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Register a new userMinimizing cones associated with isoparametric foliations
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Associated with isoparametric foliations of unit spheres, there are two classes of minimal surfaces $-$ minimal isoparametric hypersurfaces and focal submanifolds. By virtue of their rich structures, we find new series of minimizing cones. They are cones over focal submanifolds and cones over suitable products among these two classes. Except in low dimensions, all such cones are shown minimizing.
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In this paper, we study stability for harmonic foliations on locally conformal Kahler manifolds with complex leaves. We also discuss instability for harmonic foliations on compact submanifolds immersed in Euclidean spaces and compact homogeneous spaces.
This is a survey on the recent progress in several applications of isoparametric theory, including an affirmative answer to Yaus conjecture on the first eigenvalue of Laplacian in the isoparametric case, a negative answer to Yaus 76th problem in his Problem Section, new examples of Willmore submanifolds in spheres, a series of examples to Besses problem on the generalization of Einstein condition, isoparametric functions on exotic spheres, counterexamples to two conjectures of Leung, as well as surgery theory on isoparametric foliation.
The Hilberts 17th problem asks that whether every nonnegative polynomial can be a sum of squares of rational functions. It has been answered affirmatively by Artin. However, as to the question whether a given nonnegative polynomial is a sum of squares of polynomials is still a central question in real algebraic geometry. In this paper, we solve this question completely for the nonnegative polynomials associated with isoparametric polynomials (initiated by E. Cartan) which define the focal submanifolds of the corresponding isoparametric hypersurfaces.
A submanifold in space forms is isoparametric if the normal bundle is flat and principal curvatures along any parallel normal fields are constant. We study the mean curvature flow with initial data an isoparametric submanifold in Euclidean space and sphere. We show that the mean curvature flow preserves the isoparametric condition, develops singularities in finite time, and converges in finite time to a smooth submanifold of lower dimension. We also give a precise description of the collapsing.
For a closed hypersurface $M^nsubset S^{n+1}(1)$ with constant mean curvature and constant non-negative scalar curvature, the present paper shows that if $mathrm{tr}(mathcal{A}^k)$ are constants for $k=3,ldots, n-1$ for shape operator $mathcal{A}$, then $M$ is isoparametric. The result generalizes the theorem of de Almeida and Brito cite{dB90} for $n=3$ to any dimension $n$, strongly supporting Cherns conjecture.
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