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
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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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DOI: https://doi.org/10.1007/978-3-642-16286-2_5
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-16285-5
Online ISBN: 978-3-642-16286-2
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