Dynamical (in)stability of Ricci-flat ALE metrics along Ricci flow

We study the stability and instability of ALE Ricci-flat metrics around which a Lojasiewicz inequality is satisfied for a variation of Perelman’s λ functional adapted to the ALE situation and denoted λALE. This functional was introduced by the authors in a recent work and it has been proven that it satisfies a good enough Lojasiewicz inequality at least in neighborhoods of integrable ALE Ricci-flat metrics in dimension larger than or equal to 5.

[1]  Some gradient estimates for the heat equation on domains and for an equation by Perelman , 2006, math/0605518.

[2]  Remo Guidieri.  Res , 1995, RES: Anthropology and Aesthetics.

[3]  R. Bartnik.  The mass of an asymptotically flat manifold , 1986 .

[4]  Christine Guenther,et al.  The fundamental solution on manifolds with time-dependent metrics , 2002 .

[5]  Brett L. Kotschwar.  Short-time persistence of bounded curvature under the Ricci flow , 2015, 1507.08246.

[6]  Qi S. Zhang,et al.  Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature—Part II , 2015, Calculus of Variations and Partial Differential Equations.

[7]  S. Łojasiewicz.  Ensembles semi-analytiques , 1965 .

[8]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[9]  Laurent Saloff-Coste,et al.  Aspects of Sobolev-type inequalities , 2001 .

[10]  G. Perelman.  The entropy formula for the Ricci flow and its geometric applications , 2002, math/0211159.

[11]  S. Brendle,et al.  Gluing Eguchi‐Hanson Metrics and a Question of Page , 2014, 1405.0056.

[12]  T. Lamm,et al.  Weak stability of Ricci expanders with positive curvature operator , 2016, 1601.00325.

[13]  Weighted Sobolev Inequalities and Ricci Flat Manifolds , 2006, math/0602136.

[14]  Dominic Joyce,et al.  Compact Manifolds with Special Holonomy , 2000 .

[15]  Y. Choquet-Bruhat,et al.  Problèmes elliptiques du second ordre sur une variété euclidienne à l'infini , 1979 .

[16]  Alexander Grigor cprimeyan.  Gaussian upper bounds for the heat kernel on arbitrary manifolds , 1997 .

[17]  Ryosuke Takahashi.  An Ancient Solution of The Ricci Flow in Dimension 4 Converging to the Euclidean Schwarzschild Metric , 2012, 1206.5873.

[18]  Alexander Appleton.  Scalar curvature rigidity and Ricci DeTurck flow on perturbations of Euclidean space , 2016, Calculus of Variations and Partial Differential Equations.

[19]  N. Krylov,et al.  Lectures on Elliptic and Parabolic Equations in Holder Spaces , 1996 .

[20]  S. Yau,et al.  On the parabolic kernel of the Schrödinger operator , 1986 .

[21]  Dan Knopf,et al.  Stability of the Ricci flow at Ricci-flat metrics , 2002 .

[22]  Yu Li.  Ricci flow on asymptotically Euclidean manifolds , 2016, 1603.05336.

[23]  F. Schulze,et al.  Stability of hyperbolic space under Ricci flow , 2010, 1003.2107.

[24]  Thomas H. Parker,et al.  The Yamabe problem , 1987 .

[25]  M. Feldman,et al.  Rotationally symmetric shrinking and ex - panding gradient Kahler - Ricci solitons , 2003 .

[26]  S. Bando,et al.  On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth , 1989 .

[27]  L. Simon.  Asymptotics for a class of non-linear evolution equations, with applications to geometric problems , 1983 .

[28]  R. Hamilton.  Three-manifolds with positive Ricci curvature , 1982 .

[29]  Robert Haslhofer.  Perelman’s lambda-functional and the stability of Ricci-flat metrics , 2010, 1003.4633.

[30]  R. Hamilton.  A Compactness Property for Solutions of the Ricci Flow , 1995 .

[31]  Wan-Xiong Shi.  Deforming the metric on complete Riemannian manifolds , 1989 .

[32]  M. Simon.  Extending four-dimensional Ricci flows with bounded scalar curvature , 2015, Communications in Analysis and Geometry.

[33]  Mass Under the Ricci Flow , 2005, math/0510083.

[34]  Linear and dynamical stability of Ricci-flat metrics , 2004, math/0410062.

[35]  E. Calabi,et al.  Métriques kählériennes et fibrés holomorphes , 1979 .

[36]  Wan-Xiong Shi.  Ricci deformation of the metric on complete noncompact Riemannian manifolds , 1989 .

[37]  Tristan Ozuch,et al.  A Lojasiewicz inequality for ALE metrics , 2020, ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE.

[38]  Thierry Aubin,et al.  Nonlinear analysis on manifolds, Monge-Ampère equations , 1982 .

[39]  Klaus Kroencke,et al.  Stability of ALE Ricci-Flat Manifolds Under Ricci Flow , 2017, The Journal of Geometric Analysis.

[40]  Reto Müller,et al.  Dynamical stability and instability of Ricci-flat metrics , 2013, 1301.3219.

[41]  B. M. Fulk.  MATH , 1992 .

[42]  C. Ricketts.  Pages on Art , 2015 .

[43]  Alexander Appleton.  Eguchi-Hanson singularities in U(2)-invariant Ricci flow , 2019, 1903.09936.

[44]  B. Chow,et al.  Hamilton's Ricci Flow , 2018 .