Lawlor's curvature criterion and topologically trivial area-minimizing hypercones of Hsiang type

For a closed embedded minimal hypersurface in a unit sphere, we show that the existence of a uniform vanishing angle in Lawlor’s criterion suffices for its cone to minimize area. The normal-wedge condition follows from Howard’s rolling theorem. We derive explicit curvature bounds and apply them to the reported Hsiang-Sterling curvature estimates for topologically trivial cones. The resulting area-minimizing cones give an affirmative answer to the existence question raised by Ilmanen and White (Camb. J. Math., 2015).