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

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 ${\mathcal{G}}_{S_0}$ of fourth-order jets of normalized local graph solutions satisfying $\mathrm{\mid }A{\mathrm{\mid }}^2\equiv S_0$ is compact, where $A$ denotes the shape operator.