Negatively Curved Minimal Surfaces in General Closed Riemannian Manifolds

Determine whether every closed Riemannian manifold of dimension at least four contains a closed connected surface admitting a smooth minimal immersion whose induced Gaussian curvature is strictly negative at every point.

Background

The paper extends the existence problem for everywhere negatively curved minimal surfaces from round spheres to arbitrary closed Riemannian manifolds. For a minimal immersion into a general ambient manifold, the Gauss equation balances the ambient sectional curvature of the tangent plane against the squared norm of the second fundamental form, so constructing such immersions requires controlling both quantities simultaneously.

The paper notes that the question is answered for round spheres by the main construction and for certain Kähler metrics on complex projective manifolds by Mohsen’s theorem, but it remains unresolved for general closed Riemannian manifolds.

References

Let $(M,g)$ be a closed Riemannian manifold with \dim P\geq4. Does there exist a closed connected surface $\Sigma$ and a smooth minimal immersion $\iota:\Sigma\to(M,g)$ such that the Gaussian curvature of $g_\iota:=\iota*g$ is strictly negative at every point?

— Negatively Curved Minimal Surfaces in the Round Four-Sphere  (2610.12238 - Caniato, 8 Oct 2026) in Question 1 (Song), Section 1, subsection “Open problems”

Whether the immersion can always be chosen to be an embedding is a stronger question.

— Negatively Curved Minimal Surfaces in the Round Four-Sphere  (2610.12238 - Caniato, 8 Oct 2026) in Section 1, subsection “Open problems,” immediately following Question 1