Generalized Liu–Terng conjecture for equifocal hypersurfaces

Establish or refute the generalized Liu–Terng conjecture that an appropriate asymptotic behavior of a long-time backward mean curvature flow starting from a hypersurface in a simply-connected symmetric space of compact type implies that the initial hypersurface is equifocal.

Background

The paper studies equifocal hypersurfaces in simply-connected symmetric spaces of compact type as a natural generalization of isoparametric hypersurfaces in spheres. Previous work cited by the authors gives explicit asymptotic formulas for the mean curvature and squared norm of the shape operator along long-time BMCFs starting from equifocal hypersurfaces.

The authors identify a generalized Liu–Terng conjecture as a motivation for obtaining sharper asymptotic estimates. The conjecture asserts a converse-type rigidity principle: suitable asymptotic behavior of a long-time BMCF should force the initial hypersurface to be equifocal. The paper derives estimates for principal orbits of cohomogeneity-one Hermann actions but does not resolve the conjecture in general.