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Martin Franzen

Publications and source records attributed to Martin Franzen.

4 recordsLinked to original sources
arXiv2015

Pinched hypersurfaces contract to round points

We investigate the evolution of closed strictly convex hypersurfaces in $\mathbb{R}^{n+1}$, n=3, for contracting normal velocities, including powers of the mean curvature, of the norm of the second fundamental form, and of the Gauss curvature. We prove convergence to a round point for 2-pinched initial hypersurfaces. In $\mathbb{R}^{n+1}$, n=2, natural quantities exist for proving convergence to a round point for many normal velocities. Here we present their counterparts for arbitrary dimensions $n\in\mathbb{N}$.

math.DG↗
arXiv2015

When maximum-principle functions cease to exist

We consider geometric flow equations for contracting and expanding normal velocities, including powers of the Gauss curvature, of the mean curvature, and of the norm of the second fundamental form, and ask whether - after appropriate rescaling - closed strictly convex surfaces converge to spheres. To prove this, many authors use certain functions of the principal curvatures, which we call maximum-principle functions. We show when such functions cease to exist and exist, while presenting newly discovered maximum-principle functions.

math.DG↗
arXiv2013

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

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.

math.DG↗