Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity

2020/09/24 by Klaus Kroencke, Oliver Lindblad Petersen, Kroencke, Klaus +1 · 1 citation
Mathematics · Physics and Astronomy · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Advanced Differential Geometry Research

paper · pdf · doi:10.1016/j.aim.2026.111139

Abstract

We prove stability of integrable ALE manifolds with a parallel spinor under\nRicci flow, given an initial metric which is close in L^p \∩ L^\∞, for\nany p \∈ (1, n), where n is the dimension of the manifold. In particular,\nour result applies to all known examples of 4-dimensional gravitational\ninstantons. Our decay rates are strong enough to prove positive scalar\ncurvature rigidity in L^p, for each p \∈ \[1, \(n)/(n-2)\),\ngeneralizing a result by Appleton.\n

Citations

• Stability of complex hyperbolic space under curvature-normalized Ricci flow
• On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth
• The mass of an asymptotically flat manifold

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• Ricci flow and the scalar curvature rigidity of Einstein manifolds

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