Singularities of mean convex level set flow in general ambient manifolds

Robert Haslhofer; Or Hershkovits

doi:10.1016/j.aim.2018.02.008

Singularities of mean convex level set flow in general ambient manifolds

Robert Haslhofer, Or Hershkovits^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

We prove two new estimates for the level set flow of mean convex domains in Riemannian manifolds. Our estimates give control – exponential in time – for the infimum of the mean curvature, and the ratio between the norm of the second fundamental form and the mean curvature. In particular, the estimates remove a stumbling block that has been left after the work of White [16,17,20], and Haslhofer–Kleiner [9], and thus allow us to extend the structure theory for mean convex level set flow to general ambient manifolds of arbitrary dimension.

Original language	American English
Pages (from-to)	1137-1155
Number of pages	19
Journal	Advances in Mathematics
Volume	329
DOIs	https://doi.org/10.1016/j.aim.2018.02.008
State	Published - 30 Apr 2018
Externally published	Yes

Bibliographical note

Funding Information:
Acknowledgments. We thank Brian White for bringing the problem of subsequent singularities in Riemannian manifolds to our attention. This work has been partially supported by the NSF grants DMS-1406394 and DMS-1406407. The second author wishes to thank Jeff Cheeger for his generous support during the work on this project.

Publisher Copyright:
© 2018 Elsevier Inc.

Keywords

Level set flow
Mean curvature flow

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10.1016/j.aim.2018.02.008

Cite this

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title = "Singularities of mean convex level set flow in general ambient manifolds",

abstract = "We prove two new estimates for the level set flow of mean convex domains in Riemannian manifolds. Our estimates give control – exponential in time – for the infimum of the mean curvature, and the ratio between the norm of the second fundamental form and the mean curvature. In particular, the estimates remove a stumbling block that has been left after the work of White [16,17,20], and Haslhofer–Kleiner [9], and thus allow us to extend the structure theory for mean convex level set flow to general ambient manifolds of arbitrary dimension.",

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note = "Funding Information: Acknowledgments. We thank Brian White for bringing the problem of subsequent singularities in Riemannian manifolds to our attention. This work has been partially supported by the NSF grants DMS-1406394 and DMS-1406407. The second author wishes to thank Jeff Cheeger for his generous support during the work on this project. Publisher Copyright: {\textcopyright} 2018 Elsevier Inc.",

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AU - Hershkovits, Or

N1 - Funding Information: Acknowledgments. We thank Brian White for bringing the problem of subsequent singularities in Riemannian manifolds to our attention. This work has been partially supported by the NSF grants DMS-1406394 and DMS-1406407. The second author wishes to thank Jeff Cheeger for his generous support during the work on this project. Publisher Copyright: © 2018 Elsevier Inc.

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N2 - We prove two new estimates for the level set flow of mean convex domains in Riemannian manifolds. Our estimates give control – exponential in time – for the infimum of the mean curvature, and the ratio between the norm of the second fundamental form and the mean curvature. In particular, the estimates remove a stumbling block that has been left after the work of White [16,17,20], and Haslhofer–Kleiner [9], and thus allow us to extend the structure theory for mean convex level set flow to general ambient manifolds of arbitrary dimension.

AB - We prove two new estimates for the level set flow of mean convex domains in Riemannian manifolds. Our estimates give control – exponential in time – for the infimum of the mean curvature, and the ratio between the norm of the second fundamental form and the mean curvature. In particular, the estimates remove a stumbling block that has been left after the work of White [16,17,20], and Haslhofer–Kleiner [9], and thus allow us to extend the structure theory for mean convex level set flow to general ambient manifolds of arbitrary dimension.

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KW - Mean curvature flow

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JO - Advances in Mathematics

JF - Advances in Mathematics

ER -

Singularities of mean convex level set flow in general ambient manifolds

Abstract

Bibliographical note

Keywords

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