The minimal lamination closure theorem

Fan-KKM Theorem in Minimal Vector Spaces and its Applications

In this paper, after reviewing some results in minimal space, some new results in this setting are given. We prove a generalized form of the Fan-KKM typetheorem in minimal vector spaces. As some applications, the open type of matching theorem and generalized form of the classical KKM theorem in minimal vector spaces are given.

The Kuratowski Closure-complement Theorem

This remarkable little theorem and related phenomena have been the concern of many authors. Apart from the mysterious appearance of the number 14, the attraction of this theorem is that it is simple to state and can be examined and proved using concepts available after any first encounter with topology. The goal of this article is both to provide an original investigation into variations of the...

A Minimal Lamination with Cantor Set-like Singularities

Given a compact closed subset M of a line segment in R, we construct a sequence of minimal surfaces Σk embedded in a neighborhood C of the line segment that converge smoothly to a limit lamination of C away from M . Moreover, the curvature of this sequence blows up precisely on M , and the limit lamination has non-removable singularities precisely on the boundary of M .

Minimal Magnetic Energy Theorem

Thomson’s theorem states that static charge distributions in conductors only exist at the conducting surfaces in an equipotential configuration, yielding a minimal electrostatic energy. In this work we present a proof for this theorem based on the variational principle. Furthermore, an analogue statement for magnetic systems is also introduced and proven: the stored magnetic field energy reache...

Tight Closure and the Kodaira Vanishing Theorem