Complete Hypersurfaces of Constant Curvature in Hyperbolic Space with Prescribed Asymptotic Boundary.
November 03, 2005 11:00 AM to 12:00 PM
Speakers:
Spruck, Joel
|
 |
Abstract: |
In this talk we will consider the problem of constructing complete hypersurfaces in H^{n+1}with constant higher order curvature f(\kappa)=H_r(\kappa)=\frac{S_r(\kappa)}{S_r(1,\ldots,1)}=\sigma^r\in(0,1) and with prescribed boundary \Gamma at infinity. Using the half-space model for H^{n+1}, we transform this problem into a degenerate fully nonlinear elliptic Dirichlet problem: \begin{eqnarray*}
f(u\kappa +\frac1{W} \vec{1})&=&\sigma^r~~\mbox{in}~~\Omega\\u&=&0 ~~\mbox{on}~~\partial\Omega.\end{eqnarray*}Here \Gamma=\partial \Omega\subset \{x_{n+1}=0\} represents the asymptotic boundary, the \kappa[u]are the Euclidean principal curvatures of S, the graph of u, and \frac1{W}=\nu^{n+1} is the last component of the upward unit normal \nu to S. The cases r=1 (mean curvature) and r=n (Gauss curvature) have received all the attention and we shall survey the known results. We will then present new existence results for the other curvatures. |
|
|
