On maximum-principle functions for flows by powers of the Gauss curvature

Abstract

We consider flows with normal velocities equal to powers strictly larger than one of the Gauss curvature. Under such flows closed strictly convex surfaces converge to points. In his work on the square of the norm of the second fundamental form, Schnürer proposes criteria for selecting quantities that are suitable for proving convergence to a round point. Such monotone quantities exist for many normal velocities, including the Gauss curvature, some powers larger than one of the mean curvature, and some powers larger than one of the norm of the second fundamental form. In this paper, we show that no such quantity exists for any powers larger than one of the Gauss curvature.

31 pages

References in corpus (1)

• Surfaces moving by powers of Gauss curvature

Cited by in corpus (3)

• On the uniqueness of $L_p$-Minkowski problems: the constant $p$-curvature case in ${\mathbb{R}}^3$
• Equivalence of the local and global versions of the $L^p$-Brunn-Minkowski inequality
• When maximum-principle functions cease to exist