Ricci flow of negatively curved incomplete surfaces

Giesen, Gregor; Topping, Peter M.

doi:10.1007/s00526-009-0290-x

Cited by 33 publications

(59 citation statements)

References 12 publications

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“…In particular, a uniqueness result was proved in [9] and a more refined existence result can be found in [10].…”

Section: Added In Proofmentioning

confidence: 99%

Ricci flow compactness via pseudolocality, and flows with incomplete initial metrics

Topping¹

2010

J. Eur. Math. Soc.

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Abstract. By exploiting Perelman's pseudolocality theorem, we prove a new compactness theorem for Ricci flows. By optimising the theory in the two-dimensional case, and invoking the theory of quasiconformal maps, we establish a new existence theorem which generates a Ricci flow starting at an arbitrary incomplete metric, with Gauss curvature bounded above, on an arbitrary surface. The criterion we assert for well-posedness is that the flow should be complete for all positive times; our discussion of uniqueness also invokes pseudolocality.

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“…In particular, a uniqueness result was proved in [9] and a more refined existence result can be found in [10].…”

Section: Added In Proofmentioning

confidence: 99%

Ricci flow compactness via pseudolocality, and flows with incomplete initial metrics

Topping¹

2010

J. Eur. Math. Soc.

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“…Existence and uniqueness of the Ricci flow on incomplete surfaces with negative Gauss curvature was obtained by Giesen and Topping in [15]. In [15] Giesen and Topping proved the following theorem. Theorem 1.1 (Theorem 1.1 of [15]).…”

Section: Introductionmentioning

confidence: 91%

“…Global existence and uniqueness of solutions of the Ricci flow on non-compact manifold R 2 was obtained by Hsu in [3]. Existence and uniqueness of the Ricci flow on incomplete surfaces with negative Gauss curvature was obtained by Giesen and Topping in [15]. In [15] Giesen and Topping proved the following theorem.…”

Section: Introductionmentioning

confidence: 95%

Existence of solution of the logarithmic diffusion equation with bounded above Gauss curvature

Hsu¹

2013

Nonlinear Analysis: Theory, Methods & Applications

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“…The Proof (Adjustment of that in [5]). For every " > 0 consider which is well-defined since We are going to prove .v " u/ 0 on OE0; T and conclude the theorem's statement by letting " !…”

Section: B Comparison Principlementioning

confidence: 99%

Remarks on Hamilton's Compactness Theorem for Ricci flow

Topping¹

2012

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. A fundamental tool in the analysis of Ricci flow is a compactness result of Hamilton in the spirit of the work of Cheeger, Gromov and others. Roughly speaking it allows one to take a sequence of Ricci flows with uniformly bounded curvature and uniformly controlled injectivity radius, and extract a subsequence that converges to a complete limiting Ricci flow. A widely quoted extension of this result allows the curvature to be bounded uniformly only in a local sense. However, in this note we give a counterexample.

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Ricci flow of negatively curved incomplete surfaces

Cited by 33 publications

References 12 publications

Ricci flow compactness via pseudolocality, and flows with incomplete initial metrics

Ricci flow compactness via pseudolocality, and flows with incomplete initial metrics

Existence of solution of the logarithmic diffusion equation with bounded above Gauss curvature

Remarks on Hamilton's Compactness Theorem for Ricci flow

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