Minimal surfaces are among the most natural objects in Differential Geometry, and have been studied for the past 250 years ever since the pioneering work of Lagrange. The subject is characterized by a profound beauty, but perhaps even more remarkably, minimal surfaces (or minimal submanifolds) have encountered striking applications in other fields, like three-dimensional topology, mathematical physics, conformal geometry, among others. Even though it has been the subject of intense activity, many basic open problems still remain.
The time scales theory projected by Stefan Hilger in 1988 unifies the study of continuous and discrete analysis. Since then, it has been used intensively by many researchers working in different areas of mathematics. The main goal of this book is to find suitable time scale analog of minimal surfaces. This class of surfaces is considered in the context of the dynamic geometry on time scales.
The book contains four chapters. In Chapter 1 we introduce a complex integral with a real variable and some of its properties are derived. Countour integrals are defined and some of their properties are proved. Chapter 2 is devoted to the local theory of minimal surfaces on time scales. Parametric surfaces, nonparametric surfaces, first and second fundamental forms of a surface, surfaces that minimize area are introduced and developed. A time scale analog of the Bernsten theorem is formulated and proved. In Chapter 3 we introduce the global theory of minimal surfaces on time scales. They are defined \(\sigma_1\)-n-manifolds and some of their properties are deduced. The main equation of a minimal surface on time scales is derived. The Gauss map is defined and some of its basic properties are deduced. The Gauss curvature and the total curvature are introduced and investigated. Chapter 4 is devoted to a variational approach for studying minimal surfaces on time scales.
