Flow of real hypersurfaces by the trace of the Levi form (Q1574745)

The authors study a Cauchy-Riemann analogue of the mean curvature flow in Riemannian geometry. They consider the deformation of a closed real hypersurface of a complex manifold in the normal direction at a speed given at each point by the trace of the Levi form. The evolution is described by a weakly parabolic system. The main difficulty comes from the fact that the trace of the Levi form behaves like a degenerate quasi-elliptic second order differential operator. The authors prove short time existence and smoothness of the solution as long as the curvature remains uniformly bounded with its first derivatives. A barrier principle is established for distinct solution of the flow and it is shown that embeddability is preserved under the flow. In the case \(n=2\) they prove that weakly pseudoconvex hypersurfaces become stricly pseudoconvex instantaneously, providing a canonical approximation.