Equivariant harmonic maps into the sphere via isoparametric maps

By using concrete isoparametric maps we obtain some new equivariant harmonic maps between spheres and solve equivariant boundary value problems for harmonic maps from unit open ballB ^m+1 intoS ⁿ. Research partially supported by NNSFC, SFECC and ICTP.     

Harmonic Hopf constructions and isoparametric gradient maps

This note provides a complete answer to a problem of Ding-Fan-Li on the homotopy classes of harmonic Hopf constructions. Moreover, it gives applications to isoparametric gradient maps.     

New explicit solutions to the p-Laplace equation based on isoparametric foliations

In contrast to an infinite family of explicit examples of two-dimensional p-harmonic functions obtained by G. Aronsson in the late 80s, there is very little known about the higher-dimensional case. In this paper, we show how to use isoparametric polynomials to produce diverse examples of p-harmonic and biharmonic functions. Remarkably, for some distinguished values of p and the ambient dimension n this yields first examples of rational and algebraic p-harmonic functions. Moreover, we show that there are no p-harmonic polynomials of the isoparametric type. This supports a negative answer to a question proposed in 1980 by J. Lewis.     

On the singularities of foliations and of vector bundle maps

LetF be a (smooth) Γ _q ^∞ -stucture (often called a codimension-q Haefliger structure) on a compact manifoldX ⁿ . Cohomological invariants associated to the singularities ofF are defined whose vanishing is shown to be a necessary condition for deformingF to a codimension-q foliation onX ⁿ . An analagous approach to vector bundle maps is then utilized to prove a general theorem concerning the possibility of embedding a vector bundle in the tangent bundle ofX ⁿ , and applications to the planefield problem are given. In the final section geometric realizations of the singularity classes associated toF are constructed.     

Singular riemannian foliations with sections,transnormal maps and basic forms

A singular riemannian foliation on a complete riemannian manifold M is said to admit sections if each regular point of M is contained in a complete totally geodesic immersed submanifold Σ that meets every leaf of orthogonally and whose dimension is the codimension of the regular leaves of . We prove that the algebra of basic forms of M relative to is isomorphic to the algebra of those differential forms on Σ that are invariant under the generalized Weyl pseudogroup of Σ. This extends a result of Michor for polar actions. It follows from this result that the algebra of basic function is finitely generated if the sections are compact. We also prove that the leaves of coincide with the level sets of a transnormal map (generalization of isoparametric map) if M is simply connected, the sections are flat and the leaves of are compact. This result extends previous results due to Carter and West, Terng, and Heintze, Liu and Olmos. Marcos M. Alexandrino and Claudio Gorodski have been partially supported by FAPESP and CNPq.     

A general approach to index theorems for holomorphic maps and foliations

Let M be a smooth complex manifold, and S(⊂ M) be a compact irreducible subvariety with dim _C S > 0. Let be given either a holomorphic map f : M → M with f _|S  = id _S , f ≠ id _M , or a holomorphic foliation on M: we describe an approach that can be applied to both map and foliation in order to obtain index theorems. Partially supported by GNSAGA, Centro de Giorgi, M.U.R.S.T.     

Polar factorization of maps on Riemannian manifolds

Let (M,g) be a connected compact manifold, C³ smooth and without boundary, equipped with a Riemannian distance d(x,y). If s : M ? M s : M \to M is merely Borel and never maps positive volume into zero volume, we show s = t °u s = t \circ u factors uniquely a.e. into the composition of a map t(x) = exp_x[-?y(x)] t(x) = {\rm exp}_x[-\nabla\psi(x)] and a volume-preserving map u : M ? M u : M \to M , where y: M ? \bold R \psi : M \to {\bold R} satisfies the additional property that (y^c)^c = y (\psi^c)^c = \psi with y^c(y) :=inf{c(x,y) - y(x) | x ? M} \psi^c(y) :={\rm inf}\{c(x,y) - \psi(x)\,\vert\,x \in M\} and c(x,y) = d²(x,y)/2. Like the factorization it generalizes from Euclidean space, this non-linear decomposition can be linearized around the identity to yield the Hodge decomposition of vector fields.¶The results are obtained by solving a Riemannian version of the Monge--Kantorovich problem, which means minimizing the expected value of the cost c(x,y) for transporting one distribution f 3 0 f \ge 0 of mass in L¹(M) onto another. Parallel results for other strictly convex cost functions c(x,y) 3 0 c(x,y) \ge 0 of the Riemannian distance on non-compact manifolds are briefly discussed.     

