Focal Submanifolds Research Articles

The isoparametric functions on a class of Finsler spheres

In this paper, we give global expressions of geodesics and isoparametric functions on a standard Randers sphere by navigation. We obtain isoparametric families and focal submanifolds in (Sn,F,dμBH) by Cartan-Münzner polynomials. Further more, we construct some examples of closed and non-closed geodesics, isoparametric functions, isoparametric families and focal submanifolds.

Isoparametric Polynomials and Sums of Squares

Abstract Hilbert’s 17th problem asks whether every nonnegative polynomial can be a sum of squares of rational functions. It has been answered affirmatively by Artin. However, the question as to 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. Dedicated to Professor Chia-Kuei Peng on his 80th birthday

Topology and Curvature of Isoparametric Families in Spheres

An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal submanifolds. The present paper has two parts. The first part investigates topology of the isoparametric families, namely the homotopy, homeomorphism, or diffeomorphism types, parallelizability, as well as the Lusternik–Schnirelmann category. This part extends substantially the results of Wang (J Differ Geom 27:55–66, 1988). The second part is concerned with their curvatures; more precisely, we determine when they have non-negative sectional curvatures or positive Ricci curvatures with the induced metric.

The Isoparametric Functions on a Class of Finsler Spheres

In this paper, we give global expressions of geodesics and isoparametric functions on a Randers sphere by navigation. We obtain isoparametric families and focal submanifolds in (S^{n}; F; d\mu_{BH}) by Cartan-M\"unzner polynomials. Further more, we construct some examples of closed and non-closed geodesics, isoparametric functions, isoparametric families and focal submanifolds.

Isoparametric hypersurfaces in Finsler space forms

In this paper, we study isoparametric hypersurfaces in Finsler space forms by investigating focal points, tubes and parallel hypersurfaces of submanifolds. We prove that the focal submanifolds of isoparametric hypersurfaces are anisotropic-minimal and obtain a general Cartan-type formula in a Finsler space form with vanishing reversible torsion, from which we give some classifications on the number of distinct principal curvatures or their multiplicities.

Minimizing cones associated with isoparametric foliations

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.

English

We study homogeneous real hypersurfaces having no focal submanifolds in a complex hyperbolic space. They are called Lie hypersurfaces in this space. We clarify the geometry of Lie hypersurfaces in terms of their sectional curvatures, the behavior of the characteristic vector field and their holomorphic distributions.

Yau's first eigenvalue conjecture in the isoparametric case

In this paper, we first survey the development of S. T. Yau's first eigenvalue conjecture and its complete solution in the isoparametric case,and introduce its relationship with Lawson's conjecture. Furthermore, we calculate the first eigenvalues of the isoparametric focal submanifolds (minimal submanifolds of the sphere),and propose the generalization of Yau's conjecture for the higher codimension case.

Normal Scalar Curvature Inequality on the Focal Submanifolds of Isoparametric Hypersurfaces

An isoparametric hypersurface in unit spheres has two focal submanifolds. Condition A plays a crucial role in the classification theory of isoparametric hypersurfaces in [CCJ07], [Chi16] and [Miy13]. This paper determines $C_A$, the set of points with Condition A in focal submanifolds. It turns out that the points in $C_A$ reach an upper bound of the normal scalar curvature $\rho^{\bot}$ (sharper than that in DDVV inequality [GT08], [Lu11]). We also determine the sets $C_P$ (points with parallel second fundamental form) and $C_E$ (points with Einstein condition), which achieve two lower bounds of $\rho^{\bot}$.

Isoparametric foliations and critical sets of eigenfunctions

Jakobson and Nadirashvili \cite{JN} constructed a sequence of eigenfunctions on $T^2$ with a bounded number of critical points, answering in the negative the question raised by Yau \cite{Yau1} which asks that whether the number of the critical points of eigenfunctions for the Laplacian increases with the corresponding eigenvalues. The present paper finds three interesting eigenfunctions on the minimal isoparametric hypersurface $M^n$ in $S^{n+1}(1)$. The corresponding eigenvalues are $n$, $2n$ and $3n$, while their critical sets consist of $8$ points, a submanifold(infinite many points) and $8$ points, respectively. On one of its focal submanifolds, a similar phenomenon occurs.

