Dupin Hypersurfaces and the Kuiper Conjecture

设M是欧氏空间或球面上的一个超曲面。M的曲率曲面是光滑连通的子流形S，使得对于S中的每个x， S在x处的切空间等于M在x处的形状算子的主空间。这推广了欧几里德三维空间中曲面上曲率线的经典概念。如果沿每个曲率曲面，相应的主曲率是常数，则称超曲面为杜平曲面。如果该超曲面是Dupin且不同主曲率的个数g在m上是恒定的，则称为固有Dupin超曲面。一类重要的紧致固有Dupin超曲面由等参超曲面组成，它们是主曲率处处保持恒定的紧致超曲面。对于紧定Dupin超曲面，已知g= 1,2,3,4或6，其中前三种情况已被很好地理解；它们是Lie等价的，即是通过莫比乌斯映射和平行变换对等参超曲面的变换。当g=4或6时，结论不成立。然而，在这两种情况下，可以形成任意四个主曲率的交叉比。我们推测，当任意四个主曲率的交叉比为常数时，紧定Dupin超曲面与等参超曲面是等价的。当主曲率的多重度为（m,m,1,1）时，我们已经解决了g=4的情况。（一般来说，它们的形式是（m,m,n,n）。）我们建议通过研究Dupin超曲面与齐次多项式不变理论之间的有趣联系来解决g=4的整个猜想。在解决上述多重性（m,m,1,1）的猜想中，我们通过实际代数几何建立的紧定Dupin超曲面的解析性，或者更一般地说，代数性，起着至关重要的作用，因为人们可以将全局研究简化为局部研究。紧致固有Dupin超曲面被称为紧致超曲面，其本质上是所有莫尔斯距离函数具有与第k个Betti数（带Z_2系数）相同的指标k临界点数目的超曲面。柯伊伯猜想指出紧绷超曲面是代数的。我们建立了主曲率变化多重的流形的无处密集野集在适当意义上是可控的猜想。我们提出通过建立所有紧绷超曲面的这种可控性来解决柯伊伯猜想。将莫尔斯理论应用于研究曲率曲面的拓扑结构，通过野集合中的一个点来建立泛叶理似乎是一把钥匙。在气体动力学、流体动力学、化学动力学、威瑟姆平均程序等中自然出现的一类哈密顿系统，是水动力学型。值得注意的是，流体动力型弱非线性哈密顿系统的子类型与一般不需要紧致的Dupin超曲面类是一一对应的。证明了弱非线性哈密顿系统的分类完全依赖于不可约条件，该不可约条件由紧致的固有Dupin超曲面满足，这在我们的建议中是重要的。一个适当的Dupin超曲面的可解析性，或者更一般地说，代数性，与不可约条件相结合，将分类简化为研究齐次多项式的一些局部不变量的局部问题。本文的研究结果将对水动力型系统在各种应用中的理解作出重要贡献。

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.