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The Weak Maximum Principle

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

Abstract

The maximum principle is the main tool we will use to understand the behaviourof solutions to the Ricci flow. While other problems arising in geo- metric analysis and calculus of variations make strong use of techniques from functional analysis, here – due to the fact that the metric is changing – most of these techniques are not available; although methods in this direction are developed in the work of Perelman [Per02]. The maximum principle, though very simple, is also a very powerful tool which can be used to show that pointwise inequalities on the initial data of parabolic pde are preserved by the evolution. As we have already seen, when the metric evolves by Ricci flow the various curvature tensors R, Ric, and Scal do indeed satisfy systems of parabolic pde. Our main applications of the maximum principle will be to prove that certain inequalities on these tensors are preserved by the Ricci flow, so that the geometry of the evolving metrics is controlled.

Keywords

  • Vector Bundle
  • Maximum Principle
  • Sectional Curvature
  • Ricci Curvature
  • Parallel Transport

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). The Weak Maximum Principle. 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_7

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