Calculus of Variations and Partial Differential Equations

, Volume 8, Issue 1, pp 1-14

First online:

Mean curvature flow singularities for mean convex surfaces

  • Gerhard HuiskenAffiliated with Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany (e-mail: gerhard.huisken@uni-tuebingen.de)
  • , Carlo SinestrariAffiliated with Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, I-00133 Roma, Italy, (e-mail: sinestra@mat.utovrm.it)

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

Abstract.

We study the evolution by mean curvature of a smooth n–dimensional surface \({\cal M}\subset{\Bbb R}^{n+1}\), compact and with positive mean curvature. We first prove an estimate on the negative part of the scalar curvature of the surface. Then we apply this result to study the formation of singularities by rescaling techniques, showing that there exists a sequence of rescaled flows converging to a smooth limit flow of surfaces with nonnegative scalar curvature. This gives a classification of the possible singular behaviour for mean convex surfaces in the case \(n=2\).