Topology of negatively curved real affine algebraic surfaces

Abstract

We find a quartic example of a smooth embedded negatively curved surface in R 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. 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: any such component must either be topologically a plane or an annulus. In particular we prove that there exists no cubic solutions to Hadamard’s problem.