Gauss curvature flow: The fate of the rolling stones.

*(English)*Zbl 0936.35080In 1974 W. J. Firey [Mathematika 21, 1-11 (1974; Zbl 0311.52003)] introduced the motion of a convex surface by its Gauss curvature as a model for the changing shape of a tumbling stone subjected to collisions from all directions with uniform frequency. Under some assumptions concerning the existence and regularity of solutions, he showed that initial surfaces which are symmetric about the origin contract to points and become spherical in shape in the process. He also conjectured that this should be true without any symmetry assumptions on the initial surface.

In 1985 K. S. Chou (Kaising Tso) [Commun. Pure Appl. Math. 38, 867-882 (1985; Zbl 0612.53005)] resolved the existence and regularity questions for uniformly convex initial surfaces and showed contraction to a point as the final time was approached, but was unable to show the surfaces became spherical.

This question has remained open despite a great deal of progress on Gauss curvature flows. Here it is finally settled affirmatively in the two dimensional case. The key step in the proof is an estimate which says that the difference between the principal curvatures of the evolving surfaces is bounded by the maximum value of the difference for the initial surface.

In 1985 K. S. Chou (Kaising Tso) [Commun. Pure Appl. Math. 38, 867-882 (1985; Zbl 0612.53005)] resolved the existence and regularity questions for uniformly convex initial surfaces and showed contraction to a point as the final time was approached, but was unable to show the surfaces became spherical.

This question has remained open despite a great deal of progress on Gauss curvature flows. Here it is finally settled affirmatively in the two dimensional case. The key step in the proof is an estimate which says that the difference between the principal curvatures of the evolving surfaces is bounded by the maximum value of the difference for the initial surface.

Reviewer: John Urbas (Canberra)