Vortex on Surfaces and Brownian Motion in Higher Dimensions: Special Metrics

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS ACS Applied Bio Materials Pub Date : 2024-01-28 DOI:10.1007/s00332-023-10007-1

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Abstract

A single hydrodynamic vortex on a surface will in general move unless its Riemannian metric is a special “Steady Vortex Metric” (SVM). Metrics of constant curvature are SVM only in surfaces of genus zero and one. In this paper:

I show that K. Okikiolu’s work on the regularization of the spectral zeta function leads to the conclusion that each conformal class of every compact surface with a genus of two or more possesses at least one steady vortex metric (SVM).
I apply a probabilistic interpretation of the regularized zeta function for surfaces, as developed by P. G. Doyle and J. Steiner, to extend the concept of SVM to higher dimensions.

The new special metric, which aligns with the Steady Vortex Metric (SVM) in two dimensions, has been termed the “Uniform Drainage Metric” for the following reason: For a compact Riemannian manifold \( M \) , the “narrow escape time” (NET) is defined as the expected time for a Brownian motion starting at a point \( p \) in \( M {\setminus } B_\epsilon (q) \) to remain within this region before escaping through the small ball \( B_\epsilon (q) \) , which is centered at \( q \) with radius \( \epsilon \) and acts as the escape window. The manifold is said to possess a uniform drainage metric if, and only if, the spatial average of NET, calculated across a uniformly distributed set of initial points \( p \) , remains invariant regardless of the position of the escape window \( B_\epsilon (q) \) , as \( \epsilon \) approaches \( 0 \) .

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曲面上的涡旋和高维布朗运动：特殊度量

摘要除非表面的黎曼度量是一种特殊的 "稳定涡度度量"（SVM），否则表面上的单个流体动力涡一般都会移动。恒曲率度量只有在零属和一属曲面中才是 SVM。在本文中我证明了 K. Okikiolu 关于谱zeta函数正则化的工作导致了这样一个结论：每一个属大于或等于 2 的紧凑曲面的每一个共形类都拥有至少一个稳定涡度公设 (SVM)。我运用 P. G. Doyle 和 J. Steiner 对曲面正则化zeta函数的概率解释，将 SVM 的概念扩展到更高维度。新的特殊度量与二维中的稳定涡度（SVM）一致，被称为 "均匀排水度量"，原因如下：对于一个紧凑的黎曼流形（M ），"窄逃逸时间"（NET）被定义为从（ M {setminus }中的（ p ）点开始的布朗运动的预期时间。(B_epsilon(q)\)中的一个点开始的布朗运动在通过小球 \( B_epsilon (q) \)逃逸之前停留在这个区域内的预期时间。为圆心，半径为 \( \epsilon \) 的小球作为逃逸窗口。当且仅当在一组均匀分布的初始点 \( p \) 上计算的NET的空间平均值无论逃逸窗口的位置如何都保持不变时，流形被称为具有均匀排水度量。随着（ \( epsilon \)）接近（ 0 \)）。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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