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

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:

1. 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).

2. 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 \( 0 \) .