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 $\mathbb{R}^4$? 



Compact surfaces of negative curvature
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...


