5/11/2007

Jih-Hsin Cheng
Academica Sinica

Regularity of C^{1} smooth solutions to the mean curvature equation in the Heisenberg group

We consider a C^{1} smooth solution to the p(or H)-mean curvature equation in the 3-dimensional Heisenberg group. Assuming only the p-mean curvature H is C^{0}, we show that any characteristic curve is C^{2} smooth and its curvature equals -H. By introducing special coordinates and invoking the jump formulas along characteristic curves, we can prove that the Legendrian (horizontal) normal gains one more derivative. Therefore the seed curves are C^{2} smooth.