Minimal submanifolds: Encyclopedia of Mathematical Physics

Instantons: topological aspects

Short survey to be published at the Encyclopedia of Mathematical Physics.

物理学中的高阶拓扑斯理论

Higher Topos Theory in Physics

本文为《数学物理百科全书》第2版撰写，简要阐述了高阶拓扑斯理论在数学物理中的意义。

Mathematical Support to Braneworld Theory

The braneworld theory appear with the purpose of solving the problem of the hierarchy of the fundamental interactions. The perspectives of the theory emerge as a new physics, for example, deviation of the law of Newton's gravity. One of the principles of the theory is to suppose that the braneworld is local submanifold in a space of high dimension, the bulk, solution of Einstein's equations in high dimension. In this paper we approach the mathematical consistency of this theory with a new proof of the fundamental theorem of submanifolds for case of semi-Riemannian manifolds. This theorem consi…

高规范理论

Higher Gauge Theory

本文为数学物理百科全书撰写的综述，闪电式介绍高结构与高主丛联络，总结应用与动力学原理，并评论相关领域。

Volume Growth, Number of Ends and the Topology of a Complete Submanifold

Given a complete isometric immersion $ϕ: P^m \longrightarrow N^n$ in an ambient Riemannian manifold $N^n$ with a pole and with radial sectional curvatures bounded from above by the corresponding radial sectional curvatures of a radially symmetric space $M^n_w$, we determine a set of conditions on the extrinsic curvatures of $P$ that guarantees that the immersion is proper and that $P$ has finite topology, in the line of the paper "On Submanifolds With Tamed Second Fundamental Form", (Glasgow Mathematical Journal, 51, 2009), authored by G. Pacelli Bessa and M. Silvana Costa. When the ambient ma…

Geometric Langevin equations on submanifolds and applications to the stochastic melt-spinning process of nonwovens and biology

In this article we develop geometric versions of the classical Langevin equation on regular submanifolds in euclidean space in an easy, natural way and combine them with a bunch of applications. The equations are formulated as Stratonovich stochastic differential equations on manifolds. The first version of the geometric Langevin equation has already been detected before by Lelièvre, Rousset and Stoltz with a different derivation. We propose an additional extension of the models, the geometric Langevin equations with velocity of constant absolute value. The latters are seemingly new and provid…