we-solve (beta)

Math-2677 · mathematics · Geometry

Least volume of nongeodesic minimal hypersurfaces in spheres

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each integer $m\geq 2$, must a closed connected smooth m-manifold minimally immersed in the unit $S^{m+1}$, with image not totally geodesic, have induced volume at least the least volume among the Clifford hypersurfaces $S^p(\surd (p/m))×S^{m-p}(\surd ((m-p)/m))$, $1\leq p<m$?

Short English statement

For each integer $m\geq 2$, must a closed connected smooth m-manifold minimally immersed in the unit $S^{m+1}$, with image not totally geodesic, have induced volume at least the least volume among the Clifford hypersurfaces $S^p(\surd (p/m))×S^{m-p}(\surd ((m-p)/m))$, $1\leq p<m$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2677-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2677

The least-volume theorem claims the Clifford threshold for all closed connected minimal immersions in unit spheres whose images are not totally geodesic. Volume includes immersion multiplicity, so the conclusion covers the full immersion scope rather than only embedded hypersurfaces.

Least volume of nongeodesic minimal hypersurfaces in spheres

OpenAI's claim

Least volume of nongeodesic minimal hypersurfaces: The announced lower bound for a non-totally-geodesic closed minimal immersion into a round sphere is the volume of the smallest Clifford product in the same dimension. Domain volume counts any covering multiplicity.

Mathematical statement of the claimed result

Mathematical claim 1.
For an integer $m\ge2$, let $a_m=\min_{1\le k<m}\operatorname{Vol}_m\left(S^k(\sqrt{k/m})\times S^{m-k}(\sqrt{(m-k)/m})\right)$. For every closed connected smooth $m$-manifold and immersion $F:M\to S^{m+1}$ which is smooth and minimal, with unit curvature on the target sphere, if its image is not totally geodesic, the claimed sharp volume bound is \[\operatorname{Vol}(M,F^*g_{\mathrm{round}})\ge a_m.\] Volume is measured on the domain and includes immersion multiplicities. A minimizing Clifford product attains the threshold.

Outline of the argument

Least volume of nongeodesic minimal hypersurfaces: A family based on Lorentz-distance parameters detects the relevant geometric class cohomologically and feeds a min--max comparison. Average-curvature and index rigidity control the extremizing configuration, while low-density cone analysis supplies attainment. The immersion reduction preserves multiplicity, so the final volume comparison concerns the domain rather than only its image.

Formal verification

Least volume of nongeodesic minimal hypersurfaces: The pinned catalogue lists no Lean formalization for this result.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

Go to contribution →