Then, if n is greater than one, and if a given immersion, along which a smooth unit normal vector field has been chosen, has positive-definite second fundamental form, then there is a unique solution of the Gauss curvature flow with "initial data" f.[1] If n is equal to one, so that one is in the setting of the curve shortening flow, the condition on the second fundamental form is unnecessary.
[5] In the more general setting of a complete two-dimensional Riemannian manifold which has a certain convexity near infinity, Grayson proved the convergence to a closed geodesic or to a round point.
[6] Kaising Tso applied the methods of Shiu-Yuen Cheng and Shing-Tung Yau's resolution of the Minkowski problem to study the higher-dimensional version of Gage and Hamilton's result.
[9] The key of his proof was an application of the maximum principle to the quantity H2 − 4K, showing that the largest size of the point-by-point difference of the two eigenvalues of the shape operator cannot be increasing in time.
Previous results of Andrews for convex hypersurfaces of Euclidean space, as well as a Li–Yau Harnack inequality found by Bennett Chow, then applied to obtain uniform geometric control over the surfaces comprising the flow.