Non-persistence of strongly isolated singularities, and geometric applications
Alessandro Carlotto, Yangyang Li, Zhihan Wang
TL;DR
The paper proves a generic-regularity theorem for stationary integral $n$-varifolds with MSI in arbitrary ambient dimensions, showing that for a generic metric on a closed manifold every such varifold is either smooth or has a non-strongly isolated singularity (codimension one improvements strengthen the result). Central to the argument is a novel index-counting formula for the Jacobi operator in terms of the Morse indices of the singular cone links, which is nonpositive in general but becomes nonnegative for generic metrics, enabling the Sard–Smale-type perturbation analysis. This leads to a generic finiteness result for closed minimal hypersurfaces with area below a sharp threshold in nearly round $S^4$, and obstructions to persistence of Clifford football-type singularities under metric perturbations. The approach blends weighted-analytic Jacobi theory for MSI, transfer of normal sections, and a canonical-neighborhood Sard–Smale framework to handle higher codimension without relying on Banach-manifold structures, yielding robust generic-regularity conclusions with clear geometric consequences.
Abstract
We obtain a generic regularity result for stationary integral $n$-varifolds with only strongly isolated singularities inside $N$-dimensional Riemannian manifolds, in absence of any restriction on the dimension ($n\geq 2$) and codimension. As a special case, we prove that for any $n\geq 2$ and any compact $(n+1)$-dimensional manifold $M$ the following holds: for a generic choice of the background metric $g$ all stationary integral $n$-varifolds in $(M,g)$ will either be entirely smooth or have at least one singular point that is not strongly isolated. In other words, only ``more complicated'' singularities may possibly persist. This implies, for instance, a generic finiteness result for the class of all closed minimal hypersurfaces of area at most $4π^2-\varepsilon$ (for any $\varepsilon>0$) in nearly round four-spheres: we can thus give precise answers, in the negative, to the questions of persistence of the Clifford football and of Hsiang's hyperspheres in nearly round metrics. The aforementioned main regularity result is achieved as a consequence of the fine analysis of the Fredholm index of the Jacobi operator for such varifolds: we prove on the one hand an exact formula relating that number to the Morse indices of the conical links at the singular points, while on the other hand we show that the same number is non-negative for all such varifolds if the ambient metric is generic.