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Energy convexity and uniformity of the H-surface flow in ℝ3 with Dirichlet boundary condition

  • Da Rong Cheng
  • Longzhi Lin
  • Xin Zhou
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Abstract

We prove that the energy functional associated with surfaces of prescribed mean curvature in ℝ3 exhibits a convexity property when restricted to maps from the unit $2$-disk having small Dirichlet energy and a fixed boundary value, provided that the $3$-form H · volℝ3 has a primitive satisfying certain bounds. Under milder assumptions on H:ℝ3 → ℝ, we show that an analogous convexity estimate holds along the heat flow of the functional (the H-surface flow) when the initial Dirichlet energy is sufficiently small. This is done first for classical solutions, and then extended by approximation to weak solutions using a quantitative uniqueness result adapted from previous work on the harmonic map heat flow. As a consequence of the convexity estimate, we show that the H-surface flow with small-energy initial map of class C0 ∩ W1, 2 on the unit $2$-disk converges uniformly to a unique limit at infinite time, which solves the corresponding stationary problem (the H-surface system) with the same boundary value.

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Cite

BibTeXRIS
BibTeX
@misc{cheng2026energy,
  title = {{Energy convexity and uniformity of the $H$-surface flow in $\mathbbR^3$ with Dirichlet boundary condition}},
  author = {Da Rong Cheng and Longzhi Lin and Xin Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30740},
  eprint = {2609.30740},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30740},
}
RIS
TY  - UNPB
TI  - Energy convexity and uniformity of the H-surface flow in ℝ^3 with Dirichlet boundary condition
AU  - Da Rong Cheng
AU  - Longzhi Lin
AU  - Xin Zhou
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.30740
UR  - https://arxiv.org/abs/2609.30740
AB  - We prove that the energy functional associated with surfaces of prescribed mean curvature in ℝ^3 exhibits a convexity property when restricted to maps from the unit $2$-disk having small Dirichlet energy and a fixed boundary value, provided that the $3$-form H · vol_ℝ^3 has a primitive satisfying certain bounds. Under milder assumptions on H:ℝ^3 → ℝ, we show that an analogous convexity estimate holds along the heat flow of the functional (the H-surface flow) when the initial Dirichlet energy is sufficiently small. This is done first for classical solutions, and then extended by approximation to weak solutions using a quantitative uniqueness result adapted from previous work on the harmonic map heat flow. As a consequence of the convexity estimate, we show that the H-surface flow with small-energy initial map of class C^0 ∩ W^1, 2 on the unit $2$-disk converges uniformly to a unique limit at infinite time, which solves the corresponding stationary problem (the H-surface system) with the same boundary value.
ER  -