Minimal Surfaces From A Complex Analytic Viewpoint
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Minimal Surfaces from a Complex Analytic Viewpoint
This monograph offers the first systematic treatment of the theory of minimal surfaces in Euclidean spaces by complex analytic methods, many of which have been developed in recent decades as part of the theory of Oka manifolds (the h-principle in complex analysis). It places particular emphasis on the study of the global theory of minimal surfaces with a given complex structure. Advanced methods of holomorphic approximation, interpolation, and homotopy classification of manifold-valued maps, along with elements of convex integration theory, are implemented for the first time in the theory of minimal surfaces. The text also presents newly developed methods for constructing minimal surfaces in minimally convex domains of Rn, based on the Riemann–Hilbert boundary value problem adapted to minimal surfaces and holomorphic null curves. These methods also provide major advances in the classical Calabi–Yau problem, yielding in particular minimal surfaces with the conformal structure of any given bordered Riemann surface. Offering new directions in the field and several challenging open problems, the primary audience of the book are researchers (including postdocs and PhD students) in differential geometry and complex analysis. Although not primarily intended as a textbook, two introductory chapters surveying background material and the classical theory of minimal surfaces also make it suitable for preparing Masters or PhD level courses.
Cutting-Edge Mathematics
The book contains a selection of research and expository papers in pure and applied mathematics presented by various authors as plenary or invited speakers at the biennial congress of the Spanish Royal Mathematical Society held in Ciudad Real (Spain) in January 2022. The main results focus on the Yang problem and its solution proposed by Globevnik; a phylogenetic reconstruction based on algebra; the Calderon problem for local and nonlocal Schrödinger equations; some open problems in orthogonal polynomial theory; Quillen’s rational homotopy theory; Ulrich bundles and applications; and free objects in theory of Banach spaces. Researchers in these fields are potential audiences.
Differential Geometry and Its Applications
Author: John Oprea
language: en
Publisher: American Mathematical Society
Release Date: 2024-07-01
Differential Geometry and Its Applications studies the differential geometry of surfaces with the goal of helping students make the transition from the compartmentalized courses in a standard university curriculum to a type of mathematics that is a unified whole. It mixes geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences. That mix of ideas offers students the opportunity to visualize concepts through the use of computer algebra systems such as Maple. Differential Geometry and Its Applications emphasizes that this visualization goes hand in hand with understanding the mathematics behind the computer construction. The book is rich in results and exercises that form a continuous spectrum, from those that depend on calculation to proofs that are quite abstract.