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 is greater than three, few satisfactory results have been known. We attack the conjecture in the Dupin hypersurface case. Our results are: A closed proper Dupin hypersurface with constant mean curvature is isoparametric (i) if $g=3$, (ii) if $g=4$ and has constant scalar curvature, or (iii) if $g=4$ and has constant Lie curvature, and (iv) if $g=6$ and has constant Lie curvatures. These cover all the non-trivial cases for a closed proper Dupin to be isoparametric since $g$ can take only values $1,2,3,4,6$. The originality of the proof is a use of topology and geometry, which reduces assumptions needed in the algebraic argument.