IJPAM: Volume 61, No. 2 (2010)

SINGULARITY FORMATION OF EMBEDDED CURVES
EVOLVING ON SURFACES BY CURVATURE FLOW

David L. Johnson$^1$, Murugiah Muraleetharan$^2$
$^1$Department of Mathematics
Lehigh University
Bethlehem, PA 18015, USA
e-mail: david.johnson@lehigh.edu
$^2$Department of Mathematics
University of California
Riverside, CA 92521, USA
e-mail: muralee@math.ucr.edu


Abstract.In this paper, we extend Grayson's Theorem [#!gr89!#] on curvature flow of embedded curves in a compact Riemannian surface. The main result is a direct and shorter proof of a theorem of X. Zhu [#!zhu98!#] that, if a singularity develops in finite time, then the curve converges to a round point in a $C^{\infty}$ sense. The proof will extend Hamilton's isoperimetric estimates technique for curvature flow of embedded curves in the plane [#!ha95!#].

Received: February 9, 2010

AMS Subject Classification: 53C44

Key Words and Phrases: curvature flow, singularities

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 61
Issue: 2