Regularity of C^{1} smooth surfaces with prescribed p-mean curvature in the Heisenberg group

Details Regularity of C^{1} smooth surfaces with prescribed p-mean curvature in the Heisenberg group Regularity of C^{1} smooth surfaces with prescribed p-mean curvature in the Heisenberg group

2007-09-12

arxiv.org/abs/0709.1776v2

Mathematics

Differential Geometry

30 pages

Scientific paper

We consider a $C^{1}$ smooth surface with prescribed $p$(or $H$)-mean curvature in the 3-dimensional Heisenberg group. Assuming only the prescribed $p$-mean curvature $H\in C^{0},$ we show that any characteristic curve is $C^{2}$ smooth and its (line) curvature equals $-H$ in the nonsingular domain$.$ By introducing characteristic coordinates and invoking the jump formulas along characteristic curves, we can prove that the Legendrian (or horizontal) normal gains one more derivative. Therefore the seed curves are $C^{2}$ smooth. We also obtain the uniqueness of characteristic and seed curves passing through a common point under some mild conditions, respectively. These results can be applied to more general situations.

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