A course in minimal surfaces

Description:

"Minimal surfaces date back to Euler and Lagrange and the beginning of the calculus of variations. Many of the techniques developed have played key roles in geometry and partial differential equations. Examples include monotonicity and tangent cone analysis originating in the regularity theory for minimal surfaces, estimates for nonlinear equations based on the maximum principle arising in Bernstein's classical work, and even Lebesgue's definition of the integral that he developed in his thesis on the Plateau problem for minimal surfaces. This book starts with the classical theory of minimal surfaces and ends up with current research topics. Of the various ways of approaching minimal surfaces (from complex analysis, PDE, or geometric measure theory), the authors have chosen to focus on the PDE aspects of the theory. The book also contains some of the applications of minimal surfaces to other fields including low dimensional topology, general relativity, and materials science."--Publisher's description.

subjects: Minimal surfaces,  Manifolds and cell complexes -- Low-dimensional topology -- Geometric structures on low-dimensional manifolds,  Partial differential equations -- Elliptic equations and systems -- Second-order elliptic equations,  Global analysis, analysis on manifolds -- Variational problems in infinite-dimensional spaces -- Applications to minimal surfaces (problems in two independent variables),  Manifolds and cell complexes -- Topological manifolds -- Topology of general $3$-manifolds,  Calculus of variations and optimal control; optimization -- Manifolds -- Minimal surfaces,  Differential geometry -- Global differential geometry -- Immersions (minimal, prescribed curvature, tight, etc.).,  Partial differential equations -- Elliptic equations and systems -- Nonlinear elliptic equations. $2 msc,  Differential geometry -- Classical differential geometry -- Minimal surfaces, surfaces with prescribed mean curvature,  Relativity and gravitational theory -- General relativity -- Black holes,  Geometry, differential,  Calculus of variations,  Global analysis (mathematics),  Differential equations, partial,  Differential geometry -- Global differential geometry -- Immersions (minimal, prescribed curvature, tight, etc.),  Partial differential equations -- Elliptic equations and systems -- Nonlinear elliptic equations