Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 011, 25 pages      arXiv:2309.14697      https://doi.org/10.3842/SIGMA.2025.011

On Invariants of Constant $p$-Mean Curvature Surfaces in the Heisenberg Group $H_1$

Hung-Lin Chiu ^ab, Sin-Hua Lai ^c and Hsiao-Fan Liu ^de
a) Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan
^b) National Center for Theoretical Sciences, Taipei, Taiwan
^c) Fundamental Education Center, National Chin-Yi University of Technology, Taichung, Taiwan
^d) Department of Applied Mathematics and Data Science, Tamkang University, New Taipei City, Taiwan
^e) Department of Applied Mathematics, National Chung Hsing University, Taiwan

Received April 15, 2024, in final form February 04, 2025; Published online February 18, 2025

Abstract
One primary objective in submanifold geometry is to discover fascinating and significant classical examples of $H_1$. In this paper which relies on the theory we established in [Adv. Math. 405 (2022), 08514, 50 pages, arXiv:2101.11780] and utilizing the approach we provided for constructing constant $p$-mean curvature surfaces, we have identified intriguing examples of such surfaces. Notably, we present a complete description of rotationally invariant surfaces of constant $p$-mean curvature and shed light on the geometric interpretation of the energy $E$ with a lower bound.

Key words: Heisenberg group; Pansu sphere; $p$-minimal surface; Codazzi-like equation; rotationally invariant surface.

pdf (572 kb)   tex (116 kb)  

References

1. Cheng J.-H., Chiu H.-L., Hwang J.-F., Yang P., Umbilicity and characterization of Pansu spheres in the Heisenberg group, J. Reine Angew. Math. 738 (2018), 203-235, arXiv:1406.2444.
2. Cheng J.-H., Hwang J.-F., Malchiodi A., Yang P., Minimal surfaces in pseudohermitian geometry, Ann. Sc. Norm. Super. Pisa Cl. Sci. 4 (2005), 129-177, arXiv:math.DG/0401136.
3. Chiu H.-L., Huang Y.-C., Lai S.-H., An application of the moving frame method to integral geometry in the Heisenberg group, SIGMA 13 (2017), 097, 27 pages, arXiv:1509.00950.
4. Chiu H.-L., Lai S.-H., The fundamental theorem for hypersurfaces in Heisenberg groups, Calc. Var. Partial Differential Equations 54 (2015), 1091-1118.
5. Chiu H.-L., Liu H.-F., A characterization of constant $p$-mean curvature surfaces in the Heisenberg group $H_1$, Adv. Math. 405 (2022), 08514, 50 pages, arXiv:2101.11780.
6. Dragomir S., Tomassini G., Differential geometry and analysis on CR manifolds, Progr. Math., Vol. 246, Birkhäuser, Boston, MA, 2006.
7. Polyanin A.D., Zaitsev V.F., Handbook of exact solutions for ordinary differential equations, 2nd ed., Chapman & Hall/CRC, Boca Raton, FL, 2003.
8. Ritoré M., Rosales C., Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group $\mathbb H^n$, J. Geom. Anal. 16 (2006), 703-720, arXiv:math.DG/0504439.