Ricci flow neckpinches without rotational symmetry

James Isenberg; Dan Knopf; Natasa Sesum

Ricci flow neckpinches without rotational symmetry

Differential Geometry 2014-10-23 v3

Authors: James Isenberg , Dan Knopf , Natasa Sesum

Abstract

We study "warped Berger" solutions $\big(\mc S^1\times\mc S^3,G(t)\big)$ of Ricci flow: generalized warped products with the metric induced on each fiber $\{s\}\times\mathrm{SU}(2)$ a left-invariant Berger metric. We prove that this structure is preserved by the flow, that these solutions develop finite-time neckpinch singularities, and that they asymptotically approach round product metrics in space-time neighborhoods of their singular sets, in precise senses. These are the first examples of Ricci flow solutions without rotational symmetry that become asymptotically rotationally symmetric locally as they develop local finite-time singularities.

Keywords

ricci flow geometric flows

Cite

@article{arxiv.1312.2933,
  title  = {Ricci flow neckpinches without rotational symmetry},
  author = {James Isenberg and Dan Knopf and Natasa Sesum},
  journal= {arXiv preprint arXiv:1312.2933},
  year   = {2014}
}

Comments

We correct a miscalculation of some terms in equation (22) and its subsequent uses. The main results of the paper are unchanged, because the methods employed in the proofs are robust enough to give the needed estimates, with only insignificant changes of constants

Ricci flow neckpinches without rotational symmetry

Abstract

Keywords

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