Isaac Newton Institute for Mathematical Sciences

$\alpha$-Gauss curvature flows with flat sides

Presenter: Lami Kim (Hokkaido University)

Co-authors: Ki-Ahm Lee (Seoul National University), Eunjai Rhee (Duksung Womens University)


In this paper, we study the deformation of the 2-dimensional convex surfaces in $\mathbb R ^{3}$ whose speed at a point on the surface is proportional to $\alpha$-power of positive part of Gauss curvature. First, for $\frac{1}{2}<\alpha \leq 1$, we show that there is smooth solution if the initial data is smooth and strictly convex and that there is a viscosity solution with $C^{1,1}$-estimate before the collapsing time if the initial surface is only convex. Moreover, we show that there is a waiting time effect which means the flat spot of the convex surface will persist for a while. We also show the interface between the flat side and the strictly convex side of the surface remains smooth on $0 < t < T_0$ under certain necessary regularity and non-degeneracy initial conditions, where $T_0$ is the vanishing time of the flat side.