Finite index versus finite total curvature in ambient dimension seven

Determine whether every complete two-sided minimal immersion of a six-dimensional hypersurface into Euclidean seven-space with finite Morse index necessarily has finite total curvature.

Background

The paper assumes finite total curvature for its two ambient-eight results because finite index and finite total curvature are not equivalent in all dimensions. Finite total curvature implies finite index in every dimension, while the converse is known for complete two-sided minimal immersions in Euclidean spaces of dimensions three through six.

The authors explain that the converse fails in every ambient dimension at least eight, using stable cones and their associated smooth area-minimizing hypersurfaces. Ambient dimension seven is the remaining unresolved case: it is unknown whether finite Morse index forces finite total curvature for complete two-sided minimal immersions into Euclidean seven-space.

References

The remaining case $N=7$ is open in general.

— Catenoid-sharp index-topology estimates for minimal hypersurfaces  (2609.03424 - Li, 3 Sep 2026) in Section 1, Introduction, paragraph beginning “Compared to \cite{CG}, here we need to assume finite total curvature”