On the Bryant conjecture for minimal hypersurfaces in $S^4$

Abstract

We prove that a local minimal hypersurface $M^3$ in the unit sphere $S^4$ with constant scalar curvature and three distinct principal curvatures at every point must be locally congruent to the minimal Cartan isoparametric hypersurface in $S^4$, which provides a positive answer to the Bryant conjecture \cite{ChangLocal}. The key ingredient is to prove that, for each fixed \(S_0>3\), the set $\mathscr{G}_{S_0}$ of fourth-order jets of normalized local graph solutions satisfying \(|A|^2\equiv S_0\) is compact, where \(A\) denotes the shape operator.