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A Relation between Mean Curvature Flow Solitons and Minimal Submanifolds
Author(s) -
Smoczyk Knut
Publication year - 2001
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200109)229:1<175::aid-mana175>3.0.co;2-h
Subject(s) - submanifold , mathematics , mean curvature flow , conformal map , curvature , mean curvature , pointwise , geometric flow , mathematical analysis , flow (mathematics) , space (punctuation) , metric (unit) , soliton , ambient space , scalar curvature , mathematical physics , geometry , physics , nonlinear system , quantum mechanics , linguistics , philosophy , operations management , economics
We derive a one to one correspondence between conformal solitons of the mean curvature flow in an ambient space N and minimal submanifolds in a different ambient space $\tilde N$ where $\tilde N$ equals ℝ × N equipped with a warped product metric and show that a submanifold in N converges to a conformal soliton under the mean curvature flow in N if and only if its associatedsubmanifold in $\tilde N$ converges to a minimal submanifold under a rescaled mean curvature flow in $\tilde N$ . We then define a notion of stability for conformal solitons and obtain L p estimates as well as pointwise estimates for the curvature of stable solitons.