Topology of negatively curved real affine algebraic surfaces

Chris Connell, Mohammad Ghomi
2008 Journal für die Reine und Angewandte Mathematik  
We find a quartic example of a smooth embedded negatively curved surface in R 3 homeomorphic to a doubly punctured torus. This constitutes an explicit solution to Hadamard's problem on constructing complete surfaces with negative curvature and Euler characteristic in R 3 . Further we show that our solution has the optimal degree of algebraic complexity via a topological classification for smooth cubic surfaces with a negatively curved component in R 3 : any such component must either be
more » ... t either be topologically a plane or an annulus. In particular we prove that there exists no cubic solutions to Hadamard's problem.
doi:10.1515/crelle.2008.078 fatcat:xc5alqql35cj7jagyqnpcsr3la