Ancient solutions of the mean curvature flow
By Robert Haslhofer, Or Hershkovits. 2013
74 citations recorded by the source corpus.
In this short article, we prove the existence of ancient solutions of the mean curvature flow that for t -> 0 collapse to a round point, but for t -> -infinity become more and more oval: near the center they have asymptotic shrinkers modeled on round cylinders S^j x R^n-j and near the tips they have asymptotic translators modeled on Bowl^j+1 x R^n-j-1. We also give a characterization of the round shrinking sphere among ancient alpha-Andrews flows. Our proofs are based on the recent estimates of Haslhofer-Kleiner.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/266e2a0c85ca26533cfbbc1a11586ddb172308ea
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In Applied Mathematics
- Robert Haslhofer
- G. Huisken
- Bruce Kleiner
- Kyeongsu Choi
- S. Brendle
- Carlo Sinestrari
- Or Hershkovits
- N. Šešum
In Geometry and Topology
- Robert Haslhofer
- Bruce Kleiner
- G. Huisken
- Kyeongsu Choi
- S. Brendle
- Or Hershkovits
- Carlo Sinestrari
- N. Šešum
Citations in the corpus
- Uniqueness of two-convex closed ancient solutions to the mean curvature flow (2018, background): “[11] Haslhofer, R., Hershkovits, O., Ancient solutions of the mean curvature flow; Comm. Anal.”
- Convex ancient solutions of the mean curvature flow (2014, background): “Haslhofer and Hershkovitz [12] have proved a similar result (equivalence of (i), (ii), (iii) and (vii)) under the additional assumption that the solutions are α-noncollapsed in the sense of Andrews [2].”
- Mean Curvature Flow of Mean Convex Hypersurfaces (2013, result): “As an immediate consequence of (the smooth admissible version of) the convexity estimate, we obtain the original versions of the convexity estimates due to Huisken and Sinestrari [25, theorem 1.4] and White [38, theorem 8]: COROLLARY 2.10.”
- Ancient asymptotically cylindrical flows and applications (2019, background): “In [Whi03], the last author (see also [Ang13] and [HH16]) gave an example of a noncollapsed ancient, 1We recall that a mean-convex flow is α-noncollapsed [SW09, And12, HK17a] if at each point p ∈ Mt admits interior and exterior balls tangent at p of radius α/H(p).”