held June 28–July 2, 2010, in
Granada, Spain.

The goal of the Summer School was to present some of the many advances
currently taking place in the interaction between partial differential
equations and differential geometry, with special emphasis on the theory of
minimal surfaces.

This volume includes expository articles about the current state of specific
problems involving curvature and partial differential equations, with
interactions to neighboring fields such as probability. An introductory, mostly
self-contained course on constant mean curvature surfaces in Lie groups
equipped with a left invariant metric is provided.

The volume will be of interest to researchers, post-docs, and advanced PhD
students in the interface between partial differential equations and
differential geometry.

This book is published in cooperation with Real Sociedad Matemática Española (RSME)

# Table of Contents

## Geometric Analysis: Partial Differential Equations and Surfaces

- Preface vii8 free
- Geometric PDEs in the presence of isolated singularities 110 free
- 1. Introduction 110
- 2. The Hessian one equation: preliminaries 514
- 3. Isolated singularities of 𝑢ₓₓ𝑢_{𝑦𝑦}-𝑢_{𝑥𝑦}²=1: local study 615
- 4. Isolated singularities of 𝑢ₓₓ𝑢_{𝑦𝑦}-𝑢_{𝑥𝑦}²=1: global study 918
- 5. Surfaces of constant curvature in ℝ³ 1120
- 6. Isolated singularities of the constant curvature equation 1322
- 7. Immersed isolated singularities 1726
- 8. Peaked spheres in ℝ³ 1827
- 9. Further results and open problems 2029
- References 2231

- Constant mean curvature surfaces in metric Lie groups 2534
- Stochastic methods for minimal surfaces 111120
- The role of minimal surfaces in the study of the Allen-Cahn equation 137146
- On curvature estimates for constant mean curvature surfaces 165174