Interaction of Legendre curves and Lagrangian submanifolds

It is proved in [8] that there exist no totally umbilical Lagrangian submanifolds in a complex-space-form ,n≥2, except the totally geodesic ones. In this paper we introduce the notion of LagrangianH-umbilical submanifolds which are the “simplest” Lagrangian submanifolds next to the totally geodesic ones in complex-space-forms. We show that for each Legendre curve in a 3-sphereS ³ (respectively, in a 3-dimensional antide Sitter space-timeH ₁ ³ ), there associates a LagrangianH-umbilical submanifold in ℂP ⁿ (respectively, in ℂH ⁿ ) via warped products. The main part of this paper is devoted to the classification of LagrangianH-umbilical submanifolds in ℂP ⁿ and in ℂH ⁿ . Our classification theorems imply in particular that “except some exceptional classes”, LagrangianH-umbilical submanifolds of ℂP ⁿ and of ℂH ⁿ are obtained from Legendre curves inS ³ or inH ₁ ³ via warped products. This provides us an interesting interaction of Legendre curves and LagrangianH-umbilical submanifolds in non-flat complex-space-forms. As an immediate by-product, our results provide us many concrete examples of LagrangianH-umbilical isometric immersions of real-space-forms into non-flat complex-space-forms.     

A minimum residual algorithm for solving linear systems

A minimum residual algorithm for solving a large linear system (I+S)x=b, with b∈ℂ ⁿ and S∈ℂ ^n×n being readily invertible, is proposed. For this purpose Krylov subspaces are generated by applying S and S ^-1 cyclically. The iteration can be executed with every linear system once the coefficient matrix has been split into the sum of two readily invertible matrices. In case S is a translation and a rotation of a Hermitian matrix, a five term recurrence is devised. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65F10     

SO（n）-Invariant Special Lagrangian Submanifolds of C^n＋l with Fixed Loci

Abstract Let SO(n) act in the standard way on ℂⁿ and extend this action in the usual way to ℂⁿ⁺¹ = ℂ ⊕ ℂⁿ. It is shown that a nonsingular special Lagrangian submanifold L ⊂ ℂⁿ⁺¹ that is invariant under this SO(n)-action intersects the fixed ℂ ⊂ ℂⁿ⁺¹ in a nonsingular real-analytic arc A (which may be empty). If n > 2, then A has no compact component. Conversely, an embedded, noncompact nonsingular real-analytic arc A ⊂ ℂ lies in an embedded nonsingular special Lagrangian submanifold that is SO(n)-invariant. The same existence result holds for compact A if n = 2. If A is connected, there exist n distinct nonsingular SO(n)-invariant special Lagrangian extensions of A such that any embedded nonsingular SO(n)-invariant special Lagrangian extension of A agrees with one of these n extensions in some open neighborhood of A. The method employed is an analysis of a singular nonlinear PDE and ultimately calls on the work of Gérard and Tahara to prove the existence of the extension. * Project supported by Duke University via a research grant, the NSF via DMS-0103884, the Mathematical Sciences Research Institute, and Columbia University. (Dedicated to the memory of Shiing-Shen Chern, whose beautiful works and gentle encouragement have had the most profound influence on my own research)     

Minimal entropy and collapsing with curvature bounded from below

We show that if a closed manifold M admits an ℱ-structure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial S ¹-action. As a corollary we obtain that the simplicial volume of a manifold admitting an ℱ-structure is zero.?We also show that if M admits an ℱ-structure then it collapses with curvature bounded from below. This in turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is non-negative.?We show that ℱ-structures of rank zero appear rather frequently: every compact complex elliptic surface admits one as well as any simply connected closed 5-manifold.?We use these results to study the minimal entropy problem. We show the following two theorems: suppose that M is a closed manifold obtained by taking connected sums of copies of S ⁴, ℂP ², ²,S ²×S ²and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S ⁴,ℂP ²,S ²×S ²,ℂP ²#  ² or ℂP ²# ℂP ². Finally, suppose that M is a closed simply connected 5-manifold. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S ⁵,S ³×S ², then on trivial S ³-bundle over S ² or the Wu-manifold SU(3)/SO(3). Oblatum 13-III-2002 & 12-VIII-2002?Published online: 8 November 2002 G.P. Paternain was partially supported by CIMAT, Guanajuato, México.?J. Petean is supported by grant 37558-E of CONACYT.     

