Aleksandrov reflection and geometric evolution of hypersurfaces (Q5944144)

Let \(\Gamma_{t} \subset \mathbb R^{n+1}\) be a compact embedded connected \(C^{2}\) hypersurface, with each point evolving in the normal direction with speed proportional to a function of its principal curvatures \(F(\kappa_{1},\ldots,\kappa_{n},t)\), \(0 \leq t < T.\) The function \(F\) is uniformly Lipschitz continuous at each time \(t\) and non-decreasing in each \(\kappa_{i}\), so that the resulting evolution equation is weakly parabolic. Using a parabolic analog of Alexandrov's method of moving planes, the authors derive local a priori Lipschitz bounds for the evolving hypersurface \(\Gamma_{t}\), which hold outside of a convex set determined by \(\Gamma_{0}\), and inside of its convex hull. The results imply, for example, that if \(\Gamma_{t}\) leaves every compact set, then it becomes round. NEWLINENEWLINENEWLINEIt is interesting that in contrast to similar results for strictly parabolic equations, one does not first show pointwise convergence of the curvatures. NEWLINENEWLINENEWLINEThe results generalize to level set flows as well. Starting with an embedded compact (not necessarily connected) \(C^{0}\) hypersurface, one considers the evolution of the level set \(\Gamma_{t}\) of a viscosity solution of a degenerate partial differential equation defined by \(F\). The same bounds hold for the inner and outer boundaries of \(\Gamma_{t}\) in this setting as do in the hypersurface case. As an application, the authors derive new results for inverse mean curvature flow. They construct a solution which is the level set of the limit of viscosity solutions of a family of partial differential equations, and show that \(\Gamma_{t}\) converges in the Lipschitz norm to a round sphere or annulus. They compare these ``extended'' solutions to those in the work of \textit{G. Huisken} and \textit{T. Ilmanen} [``The inverse mean-curvature flow and the Riemannian Penrose inequality'', J. Differ. Geom. 59, 353-437 (2001)], which result from a non-local variational problem.