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PreprintCC BY 4.0arXiv

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 C2-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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Cite

BibTeXRIS
BibTeX
@misc{bombardieri2026meanconvex,
  title = {{Mean-convex horizontal mean curvature flow in the Heisenberg group}},
  author = {Gaia Bombardieri and Mattia Fogagnolo and Valentina Franceschi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38929},
  eprint = {2610.38929},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38929},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Mean-convex horizontal mean curvature flow in the Heisenberg group
AU  - Gaia Bombardieri
AU  - Mattia Fogagnolo
AU  - Valentina Franceschi
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38929
UR  - https://arxiv.org/abs/2610.38929
AB  - 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.
ER  -