{"title":"Vortex on Surfaces and Brownian Motion in Higher Dimensions: Special Metrics","authors":"","doi":"10.1007/s00332-023-10007-1","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>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: <ol> <li> <p>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).</p> </li> <li> <p>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.</p> </li> </ol> 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 <span> <span>\\( M \\)</span> </span>, the “narrow escape time” (NET) is defined as the expected time for a Brownian motion starting at a point <span> <span>\\( p \\)</span> </span> in <span> <span>\\( M {\\setminus } B_\\epsilon (q) \\)</span> </span> to remain within this region before escaping through the small ball <span> <span>\\( B_\\epsilon (q) \\)</span> </span>, which is centered at <span> <span>\\( q \\)</span> </span> with radius <span> <span>\\( \\epsilon \\)</span> </span> 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 <span> <span>\\( p \\)</span> </span>, remains invariant regardless of the position of the escape window <span> <span>\\( B_\\epsilon (q) \\)</span> </span>, as <span> <span>\\( \\epsilon \\)</span> </span> approaches <span> <span>\\( 0 \\)</span> </span>.</p>","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":"6 1","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2024-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Nonlinear Science","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00332-023-10007-1","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
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 \( 0 \).
期刊介绍：
The mission of the Journal of Nonlinear Science is to publish papers that augment the fundamental ways we describe, model, and predict nonlinear phenomena. Papers should make an original contribution to at least one technical area and should in addition illuminate issues beyond that area''s boundaries. Even excellent papers in a narrow field of interest are not appropriate for the journal. Papers can be oriented toward theory, experimentation, algorithms, numerical simulations, or applications as long as the work is creative and sound. Excessively theoretical work in which the application to natural phenomena is not apparent (at least through similar techniques) or in which the development of fundamental methodologies is not present is probably not appropriate. In turn, papers oriented toward experimentation, numerical simulations, or applications must not simply report results without an indication of what a theoretical explanation might be.
All papers should be submitted in English and must meet common standards of usage and grammar. In addition, because ours is a multidisciplinary subject, at minimum the introduction to the paper should be readable to a broad range of scientists and not only to specialists in the subject area. The scientific importance of the paper and its conclusions should be made clear in the introduction-this means that not only should the problem you study be presented, but its historical background, its relevance to science and technology, the specific phenomena it can be used to describe or investigate, and the outstanding open issues related to it should be explained. Failure to achieve this could disqualify the paper.
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