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Tytuł artykułu

Metric Perspectives of the Ricci Flow Applied to Disjoint Unions

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this paper we consider compact, Riemannian manifolds M1, M2 each equipped with a oneparameter family of metrics g1(t), g2(t) satisfying the Ricci flow equation. Adopting the characterization of super-solutions to the Ricci flow developed by McCann-Topping, we define a super Ricci flow for a family of distance metrics defined on the disjoint union M1 ⊔ M2. In particular, we show such a super Ricci flow property holds provided the distance function between points in M1 and M2 is itself a super solution of the heat equation on M1 × M2. We also discuss possible applications and examples.
Słowa kluczowe
Wydawca
Rocznik
Tom
2
Numer
1
Opis fizyczny
Daty
otrzymano
2014-03-03
zaakceptowano
2014-10-14
online
2014-11-28
Twórcy
  • Institutfür Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115 Bonn,Germany
autor
  • University of Missouri, Dept. of Mathematics, Columbia, MO 65201, USA, munnm@missouri.edu
Bibliografia
  • [1] Angenent, S. and Knopf, D., An example of neck-pinching for Ricci flow on Sn+1. Math. Res. Lett. 11 (2004), no. 4, 493-518.
  • [2] Angenent, S. and Knopf, D., Precise asymptotics of the Ricci flow neckpinch. Comm. Anal. Geom. 15 (2007), no. 4, 773-844.
  • [3] Angenent, S., Caputo, C., and Knopf, D. Minimally invasive surgery for Ricci flow singularities. J. Reine Angew. Math. 672(2012), 39–87.
  • [4] Bakry, D. and Emery, M. Diffusions hypercontractives. In Séminaire de probabilités, XIX, 1983/84, volume 1123 of LectureNotes in Math., pp 177–206. Springer, Berlin, 1985.
  • [5] Böhm, C. and Wilking, B. Manifolds with positive curvature operators are space forms. Ann. of Math. (2) 167 (2008), no. 3,1079–1097.
  • [6] Cordero-Erausquin, D., McCann, R., and Schmuckenschläger, M. Prèkopa-Leindler type inequalities on Riemannian manifolds,Jacobi fields, and optimal transport Ann. Fac. Sci. Toulouse Math. (6) 15 (2006), no. 4, 613–635.
  • [7] Cordero-Erausquin, D.,McCann, R., and Schmuckenschläger, M. A Riemannian interpolation inequality à la Borell, Brascampand Lieb, Invent. Math. 146 (2001), no. 2, 219–257.
  • [8] Ethier, S.N. and Kurtz, T. G. Markov processes. Characterization and convergence. Wiley Series in Probability and MathematicalStatistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986.
  • [9] Feldman, M., Ilmanen, T. and Knopf, D., Rotationally symmetric shrinking and expanding gradient Kahler-Ricci solitons. J.Differential Geom. 65 (2003), no. 2, 169-209.
  • [10] Hamilton, R. Three-manifolds with positive Ricci curvature. J. Differential Geom. 17 (1982), no. 2, 255-306.
  • [11] Hamilton, R. Four-manifolds with positive curvature operator. J. Differential Geom. 24 (1986), no. 2, 153–179.
  • [12] Kleiner, B. and Lott, J. Singular Ricci flows I. arXiv:1408.2271v1
  • [math.DG].
  • [13] Lott, J. and Villani, C., Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2) 169 (2009), no. 3,903-991.
  • [14] Topping, P. and McCann, R., Ricci flow, entropy and optimal transportation. Amer. J. Math. 132 (2010), no. 3, 711-730.
  • [15] Perelman, G. The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159v1 [math.DG].
  • [16] Sesum, N., Curvature tensor under the Ricci flow. Amer. J. Math. 127 (2005), no. 6, 1315-1324.
  • [17] Simon, M. A class of Riemannian manifolds that pinch when evolved by Ricci flow. Manuscripta Math. 101 (2000), no. 1, 89-114.
  • [18] Sturm, K.T. On the geometry of metric measure spaces. I. Acta Math. 196 (2006), no. 1, 65-131.
  • [19] Sturm, K.T. On the geometry of metric measure spaces. II. Acta Math. 196 (2006), no. 1, 133-177.
  • [20] Sturm, K.T. and von Renesse, M., Transport inequalities, gradient estimates, entropy, and Ricci curvature. Comm. Pure Appl.Math. 58 (2005), no. 7, 923-940.
  • [21] Topping, P. Lectures on the Ricci flow. London Mathematical Society Lecture Note Series, 325. Cambridge University Press,Cambridge, 2006.
  • [22] Vuillermot, P. A generalization of Chernoff’s product formula for time-dependent operators. J. Funct. Anal. 259 (2010), no. 11,2923-2938.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_agms-2014-0011
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