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Short-Time Existence

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2011)

Abstract

An important foundational step in the study of any system of evolutionary partial differential equations is to show short-time existence and uniqueness. For the Ricci flow, unfortunately, short-time existence does not follow from standard parabolic theory, since the flow is only weakly parabolic. To overcome this, Hamilton's seminal paper [Ham82b] employed the deep Nash –Moser implicit function theorem to prove short-time existence and uni- queness. A detailed exposition of this result and its applications can be found in Hamilton's survey [Ham82a]. DeTurck [DeT83]later found a more direct proof by modifying the flow by a time-dependent change of variables to make it parabolic. It is this method that we will follow.

Keywords

  • Bianchi Identity
  • Ricci Tensor
  • Principal Symbol
  • Ricci Flow
  • Geometric Invariance

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Correspondence to Ben Andrews .

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© 2011 Springer-Verlag Berlin Heidelberg

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Andrews, B., Hopper, C. (2011). Short-Time Existence. In: The Ricci Flow in Riemannian Geometry. Lecture Notes in Mathematics(), vol 2011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16286-2_5

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