Embedding Certain Infinitely Connected Subsets of Bordered Riemann Surfaces Properly into ℂ2

We prove that certain infinitely connected domains D in a bordered Riemann surface, which admits a holomorphic embedding into C ², admit a proper holomorphic embedding into C ². We also prove that certain infinitely connected open subsets D⊂ℂ admit a proper holomorphic embedding into ℂ².      

On Levi-flat hypersurfaces with given boundary in ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let S ⊂ ℂ ⁿ be a compact connected 2-codimensional submanifold. If n ⩾ 3, essentially local conditions and the assumption: every complex point of S is elliptic imply the existence of a projection in ℂ ⁿ of a Levi-flat (2n−1)-subvariety whose boundary is S (Dolbeault, Tomassini, Zaitsev, 2005). We extend the result when S is homeomorphic to a sphere and has one hyperbolic point. For n = 2 many results are known since the 1980’s and a new result with a very technical hypothesis is announced. Dedicated to Professor LU QiKeng on the occasion of his 80th birthday     

The lagrange intersections for (ℂP n , ℝP n )

Summary If (M, ω) is a compact symplectic manifold andL ⊂M a compact Lagrangian submanifold and if φ is a Hamiltonian diffeomorphism ofM then the V. Arnold conjecture states (possibly under additional conditions) that the number of intersection section points ofL and φ (L) can be estimated by #{Lϒφ (L)}≥ cuplength +1. We shall prove this conjecture for the special case (L, M)=(ℝP ⁿ , ℂP ⁿ ) with the standard symplectic structure.     

Laguerre expansion on the Heisenberg group and Fourier-Bessel transform on ?n

Given a principal value convolution on the Heisenberg group H _n = ℂ ⁿ × ℝ, we study the relation between its Laguerre expansion and the Fourier-Bessel expansion of its limit on ℂ ⁿ . We also calculate the Dirichlet kernel for the Laguerre expansion on the group H _n . Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

A minimal triangulation of complex projective plane admitting a chess colouring of four-dimensional simplices

We construct and study a new 15-vertex triangulation X of the complex projective plane ℂP². The automorphism group of X is isomorphic to S ₄ × S ₃. We prove that the triangulation X is the minimal (with respect to the number of vertices) triangulation of ℂP² admitting a chess colouring of four-dimensional simplices. We provide explicit parametrizations for the simplices of X and show that the automorphism group of X can be realized as a group of isometries of the Fubini-Study metric. We find a 33-vertex subdivision $ \bar X $ \bar X of the triangulation X such that the classical moment mapping μ: ℂP² → Δ² is a simplicial mapping of the triangulation $ \bar X $ \bar X onto the barycentric subdivision of the triangle Δ². We study the relationship of the triangulation X with complex crystallographic groups.     

Certain 4-manifolds with non-negative sectional curvature

In this paper, we study certain compact 4-manifolds with non-negative sectional curvature K. If s is the scalar curvature and W. is the self-dual part of Weyl tensor, then it will be shown that there is no metric g on S ^• × S ^• with both (i) K > 0 and (ii) ÷ s − W _• ⩾ 0. We also investigate other aspects of 4-manifolds with non-negative sectional curvature. One of our results implies a theorem of Hamilton: “If a simply-connected, closed 4-manifold M^• admits a metric g of non-negative curvature operator, then M^• is one of S^•, ℂP^• and S^•×S^•”. Our method is different from Hamilton’s and is much simpler. A new version of the second variational formula for minimal surfaces in 4-manifolds is proved.      

