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 , 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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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.
© 2018 Elsevier Inc.
- Level set flow
- Mean curvature flow