Columbia Mathematics Department Colloquium

"  Locally convex hypersurfaces of constant curvature with boundary "


Joel Spruck

(Johns Hopkins University)


In this talk we will discuss the problem of finding a locally convexhypersurface S in R^{n+1}
of constant curvature f(\kappa)=\sigma>0 with prescribed boundary \Gamma. Here f is a smooth
symmetric function and \kappa=(\kappa_1, ... , \kappa_n) are the principal curvatures of S.
The function f(\kappa) must satisfy additional conditions so that theresulting fully nonlinear equation
for S is elliptic. A classical example is f(\kappa)=\sigma_n(\kappa)=K the (extrinsic) Gauss curvature.
This question is quite subtle for even deciding when a smooth Jordancurve in R^3 bounds a surface
of positive curvature is open.

  February 6th, Wednesday, 5:00-6:00 pm
Mathematics 520
Tea will be served at 4:30pm