Isoparametric foliations, diffeomorphism groups and exotic smooth structures

In this paper, we are concerned with interactions between isoparametric theory and differential topology. Two foliations are called equivalent if there exists a diffeomorphism between the foliated manifolds mapping leaves to leaves. Using differential topology, we obtain several results towards the classification problem of isoparametric foliations up to equivalence. In particular, we show that each homotopy n-sphere has the “same” isoparametric foliations as the standard sphere Sn has except for n=4, reducing the classification problem on homotopy spheres to that on the standard sphere. Moreover, we prove the uniqueness up to equivalence of isoparametric foliations with two points as the focal submanifolds on each sphere Sn except for n=5. Besides, we show that the uniqueness holds on S5 if and only if π0(Diff(S4))≃Z2, i.e., pseudo-isotopy implies isotopy for diffeomorphisms on S4. At last, some ideas behind the proofs enable us to discover new exotic smooth structures on certain manifolds.

Isoparametric foliations, a problem of Eells–Lemaire and conjectures of Leung

In this paper, two sequences of minimal isoparametric hypersurfaces are constructed via representations of Clifford algebras. Based on these, we give estimates on eigenvalues of the Laplacian of the focal submanifolds of isoparametric hypersurfaces in unit spheres. This improves results of [TY13] and [TXY14]. Eells and Lemaire [EL83] posed a problem to characterize the compact Riemannian manifold M for which there is an eigenmap from M to S^n. As another application of our constructions, the focal maps give rise to many examples of eigenmaps from minimal isoparametric hypersurfaces to unit spheres. Most importantly, by investigating the second fundamental forms of focal submanifolds of isoparametric hypersurfaces in unit spheres, we provide infinitely many counterexamples to two conjectures of Leung [Le91] (posed in 1991) on minimal submanifolds in unit spheres. Notice that these conjectures of Leung have been proved in the case that the normal connection is flat [HV01].

Singular critical points for variational free boundary problem from isoparametric hypersurfaces

We construct three families of singular critical points for a variational free boundary problem. These critical points are homogeneous solutions of degree one to some overdetermined boundary value problem. The intersections of the level sets of these solutions with the unit sphere are isoparametric hypersurfaces and their focal submanifolds.

On submanifolds whose tubular hypersurfaces have constant higher order mean curvatures

Motivated by the theory of isoparametric hypersurfaces, we study submanifolds whose tubular hypersurfaces have some constant higher order mean curvatures. Here a k-th order mean curvature Qkν (k ≥ 1) of a submanifold Mn is defined as the k-th power sum of the principal curvatures, or equivalently, of the shape operator with respect to the unit normal vector ν. We show that if all nearby tubular hypersurfaces of M have some constant higher order mean curvatures, then the submanifold M itself has some constant higher order mean curvatures Qkν independent of the choice of ν. Many identities involving higher order mean curvatures and Jacobi operators on such submanifolds are also obtained. In particular, we generalize several classical results in isoparametric theory given by E. Cartan, K. Nomizu, H. F. M¨unzner, Q. M. Wang, et al. As an application, we finally get a geometrical filtration for the focal submanifolds of isoparametric functions on a complete Riemannian manifold.

Willmore submanifolds in the unit sphere via isoparametric functions

In this paper, we show that both focal submanifolds of each isoparametric hypersurface in the sphere with six distinct principal curvatures are Willmore, hence all focal submanifolds of isoparametric hypersurfaces in the sphere are Willmore.

Clifford systems, Cartan hypersurfaces and Riemannian submersions

Cartan hypersurfaces are minimal isoparametric hypersurfaces with 3 distinct constant principal curvatures in unit spheres. In this article, we firstly build a relationship between the focal submanifolds of Cartan hypersurfaces and the Hopf fiberations and give a new proof of the classification result on Cartan hypersurfaces. Nextly, we show that there exists a Riemannian submersion with totally geodesic fibers from each Cartan hypersurface M3m to the projective planes \({{\mathbb{F}}P^2}\) (\({{\mathbb{F}}={\mathbb{R}},{\mathbb{C}},{\mathbb{H}},{\mathbb{O}}}\) for m = 1, 2, 4, 8, respectively) endowed with the canonical metrics. As an application, we give several interesting examples of Riemannian submersions satisfying a basic equality due to Chen (Proc Jpn Acad Ser A Math Sci 81:162–167, 2005).

