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arXiv · 2610.38929

Mean-convex horizontal mean curvature flow in the Heisenberg group

Gaia Bombardieri·Mattia Fogagnolo·Valentina Franceschi
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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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Gaia Bombardieri, Mattia Fogagnolo, Valentina Franceschi. 2026-10-28. Mean-convex horizontal mean curvature flow in the Heisenberg group. https://arxiv.org/abs/2610.38929

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