Special points on products of two Shimura curves

Let S ₁ and S ₂ be two Shimura curves over ℚ attached to rational indefinite quaternion algebras B _{1 ℚ} and B _{1 ℚ} with maximal orders B ₁ and B ₂ respectively. We consider an irreducible closed algebraic curve C in the product (S ₁×S ₂)_ℂ such that C(ℂ) ∩ (S ₁×S ₂)(ℂ) contains infinitely many complex multiplication points. We prove, assuming the Generalized Riemann Hypothesis (GRH) for imaginary quadratic fields, that C is of Hodge type. Received: 3 January 2000 / Revised version: 2 October 2000     

Minimal Lagrangian submanifolds in ℂP<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> with diagonal metric

We propose a method for constructing minimal Lagrangian submanifolds in ℂP ⁿ using the Baker-Akhiezer functions of algebraic spectral curves.     

Congruence criteria for finite subsets of complex projective and complex hyperbolic spaces

We investigate congruence classes and direct congruence classes of m-tuples in the complex projective space ℂP ⁿ . For direct congruence one allows only isometries which are induced by linear (instead of semilinear) mappings. We establish a canonical bijection between the set of direct congruence classes of m-tuples of points in ℂP ⁿ and the set of equivalence classes of positive semidefinite Hermitean m×m-matrices of rank at most n+1 with 1's on the diagonal. As a corollary we get that the direct congruence class of an m-tuple is uniquely determined by the direct congruence classes of all of its triangles, provided that no pair of points of the m-tuple has distance π/2. Examples show that the situation changes drastically if one replaces direct congruence classes by congruence classes or if distances π/2 are allowed. Finally we do the same kind of investigation also for the complex hyperbolic space ℂH ⁿ . Most of the results are completely analogous, however, there are also some interesting differences. Received: 15 January 1996     

Lagrangian embeddings into subcritical Stein manifolds

In this paper we obtain new topological restrictions on Lagrangian embeddings into subcritical Stein manifolds. We also extend previous results of Gromov, Oh, Polterovich and Viterbo on Lagrangian submanifolds of ℂ ⁿ to the more general case of subcritical Stein manifolds. Research partially supported by the US-Israel Binational Science Foundation grant 1999086.     

On the weak non-defectivity of veronese embeddings of projective spaces

Fix integers n, x, k such that n≥3, k>0, x≥4, (n, x)≠(3, 4) and k(n+1)<( _n ^n+x ). Here we prove that the order x Veronese embedding ofP ⁿ is not weakly (k−1)-defective, i.e. for a general S⊃P ⁿ such that #(S) = k+1 the projective space | I _2S (x)| of all degree t hypersurfaces ofP ⁿ singular at each point of S has dimension ( _n ^/n+x )−1− k(n+1) (proved by Alexander and Hirschowitz) and a general F∈| I _2S (x)| has an ordinary double point at each P∈ S and Sing (F)=S. The author was partially supported by MIUR and GNSAGA of INdAM (Italy).     

A bilinear algorithm for sparse representations

We consider the following sparse representation problem: represent a given matrix X∈ℝ ^m×N as a multiplication X=AS of two matrices A∈ℝ ^m×n (m≤n<N) and S∈ℝ ^n×N , under requirements that all m×m submatrices of A are nonsingular, and S is sparse in sense that each column of S has at least n−m+1 zero elements. It is known that under some mild additional assumptions, such representation is unique, up to scaling and permutation of the rows of S. We show that finding A (which is the most difficult part of such representation) can be reduced to a hyperplane clustering problem. We present a bilinear algorithm for such clustering, which is robust to outliers. A computer simulation example is presented showing the robustness of our algorithm.     

A combinatorial proof of the Borsuk-Ulam antipodal point theorem

We give a proof of Tucker’s Combinatorial Lemma that proves the fundamental nonexistence theorem: There exists no continuous map fromB ⁿ toS ^{n − 1} that maps antipodal points of∂B ⁿ to antipodal points ofS ^{n − 1}.     

Curvature surfaces of Hopf hypersurfaces in complex space forms

We study the principal curvatures of a Hopf hypersurfaceM in ℂP ⁿ or ℂH ⁿ . The respective eigenspaces of the shape operator often turn out to induce totally real foliations ofM, whose leaves are spherical in the ambient space. Finally we classify the Hopf hypersurfaces with three distinct principal curvatures satisfying a certain non-degeneracy condition.