On Focal Submanifolds of Isoparametric Hypersurfaces and Simons Formula

The focal submanifolds of isoparametric hypersurfaces in spheres are all minimal Willmore submanifolds, mostly being $${\mathcal{A}}$$ -manifolds in the sense of A.Gray but rarely Ricci-parallel, see Li et al. (Sci China Math 58, 2015), Qian et al. (Ann Glob Anal Geom 43:47–62, 2013), Tang and Yan (Isoparametric foliation and a problem of Besse on generalizations of Einstein condition arXiv:1307.3807 , 2013). In this paper we study the geometry of the focal submanifolds via Simons formula. We show that all the focal submanifolds with g ≥ 3 are not normally flat by estimating the normal scalar curvatures. Moreover, we give a complete classification of the semiparallel submanifolds among the focal submanifolds.

On Ricci tensor of focal submanifolds of isoparametric hypersurfaces

A-manifolds and B-manifolds, introduced by Gray (1978), are two significant classes of Einstein-like Riemannian manifolds. A Riemannian manifold is Ricci parallel if and only if it is simultaneously an A-manifold and a B-manifold. The present paper proves that both focal submanifolds of each isoparametric hypersurface in unit spheres with g = 4 distinct principal curvatures are A-manifolds. As for the focal submanifolds with g = 6, m = 1 or 2, only one is an A-manifold, and neither is a B-manifold.

Isoparametric foliation and Yau conjecture on the first eigenvalue, II

This is a continuation of [23], which investigated the first eigenvalues of minimal isoparametric hypersurfaces with g=4 distinct principal curvatures and focal submanifolds in unit spheres. For the focal submanifolds with g=6, the present paper obtains estimates on all the eigenvalues, among others, giving an affirmative answer in one case to the problem posed in [23], which may be regarded as a generalization of Yau's conjecture. In two of the four unsettled cases in [23] for focal submanifolds M1 of OT-FKM-type, we prove the first eigenvalues to be their respective dimensions.

Isoparametric functions and related topics

The famous Yau conjecture asserts that the rst eigenvalue of every closed minimal embedded hypersurface in the unit sphere is just its dimension. Over several decades, research on the eigenvalues of the Laplace operator has always been a core issue in the study of geometry, many geometricians are committed to the study of Yau conjecture in recent years. However, the results that ever known are only some inequalities on the estimate of the rst eigenvalue. As a main result of this paper, we show that the rst eigenvalue of a closed minimal isoparametric hypersurface in the unit sphere is just its dimension. Furthermore, we show that under some dimensional conditions, the focal submanifolds of an isoparametric hypersurface in the unit sphere also have their dimensions as the rst eigenvalues. As the second main result of this paper, motivated by the famous Schoen-Yau-Gromov-Lawson surgery theory on scalar curvature, we make a surgery at the embedding hypersurface in a Riemannian manifold, constructing a new manifold with good geometry properties, which is called a double manifold. In particular, we construct a double manifold associated with a minimal isoparametric hypersurface in the unit sphere. The resulting double manifold has complicated topological properties (but its characteristic classes can be precisely described) and carries a metric of positive scalar curvature, more importantly, the isoparametric foliation is kept. A more extensive de nition of Willmore surface is the Willmore submanifold, which is an extremal submanifold of Willmore functional in spheres. Examples of Willmore submanifolds in the unit sphere are scarce in the literature. As the last two main results of this paper, by taking advantage of isoparametric functions of OT-FKM-type, we give a series of new examples of Willmore submanifolds in the unit sphere; hereafter, we give a uni ed geometric proof that both of focal submanifolds of every isoparametric hypersurface in spheres with four distinct principal curvatures are Willmore. These new examples of Willmore submanifolds are all minimal in spheres, but in general not Einstein.