Surfaces of prescribed $p$-mean curvature in the Heisenberg group


Jih-Hsin Cheng 

14:00 - 14:50 

308 , Mathematics Research Center Building (ori. New Math. Bldg.)

I will report on the study of the prescribed $p$% -mean curvature equation in recent years. We interpreted the $p$-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve (in dimension $2$), and as a quantity in terms of calibration geometry. As a differential equation, the prescribed $p$-mean curvature equation is degenerate (hyperbolic and elliptic) in dimension 2 while subelliptic in the nonsingular domain for higher dimensions. Also in dimension 2 with zero $p$-mean curvature, it is a special type of scalar conservation law in the variable of the so-called characteristic angle. We analyzed the singular set and formulated an extension theorem. This allowed us to classify the entire solutions to this equation and to solve a Bernstein-type problem. As a geometric application, we proved the nonexistence of $C^{2}$ smooth hyperbolic surfaces having bounded $p$-mean curvature, immersed in a pseudohermitian 3-manifold. From the variational formulation of the equation, we studied the Dirichlet problem by proving the existence and the uniqueness of the ($p$-)minimizers. An intriguing point is that, in dimension 2, a $C^{2}$ smooth solution from the PDE viewpoint may not be a minimizer. As an application to neuro-biology, we could use such a $p$-minimal surface theory as an image completion model for missing visual data in the first layer of the visual cortex. About the regularity of a $C^{1}$ smooth surface with continuous prescribed $p$-mean curvature, we proved that the horizontal normal gains one more derivative by invoking the jump formulas along characteristic curves. Near a singular (or shock) curve we depicted the Rankine-Hugoniot jump condition as characteristic incident and reflected angles being equal. In the last section we reviewed the second variation formula for a $C^{2}$ smooth $p$-minimal surface in an arbitrary $3$-dimensional pseudohermitian manifold.