On the evolution of convex hypersurfaces by the $Q_k$ flow

Caputo, M. Cristina; Daskalopoulos, Panagiota; Sesum, Natasa

On the evolution of convex hypersurfaces by the $Q_k$ flow

Authors

Caputo, M. Cristina

Daskalopoulos, Panagiota

Sesum, Natasa

Description

We prove the existence and uniqueness of a $C^{1,1}$ solution of the $Q_k$ flow in the viscosity sense for compact convex hypersurfaces $Σ_t$ embedded in $R^{n+1}$ ($n \geq 2$) . In particular, for compact convex hypersurfaces with flat sides we show that, under a certain non-degeneracy initial condition, the interface separating the flat from the strictly convex side, becomes smooth, and it moves by the $Q_{k-1}$ flow at least for a short time.