Conic manifolds under the Yamabe flow

Abstract

We consider the unnormalized Yamabe flow on manifolds with conical singularities. Under certain geometric assumption on the initial cross section, we show well-posedness of the short-time solution in the \(L^q\)-setting. Moreover, we give a picture of the deformation of the conical tips under the flow by providing an asymptotic expansion of the evolving metric close to the boundary in terms of the initial local geometry. Due to the blowup of the scalar curvature close to the singularities, we use maximal \(L^q\)-regularity theory for conically degenerate operators.

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Correspondence to Nikolaos Roidos.

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The author was supported by Deutsche Forschungsgemeinschaft, Grant SCHR 319/9-1.

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Roidos, N. Conic manifolds under the Yamabe flow. J. Evol. Equ. 20, 321–334 (2020). https://doi.org/10.1007/s00028-019-00521-9

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Mathematics Subject Classification

  • 35K59
  • 35K65
  • 35R01
  • 53C44