Foliated plateau problem,part II: Harmonic maps of foliations

Our basic results concerning harmonic maps are parallel to those in Part I [Gro11] about minimal subvarieties. First we produce compact harmonic foliations by solving in some cases the asymptotic Dirichlet problem. Then we construct transversal measures by adopting the parabolic equation method of Eells and Sampson. Finally we indicate some applications to the rigidity and the pinching problems.     

On realizing measured foliations via quadratic differentials of harmonic maps toR-trees

We give a brief, elementary and analytic proof of the theorem of Hubbard and Masur [HM] (see also [K], [G]) that every class of measured foliations on a compact Riemann surfaceR of genusg can be uniquely represented by the vertical measured foliation of a holomorphic quadratic differential onR. The theorem of Thurston [Th] that the space of classes of projective measured foliations is a 6g—7 dimensional sphere follows immediately by Riemann-Roch. Our argument involves relating each representative of a class of measured foliations to an equivariant map from to anR-tree, and then finding an energy minimizing such map by the direct method in the calculus of variations. The normalized Hopf differential of this harmonic map is then the desired differential. Partially supported by NSF grant DMS9300001; Alfred P. Sloan Research Fellow.     

On degeneracy schemes of maps of vector bundles and applications to holomorphic foliations

In this paper we provide sufficient conditions for maps of vector bundles on smooth projective varieties to be uniquely determined by their degeneracy schemes. We then specialize to holomorphic distributions and foliations. In particular, we provide sufficient conditions for foliations of arbitrary rank on $\mathbb P ^n$ to be uniquely determined by their singular schemes.     

Totally geodesic foliations close to Riemannian foliations

The relation of the curvature and topology of totally geodesic foliations close to Riemannian ones is studied. The main result complements Ferus's famous theorem on totally geodesic foliations.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 114–118, 1992.     

Commuting foliations

The aim of this paper is to extend the notion of commutativity of vector fields to the category of singular foliations using Nambu structures, i.e., integrable multi-vector fields. We will classify the relationship between singular foliations and Nambu structures and show some basic results about commuting Nambu structures.     

Metric foliations and curvature

We prove a rigidity theorem for Riemannian fibrations of flat spaces over compact bases and give a metric classification of compact four-dimensional manifolds of nonnegative curvature that admit totally geodesic Riemannian foliations.     

Affine isoparametric hypersurfaces

Research supported by an NSERC Postdoctoral Fellowship and NSERC Operating Grant OGP0002501     

Seshadri-exceptional foliations

 The maximal Seshadri number μ(L) of an ample line bundle L on a smooth projective variety X measures the local positivity of the line bundle L at a general point of X. By refining the method of Ein-Küchle-Lazarsfeld, lower bounds on μ(L) are obtained in terms of L ⁿ , n=dim(X), for a class of varieties. The main idea is to show that if a certain lower bound is violated, there exists a non-trivial foliation on the variety whose leaves are covered by special curves. In a number of examples, one can show that such foliations must be trivial and obtain lower bounds for μ(L). The examples include the hyperplane line bundle on a smooth surface in ℙ³ and ample line bundles on smooth threefolds of Picard number 1. Received: 29 June 2001 / Published online: 16 October 2002 RID="⋆" ID="⋆" Supported by Grant No. 98-0701-01-5-L from the KOSEF. RID="⋆⋆" ID="⋆⋆" Supported by Grant No. KRF-2001-041-D00025 from the KRF.     

Diophantine approximations and foliations

In this paper we indicate the proof of an effective version of the Green-Griffiths conjecture for surfaces of general type and positive second Segre class (i.e.c ₁ ² >c ₂). Naturally this effective version is stronger than the Green-Griffiths conjecture itself.