Preprint

Inserted: 28 oct 2026
Last Updated: 28 oct 2026

Year: 2026

Abstract:

We study the arrival time equation of the mean-convex horizontal mean curvature flow in the Heisenberg group. We prove existence of viscosity solutions when the initial set is horizontally mean-convex and either non-characteristic, or axisymmetric and with horizontal mean curvature extending continuously across the characteristic set. A key ingredient is an approximation scheme by sets with prescribed Riemannian mean curvature, which preserves the mean-curvature bounds up to an arbitrarily small error. We further show that, when the evolving level sets form a generalized \(C^2\)-foliation, the horizontal mean curvature blows up around characteristic points. This provides an obstruction to self-similar evolution in the axisymmetric class considered above and, in particular, shows that the Pansu sphere cannot evolve self-similarly.

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• HMCF.pdf