Vortex on surfaces and Brownian-motion in higher dimensions: special metrics

Abstract

A single hydrodynamic vortex on a surface will in general moves 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_ε(q) $ to remain within this region before escaping through the small ball $ B_ε(q) $, which is centered at $ q $ with radius $ε$ 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_ε(q) $, as $ ε$ approaches $ 0 $.

Figures (3)

• Figure 1: LEFT: Difference $R_1-R_0$ as a function of $a$, where $R_1$ ($R_0$) is the Robin function of the non flat torus $\{ {\mathbb R}^2/(a{\mathbb Z}\times a^{-1}{\mathbb Z}),g_1\}$ (flat torus $\{ {\mathbb R}^2/(a{\mathbb Z}\times a^{-1}{\mathbb Z}),g_0\}$). RIGHT: Graphs of $R_1$ and $R_0$ as a function of $a$. The horizontal line represents the value of the Robin function $R_S$ for a round sphere of area 1. According to okitorus (Appendix): $R_0(a)=-\frac{\log (2 \pi)}{2\pi} - \frac{\log (|\eta(i a^2)|^4 a^2)}{4\pi}$ and $R_S=-\frac{1+\log\pi}{4\pi}$, where $\eta$ is the Dedekind eta function.
• Figure 2: Generating functions of four periodic cylinders (each cylinder is constructed rotating the graph of $X\to Z(X)$ about the $X$-axis). The quotient of a cylinder by the group of periodic translations gives a torus that is isometric to a non-flat torus with a steady-vortex metric. The value of the period $a$ of each torus is shown in the corresponding figure. There are two different tori with $a=3$: one for which the minimal period of $f$ is $3$ and another for which the minimal period of $f$ is $1.5$, and so $f$ oscillates twice inside a fundamental cell.
• Figure 3: Three-dimensional representation of the tori whose generators are shown in Figure \ref{['fig1a']}. See the caption of Figure \ref{['fig1a']} for explanations.

Theorems & Definitions (11)

• Theorem 2.1: Steady Vortex Metric
• Theorem 2.2
• Theorem 2.3
• Theorem 3.1
• Theorem 4.1
• Proposition 4.1
• Lemma A.1
• Lemma A.2
• Lemma A.3
• Theorem A.1: Ding, Jost, Li,and Wang