"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 re…
Can one set of basic laws account for both the recurring themes and the infinite variety of nature's designs? When it comes to shape and form, does nature simply proceed in the easiest, most efficient way? Complete answers to these questions are likely never to be discovered. St…
Global theory of minimal surfaces: proceedings of the Clay Mathematics Institute 200…
by Clay Mathematics Institute, Hoffman, David A., Summer School on the Global Theory of Minimal Surfaces, Berkeley, Calif, Clay Mathematics Institute Summer School, Berkeley, Calif.), Summer School on the Global Theory of Minimal Surfaces, Berkeley, Calif.)
fromClay Mathematics Proceedings
"The mean curvature of a surface is an extrinsic parameter measuring how the surface is curved in the three-dimensional space. A surface whose mean curvature is zero at each point is a minimal surface, and it is known that such surfaces are models for soap film. A surface whose …
One of the fundamental objects of study in geometric measure theory is an "area-minimizing surface." A compact, k-dimensional surface (with boundary) is called area-minimizing if no other surface with the same boundary has less surface area. An area-minimizing surface can have s…
A multiple solution theory to the Plateau problem in a Riemannian manifold is established. In [italic capital]S[superscript italic]n, the existence of two solutions to this problem is obtained. The Morse-Tompkins-Shiffman Theorem is extended to the case when the ambient space ad…
This paper is devoted primarily to the study of properties of the Grassmannian of oriented 2-planes in [double-struck capital]ℝ[superscript]n and to applications of these properties to understanding minimal surfaces in [double-struck capital]ℝ[superscript]n via the generalized G…
This memoir is devoted to the case of constant mean curvature surfaces immersed in [bold]R³. We reduce this geometrical problem to finding certain integrable solutions to the Gauss equation. Many new and interesting examples are presented, including immersed cylinders in [bold]R…