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Neumann and second boundary value problems for Hessian and Gauß curvature flows. (English) Zbl 1032.53058

This paper has two main parts. In the first part the authors consider the flow of a strictly convex graphical hypersurface by its Gauss curvature. They show that for the Neumann boundary condition and for the second boundary condition the flow has a smooth solution for all time and as $t\to \infty$ it converges to a solution of the prescribed Gauss curvature equation. More general Monge-Ampère equations are also considered.

In the second part they consider Hessian flows in conjunction with the second boundary condition and prove long time existence and convergence to a stationary solution.

The results can be viewed as parabolic versions of results proved by P.-L. Lions, N. S. Trudinger and J. I. E. Urbas [Commun. Pure Appl. Math. 39, 539–563 (1986; Zbl 0604.35027)] and by the reviewer J. Urbas [Commun. Partial Differ. Equ. 26, 859–882 (2001; Zbl 1194.35158)] for the corresponding elliptic equations.

MSC:
 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 35K55 Nonlinear parabolic equations 35K60 Nonlinear initial value problems for linear parabolic equations 53C42 Immersions (differential geometry)