Minimal surfaces in pseudohermitian geometry
| dc.creator | Cheng, Jih-Hsin | |
| dc.creator | Hwang, Jenn-Fang | |
| dc.creator | Malchiodi, Andrea | |
| dc.creator | Yang, Paul | |
| dc.date | 2004-01-14 | |
| dc.date | 2004-06-07 | |
| dc.date.accessioned | 2026-07-07T09:32:45Z | |
| dc.date.available | 2026-07-07T09:32:45Z | |
| dc.description | We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve, and as a quantity in terms of calibration geometry. As a differential equation, the p-minimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set, we formulate some {\em extension} theorems, which describe how the characteristic curves meet the singular set. This allows us to classify the entire solutions to this equation and to solve a Bernstein-type problem (for graphs over the $xy$-plane) in the Heisenberg group $H_1$. In $H_{1}$, identified with the Euclidean space $R^{3}$, the p-minimal surfaces are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set in two dimensions, and generalize to higher dimensions without any size control condition. We also show that there are no closed, connected, $C^{2}$ smoothly immersed constant p-mean curvature or p-minimal surfaces of genus greater than one in the standard $S^{3}.$ This fact continues to hold when $S^{3}$ is replaced by a general spherical pseudohermitian 3-manifold. | |
| dc.description | 45 pages | |
| dc.identifier | https://arxiv.org/abs/math/0401136 | |
| dc.identifier | http://arxiv.org/abs/math/0401136 | |
| dc.identifier | Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (5), 4 (2005) 129-177. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/158896 | |
| dc.subject | Differential Geometry | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 35L80; 35J70; 32V20; 53A10; 49Q10 | |
| dc.title | Minimal surfaces in pseudohermitian geometry | |
| dc.type | text |