Compact surfaces of negative curvature - MathOverflow

Compact surfaces of negative curvature

2010-07-20T05:22:31Z

John Hubbard recently told me that he has been asking people if there are compact surfaces of negative curvature in $\mathbb{R}^4$ without getting any definite answers. I had assumed it was possible, but couldn't come up with an easy example off the top of my head.

In $\mathbb{R}^3$ it is easy to show that surfaces of negative curvature can't be compact: throw planes at your surface from very far away. At the point of first contact, your plane and the surface are tangent. But the surface is everywhere saddle-shaped, so it cannot be tangent to your plane without actually piercing it, contradicting first contact.

This easy argument fails in $\mathbb{R}^4$. Can the failure of the easy argument be used to construct an example? Is there a simple source of compact negative curvature surfaces in $\mathbb{R}^4$?

Answer by Mariano Suárez-Alvarez for Compact surfaces of negative curvature

2010-07-20T06:00:46Z

You will find examples (topologically, spheres with seven handles) in section 5.5 of Surfaces of Negative Curvature by E. R. Rozendorn, in Geometry III: Theory of surfaces, Yu. D. Burago VI A. Zalgaller (Eds.) EMS 48.

Rozendorn tells us that «from the visual point of view, their construction seems fairly simple.» Well...

Answer by Louis for Compact surfaces of negative curvature

2011-12-03T23:06:51Z

In "Y. Martinez-Maure, A counter-example to a conjectured characterization of the sphere. (Contre-exemple à une caractérisation conjecturée de la sphère.) (French), C. R. Acad. Sci., Paris, Sér. I, Math. 332, 41-44 (2001), the author disproves an old characterization of the 2-sphere by giving an exemple of a "hyperbolic hedgehog" of R^3 (a sphere-homeomorphic envelope parametrized by its Gauss map whose Gaussian curvature K is everywhere negative excepted at four singular points where K is infinite).

By projective duality, this implies the existence of a 2-sphere C^2 embedded in the 3-sphere with a nonpositive extrinsic curvature but not totally geodesic.