Stable Minimal Hypersurfaces in Positively Curved $4$-Manifolds

Let $M^3\to X^4$ be a complete, connected, two-sided stable minimal immersion. We prove that if the ambient sectional curvature is nonnegative and the ambient scalar curvature has a positive uniform lower bound, then $M$ is totally geodesic and its normal Ricci curvature vanishes. No weak bounded geometry assumption and no upper curvature bound are imposed. We also construct a complete metric of strictly positive sectional curvature on $\mathbb{R}^4$ admitting a complete, embedded, one-ended, nonparabolic, two-sided stable minimal hypersurface diffeomorphic to $\mathbb{R}^3$ which is not totally geodesic. The rigidity proof combines spectral splitting theory, a warped $\mu$-bubble construction, and a harmonic function level set argument. The example is obtained by a compactly supported deformation of an example of Chodosh–Li–Stryker.

Main results

This paper studies complete two-sided stable minimal hypersurfaces of dimension three in positively curved four-manifolds. It identifies the curvature gap that forces rigidity and shows, by an explicit construction, why that gap is essential.

Rigidity

Positive scalar curvature forces total geodesicity

If the ambient sectional curvature is nonnegative and the ambient scalar curvature satisfies $\overline R\geq\kappa>0$, then every complete, connected, two-sided stable minimal immersion $M^3\to X^4$ has $A\equiv0$ and $\overline{\operatorname{Ric}}(\nu,\nu)\equiv0$.

Sharpness

The uniform scalar curvature gap is necessary

There is a complete metric with strictly positive sectional curvature on $\mathbb R^4$ containing a complete, embedded, one-ended, nonparabolic, two-sided stable minimal hypersurface $\Sigma^3\cong\mathbb R^3$ that is not totally geodesic.

Improvement

No bounded-geometry or upper-curvature hypothesis

The rigidity theorem removes the weakly bounded geometry assumption from the earlier result of Chodosh--Li--Stryker. It also requires no uniform upper bound for the ambient sectional curvature.

Proof architecture

The argument replaces extrinsic volume comparison by an intrinsic spectral and potential-theoretic route.

Step 1

Spectral and topological reduction

A positive Jacobi function turns stability into spectral Ricci and scalar curvature inequalities. Sharp splitting and topology theorems reduce the only nontrivial case to a one-ended manifold diffeomorphic to $\mathbb R^3$.

Step 2

Separating warped $\mu$-bubbles

Escaping warped $\mu$-bubble separators are constructed with a uniform integral estimate for their mean curvature, the prescribed curvature term, and the normal derivative of the Jacobi weight.

Step 3

Harmonic functions and flux

A level-set identity gives a uniform $L^1$ bound for capacitor gradients. Nonparabolicity would contradict conservation of flux, so the hypersurface is parabolic; stability cutoffs then force the Jacobi potential to vanish identically.

The counterexample mechanism

Starting from the Chodosh–Li–Stryker positively curved model, the ambient metric is changed only near a compact subset of a totally geodesic stable hypersurface. The perturbation introduces a nonzero trace-free second fundamental form while preserving both the induced metric and the Jacobi potential

$$ |A|^2+\overline{\operatorname{Ric}}(\nu,\nu). $$

Thus completeness, minimality, and stability remain unchanged, while the hypersurface ceases to be totally geodesic. Because the deformation is compactly supported and small in $C^2$, strict positivity of sectional curvature is preserved.