The Ricci flow of the `RP3 geon' and noncompact manifolds with essential minimal spheres

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Sammanfattning

It is well-known that the Ricci flow of a closed 3-manifold containing an essential minimal 2-sphere will fail to exist after a finite time. Conversely, the Ricci flow of a complete, rotationally symmetric, asymptotically flat manifold containing no minimal spheres is immortal. We discuss an intermediate case, that of a complete, noncompact manifold with essential minimal hypersphere. For 3-manifolds, if the scalar curvature vanishes on asymptotic ends and is bounded below initially by a negative constant (that depends on the initial area of the minimal sphere), we show that a singularity develops in finite time. In particular, this result applies to asymptotically flat manifolds, which are a boundary case with respect to the neckpinch theorem of M Simon. We provide numerical evolutions to explore the case where the initial scalar curvature is less than the bound.
Originalspråkengelska
TidskriftarXiv.org
ISSN2331-8422
Status!!Unpublished - 2010

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title = "The Ricci flow of the `RP3 geon' and noncompact manifolds with essential minimal spheres",
abstract = "It is well-known that the Ricci flow of a closed 3-manifold containing an essential minimal 2-sphere will fail to exist after a finite time. Conversely, the Ricci flow of a complete, rotationally symmetric, asymptotically flat manifold containing no minimal spheres is immortal. We discuss an intermediate case, that of a complete, noncompact manifold with essential minimal hypersphere. For 3-manifolds, if the scalar curvature vanishes on asymptotic ends and is bounded below initially by a negative constant (that depends on the initial area of the minimal sphere), we show that a singularity develops in finite time. In particular, this result applies to asymptotically flat manifolds, which are a boundary case with respect to the neckpinch theorem of M Simon. We provide numerical evolutions to explore the case where the initial scalar curvature is less than the bound.",
author = "Balehowsky, {Tracey Jean}",
year = "2010",
language = "English",
journal = "arXiv.org",
issn = "2331-8422",
publisher = "Cornell University",

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The Ricci flow of the `RP3 geon' and noncompact manifolds with essential minimal spheres. / Balehowsky, Tracey Jean.

I: arXiv.org , 2010.

Forskningsoutput: TidskriftsbidragAndra artiklarVetenskaplig

TY - JOUR

T1 - The Ricci flow of the `RP3 geon' and noncompact manifolds with essential minimal spheres

AU - Balehowsky, Tracey Jean

PY - 2010

Y1 - 2010

N2 - It is well-known that the Ricci flow of a closed 3-manifold containing an essential minimal 2-sphere will fail to exist after a finite time. Conversely, the Ricci flow of a complete, rotationally symmetric, asymptotically flat manifold containing no minimal spheres is immortal. We discuss an intermediate case, that of a complete, noncompact manifold with essential minimal hypersphere. For 3-manifolds, if the scalar curvature vanishes on asymptotic ends and is bounded below initially by a negative constant (that depends on the initial area of the minimal sphere), we show that a singularity develops in finite time. In particular, this result applies to asymptotically flat manifolds, which are a boundary case with respect to the neckpinch theorem of M Simon. We provide numerical evolutions to explore the case where the initial scalar curvature is less than the bound.

AB - It is well-known that the Ricci flow of a closed 3-manifold containing an essential minimal 2-sphere will fail to exist after a finite time. Conversely, the Ricci flow of a complete, rotationally symmetric, asymptotically flat manifold containing no minimal spheres is immortal. We discuss an intermediate case, that of a complete, noncompact manifold with essential minimal hypersphere. For 3-manifolds, if the scalar curvature vanishes on asymptotic ends and is bounded below initially by a negative constant (that depends on the initial area of the minimal sphere), we show that a singularity develops in finite time. In particular, this result applies to asymptotically flat manifolds, which are a boundary case with respect to the neckpinch theorem of M Simon. We provide numerical evolutions to explore the case where the initial scalar curvature is less than the bound.

UR - https://arxiv.org/abs/1004.1833

M3 - Other articles

JO - arXiv.org

JF - arXiv.org

SN - 2331-8422

ER -