Chern's conjecture in the Dupin case

Chern's conjecture states that a closed minimal hypersurface in the Euclidean sphere is isoparametric if it has constant scalar curvature. When the number $g$ of distinct principal curvatures exceeds three, only limited results have been established. In this work, we examine the conjecture for Dupin hypersurfaces, and establish the following results: a closed proper Dupin hypersurface with constant mean curvature is isoparametric (i) if $g = 3$; (ii) if $g = 4$ and it has constant scalar curvature; (iii) if $g = 4$ and it has constant Lie curvature; and (iv) if $g = 6$ and it has constant Lie curvatures. These cases cover all nontrivial possibilities for closed proper Dupin hypersurfaces. Our proof employs topological and geometric methods, in contrast to earlier algebraic and analytic approaches.