摘要
This work reports on the author's recent study about regularity and the singular set of a C 1 smooth surface with prescribed p (or H)-mean curvature in the 3-dimensional Heisenberg group.As a differential equation,this is a degenerate hyperbolic and elliptic PDE of second order,arising from the study of CR geometry.Assuming only the p-mean curvature H ∈ C 0,it is shown that any characteristic curve is C 2 smooth and its (line) curvature equals-H.By introducing special coordinates and invoking the jump formulas along characteristic curves,it is proved that the Legendrian (horizontal) normal gains one more derivative.Therefore the seed curves are C 2 smooth.This work also obtains the uniqueness of characteristic and seed curves passing through a common point under some mild conditions,respectively.In an on-going project,it is shown that the p-area element is in fact C 2 smooth along any characteristic curve and satisfies a certain ordinary differential equation of second order.Moreover,this ODE is analyzed to study the singular set.
This work reports on the author’s recent study about regularity and the singular set of a C 1 smooth surface with prescribed p (or H)-mean curvature in the 3-dimensional Heisenberg group. As a differential equation, this is a degenerate hyperbolic and elliptic PDE of second order, arising from the study of CR geometry. Assuming only the p-mean curvature H ∈ C 0, it is shown that any characteristic curve is C 2 smooth and its (line) curvature equals ?H. By introducing special coordinates and invoking the jump formulas along characteristic curves, it is proved that the Legendrian (horizontal) normal gains one more derivative. Therefore the seed curves are C 2 smooth. This work also obtains the uniqueness of characteristic and seed curves passing through a common point under some mild conditions, respectively. In an on-going project, it is shown that the p-area element is in fact C 2 smooth along any characteristic curve and satisfies a certain ordinary differential equation of second order. Moreover, this ODE is analyzed to study the singular set.
基金
supported by the "Science Council" of Taiwan 11529,China (Grant No. 97-2115-M-001-016-MY3)
