Hypersurface geometry and hamiltonian systems of hydrodynamic type

Around 1982–1988, Dobrovin and Novikov gave a mathematical formulation of Hamiltonian systems of hydrodynamic type, which appear in gas dynamics, gas chromatography, etc. In 1991, Ferapontov showed one to one correspondence between these systems and the hypersurfaces in the (pseudo-)Euclidean space. Using this, he obtained the classification of the integrable non-diagonalizable $3\times 3$ Hamiltonian systems of hydrodynamic type via the classification of Dupin hypersurfaces in $E^4$ due to Pinkall. Furthermore, he showed that the systems corresponding to the homogeneous isoparametric hypersurfaces in $S^n$ with more than two principal curvatures are non-diagonalizable but integrable, by deforming the systems to the $n$-wave systems which are known to be integrable.
Isoparametric hypersurfaces with less than four principal curvatures are known to be all homogeneous, while there exist infinitely many non-homogeneous examples with four principal curvatures. S. T. Yau posed a question to classify isoparametric hypersurfaces with four or six principal curvatures, which is the remaining possible cases. Recently, the author proved that isoparametric hypersurfaces with six principal curvatures are homogeneous. The idea of the proof is a use of “isospectral principle”, which holds for a one-parameter family of (self-adjoint) operators satisfying the Lax equation. This is a basic principle to solve the Cauchy problem of the KdV equation, the non-linear Schrödinger equation, the sine-Gordon equation and so forth, via the inverse scattering method. Now, it turned out that all isoparametric hypersurfaces other than those with four principal curvatures correspond to the integrable systems.
Moreover, Ferapontov showed that the Lie transformation of hypersurfaces corresponds to the “reciprocal transformation” of the Hamiltonian systems, which is a transformation preserving the diagonalizability, integrability and linearly degeneracy of the isoparametric hypersurfaces under the Lie transformations correspond also to the integrable systems. These hypersurfaces belong to the hypersurface class called Dupin. Therefore the classification of Dupin hypersurfaces is also interesting and important.