A 40-year-old math problem about the curviness of surfaces in complex spaces has finally been solved.
April 29, 2026
Original Paper
Total absolute curvature and rigidity of surfaces in Cartan-Hadamard manifolds
arXiv · 2604.25024
The Takeaway
Closed surfaces in specialized curved spaces called Cartan-Hadamard manifolds must enclose a flat, convex body if their total curvature is minimal. This rule extends a fundamental law of simple Euclidean geometry to much more exotic and warped environments. The solution addresses a challenge first proposed by the famous mathematician Mikhail Gromov in the 1980s. It proves that even in wildly curved universes, certain geometric shapes are forced to remain rigid and predictable. This result provides a new foundation for understanding the structural limits of surfaces in general relativity and high-dimensional physics.
From the abstract
We show that closed surfaces with minimal total absolute curvature in Cartan-Hadamard 3-manifolds bound flat convex bodies. This generalizes Chern-Lashof's theorem for surfaces in Euclidean space and solves a problem posed by Gromov in 1985. Our proof is based on an isometric embedding construction via holonomy, and uses Pogorelov's theory of surfaces with bounded extrinsic curvature. Along the way, we obtain a regularity result for convex hulls and a Schur-type comparison theorem for curves in