Let M be a hypersurface in Euclidean space or sphere. A curvature surface of M is a smooth connected submanifold S such that for each x in S, the tangent space to S at x is equal to a principal space of the shape operator of M at x. This generalizes the classical notion of a line of curvature on a surface in the Euclidean 3-space. The hypersurface is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. The hypersurface is called proper Dupin if it is Dupin and the number g of distinct principal curvatures is constant on M. An important class of compact proper Dupin hypersurfaces consists of isoparametric hypersurfaces, which are compact hypersurfaces whose principal curvatures remain constant everywhere. It is known that g=1, 2, 3, 4, or 6 for a compact proper Dupin hypersurface, where the first three cases have been well understood; they are Lie equivalent to, i.e., are transforms of, isoparametric hypersurfaces via Mobius maps and parallel transforms. The conclusion is not true when g=4, or 6. However, in these two cases one can form the cross ratio of any four principal curvatures. It has been conjectured that a compact proper Dupin hypersurface is Lie equivalent to an isoparametric hypersurface if the cross ratio of any four principal curvatures is constant. We have settled the case for g=4 when the multiplicities of the principal curvatures are of the form (m,m,1,1). (In general, they are of the form (m,m,n,n).) We propose to settle the entire conjecture, for g=4, by investigating an intriguing link between Dupin hypersurfaces and the invariant theory of homogeneous polynomials. In the resolution of the aforementioned conjecture for multiplicities (m,m,1,1), the analyticity, or more generally, algebraicity, of a compact proper Dupin hypersurface that we established via real algebraic geometry, palys a crucial role, in that one can reduce the global study to a local one. A compact proper Dupin hypersurface is known to be a taut hypersurface, which is essentially one for which all Morse distance functions have the same number of critical points of index k as the k-th Betti number (with Z_2 coefficients). The Kuiper conjecture states that a compact taut hypersurface is algebraic. We have established the conjecture when the nowhere dense wild set of the manifold, where the principal curvatures change multiplicities, is controllable in an appropriate sense. We propose to settle the Kuiper conjecture by establishing this controllability for all compact taut hypersurfaces. Morse theory applied to studying the topology of a curvature surface through a point in the wild set to establish a generic foliation seems to hold the key.
A type of Hamiltonian systems naturally arises in gas dynamics, hydrodynamics, chemical kinetics, Whitham averaging procedure, etc., is the hydrodynamic type. Remarkably, the subtype of the weakly nonlinear Hamiltonian systems of hydrodynamic type is in one-to-one correspondence with the class of Dupin hypersurfaces, which need not be compact in general. It turns out that the classification of weakly nonlinear Hamiltonian systems rests exactly on the irreducibility condition, satisfied by a compact proper Dupin hypersurface, which is important in our proposal. The analyticity, or more generally, algebraicity, of a proper Dupin hypersurface, compact or not, joins hand in hand with the irreducibility condition to reduce the classification to a local problem of investigating some local invariants of homogeneous polynomials. The outcome of our proposal would contribute significantly to the understanding of the system of hydrodynamic type in various applications.
