Geometric FlowsPub Date : 2013-07-17DOI: 10.1515/geofl-2015-0007
Eric Bahuaud, R. Mazzeo, E. Woolgar
{"title":"Renormalized volume and the evolution of APEs","authors":"Eric Bahuaud, R. Mazzeo, E. Woolgar","doi":"10.1515/geofl-2015-0007","DOIUrl":"https://doi.org/10.1515/geofl-2015-0007","url":null,"abstract":"Abstract We study the evolution of the renormalized volume functional for even-dimensional asymptotically Poincaré-Einstein metrics (M, g) under normalized Ricci flow. In particular, we prove that where S(g(t)) is the scalar curvature for the evolving metric g(t). This implies that if S +n(n − 1) ≥ 0 at t = 0, then RenV(Mn , g(t)) decreases monotonically. For odd-dimensional asymptotically Poincaré-Einstein metrics with vanishing obstruction tensor,we find that the conformal anomaly for these metrics is constant along the flow. We apply our results to the Hawking-Page phase transition in black hole thermodynamics.We compute renormalized volumes for the Einstein 4-metrics sharing the conformal infinity of an AdS-Schwarzschild black hole. We compare these to the free energies relative to thermal hyperbolic space, as originally computed by Hawking and Page using a different regularization technique, and find that they are equal.","PeriodicalId":310900,"journal":{"name":"Geometric Flows","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123937577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Geometric FlowsPub Date : 2012-03-20DOI: 10.1515/geofl-2017-0001
C. Mantegazza, Samuele Mongodi, Michele Rimoldi
{"title":"The Cotton Tensor and the Ricci Flow","authors":"C. Mantegazza, Samuele Mongodi, Michele Rimoldi","doi":"10.1515/geofl-2017-0001","DOIUrl":"https://doi.org/10.1515/geofl-2017-0001","url":null,"abstract":"Abstract We compute the evolution equation of the Cotton and the Bach tensor under the Ricci flow of a Riemannian manifold, with particular attention to the three dimensional case, and we discuss some applications.","PeriodicalId":310900,"journal":{"name":"Geometric Flows","volume":"84 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123154399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}