Skip to main content

Non-collapsing in fully nonlinear curvature flows

Journal Article


Download full-text (Open Access)

Abstract


  • We consider embedded hypersurfaces evolving by fully nonlinear flows in which the normal speed of motion is a homogeneous degree one, concave or convex function of the principal curvatures, and prove a non-collapsing estimate: Precisely, the function which gives the curvature of the largest interior sphere touching the hypersurface at each point is a subsolution of the linearized flow equation if the speed is concave. If the speed is convex then there is an analogous statement for exterior spheres. In particular, if the hypersurface moves with positive speed and the speed is concave in the principal curvatures, then the curvature of the largest touching interior sphere is bounded by a multiple of the speed as long as the solution exists. The proof uses a maximum principle applied to a function of two points on the evolving

    hypersurface. We illustrate the techniques required for dealing with such functions in a proof of the known containment principle for flows of hypersurfaces.

Authors


  •   Andrews, Ben H. (external author)
  •   Langford, Mat (external author)
  •   McCoy, James A.

Publication Date


  • 2013

Citation


  • Andrews, B. H., Langford, M. & McCoy, J. (2013). Non-collapsing in fully nonlinear curvature flows. Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire, 30 (1), 23-32.

Scopus Eid


  • 2-s2.0-84872897439

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=1142&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/137

Has Global Citation Frequency


Number Of Pages


  • 9

Start Page


  • 23

End Page


  • 32

Volume


  • 30

Issue


  • 1

Place Of Publication


  • France

Abstract


  • We consider embedded hypersurfaces evolving by fully nonlinear flows in which the normal speed of motion is a homogeneous degree one, concave or convex function of the principal curvatures, and prove a non-collapsing estimate: Precisely, the function which gives the curvature of the largest interior sphere touching the hypersurface at each point is a subsolution of the linearized flow equation if the speed is concave. If the speed is convex then there is an analogous statement for exterior spheres. In particular, if the hypersurface moves with positive speed and the speed is concave in the principal curvatures, then the curvature of the largest touching interior sphere is bounded by a multiple of the speed as long as the solution exists. The proof uses a maximum principle applied to a function of two points on the evolving

    hypersurface. We illustrate the techniques required for dealing with such functions in a proof of the known containment principle for flows of hypersurfaces.

Authors


  •   Andrews, Ben H. (external author)
  •   Langford, Mat (external author)
  •   McCoy, James A.

Publication Date


  • 2013

Citation


  • Andrews, B. H., Langford, M. & McCoy, J. (2013). Non-collapsing in fully nonlinear curvature flows. Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire, 30 (1), 23-32.

Scopus Eid


  • 2-s2.0-84872897439

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=1142&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/137

Has Global Citation Frequency


Number Of Pages


  • 9

Start Page


  • 23

End Page


  • 32

Volume


  • 30

Issue


  • 1

Place Of Publication


  • France