Deforming a map into a harmonic map

This paper presents some existence and uniqueness theorems for harmonic maps between complete noncompact Riemannian manifolds. In particular, we obtain as a corollary a recent result of Hardt-Wolf on the existence of harmonic quasi-isometries of the hyperbolic plane.

     

On a map from the splines into a positive cone with applications

Résumé Dans cet article nous généralisons quelques caractérisations du meilleur approximant dans l'espace des fonctions splines, qui ont été récemment obtenues, indépendemment par John Rice et Larry Schumaker. Cette généralisation englobe quelques aspects théoriques et appliqués de l'approximation simultanée qui ont été étudiés dans des cadres différents par A. Bacopoulos. De plus, une estimation des vitesses de convergence des meilleures approximations splines est donnée en employant quelques résultats obtenus par M. Marsden et M. Marsden-I. J. Shoenberg.

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This research was partially supported by NRC grant No. A 8108.     

Biharmonic map heat flow into manifolds of nonpositive curvature

Let M ^m and N be two compact Riemannian manifolds without boundary. We consider the L ² gradient flow for the energy . If and N has nonpositive sectional curvature we show that the biharmonic map heat flow exists for all time, and that the solution subconverges to a smooth harmonic map as time goes to infinity. This reproves the celebrated theorem of Eells and Sampson [6] on the existence of harmonic maps in homotopy classes for domain manifolds with dimension less than or equal to 4.Received: 27 March 2003, Accepted: 5 April 2004, Published online: 16 July 2004Mathematics Subject Classification (2000): 58E20, 58J35     

Constraint preserving implicit finite element discretization of harmonic map flow into spheres

Discretization of the harmonic map flow into spheres often uses a penalization or projection strategy, where the first suffers from the proper choice of an additional parameter, and the latter from the lack of a discrete energy law, and restrictive mesh-constraints. We propose an implicit scheme that preserves the sphere constraint at every node, enjoys a discrete energy law, and unconditionally converges to weak solutions of the harmonic map heat flow.

     

On the number of hexagons in a map

A lower bound is given for the number of hexagonal faces in a simple map on a closed surface whose graph is 3-connected depending on the numbers of i-gonal faces, i ≠ 6, and the genus of the surface.     

Selfsimilar expanders of the harmonic map flow

We study the existence, uniqueness, and stability of self-similar expanders of the harmonic map heat flow in equivariant settings. We show that there exist selfsimilar solutions to any admissible initial data and that their uniqueness and stability properties are essentially determined by the energy-minimising properties of the so-called equator maps.     

On a property specific to the tent map

Let a set {X_λ; λ  Λ} of subspaces of a topological space X be a cover of X. Mathematical conditions are proposed for each subspace X_λ to define a map g_Xλ:X_λ→X which has the following property specific to the tent map known in the baker’s transformation. Namely, for any infinite sequence ω₀, ω₁, ω₂, … of X_λ, λ  Λ, we can find an initial point x₀  ω₀ such that g_ω0(x₀)ω₁,g_ω1(g_ω0(x₀))ω₂,…. The conditions are successfully applied to a closed cover of a weak self-similar set.     

On the bicanonical map of a surface section of a threefold

Let (ℳ, ℒ) be a 3-fold of log-general type polarized by a very ample line bundle ℒ. We study the pairs (ℳ, ℒ) in the case when there exists at least one smooth surface Ŝ ∈ |ℒ| such that the bicanonical map associated to |2K_Ŝ| is not birational. As one consequence of our classification we obtain the result:if a smooth projective threefold has non- negative Kodaira dimension, then given any smooth very ample divisor Ŝon the threefold, the bicanonical map associated to |2K_Ŝ|is birational.     

On the invariant faces associated with a cone-preserving map

For an nonnegative matrix , an isomorphism is obtained between the lattice of initial subsets (of ) for and the lattice of -invariant faces of the nonnegative orthant . Motivated by this isomorphism, we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a cone-preserving map on a polyhedral cone, formulated in terms of its invariant faces. In particular, we obtain the following extension of the famous Rothblum index theorem for a nonnegative matrix: If leaves invariant a polyhedral cone , then for each distinguished eigenvalue of for , there is a chain of distinct -invariant join-irreducible faces of , each containing in its relative interior a generalized eigenvector of corresponding to (referred to as semi-distinguished -invariant faces associated with ), where is the maximal order of distinguished generalized eigenvectors of corresponding to , but there is no such chain with more than members. We introduce the important new concepts of semi-distinguished -invariant faces, and of spectral pairs of faces associated with a cone-preserving map, and obtain several properties of a cone-preserving map that mostly involve these two concepts, when the underlying cone is polyhedral, perfect, or strictly convex and/or smooth, or is the cone of all real polynomials of degree not exceeding that are nonnegative on a closed interval. Plentiful illustrative examples are provided. Some open problems are posed at the end.

     

On the Betti number of the image of a generic map

Let be a differentiable map of a closed m-dimensional manifold into an (m + k)-dimensional manifold with k > 0. We show, assuming that f is generic in a certain sense, that f is an embedding if and only if the (m - k + 1)-th Betti numbers with respect to the Čech homology of M and f(M) coincide, under a certain condition on the stable normal bundle of f. This generalizes the authors' previous result for immersions with normal crossings [BS1]. As a corollary, we obtain the converse of the Jordan-Brouwer theorem for codimension-1 generic maps, which is a generalization of the results of [BR, BMS1, BMS2, Sae1] for immersions with normal crossings. Received: January 3, 1996     

On the tropical Torelli map

We construct the moduli spaces of tropical curves and tropical principally polarized abelian varieties, working in the category of (what we call) stacky fans. We define the tropical Torelli map between these two moduli spaces and we study the fibers (tropical Torelli theorem) and the image of this map (tropical Schottky problem). Finally we determine the image of the planar tropical curves via the tropical Torelli map and we use it to give a positive answer to a question raised by Namikawa on the compactified classical Torelli map.     

On the moving frame of a conformal map from 2-disk into {\mathbb{R}^n}

Let f be a conformal map from the 2-disk into ${\mathbb{R}^n}$ . We prove that the image f(B) have a normal tangent vector basis (e ₁, e ₂) with ${\|d(e_{1}, e_{2})\|_{L^2(B)} \leq C\|A\|_{L^2(B)}}$ when the total Gauss curvature ${\int_B |K_{f}| d\mu_f < 2\pi}$ .     

On the smooth Feshbach-Schur map

A new variant of the Feshbach map, called smooth Feshbach map, has been introduced recently by Bach et al., in connection with the renormalization analysis of non-relativistic quantum electrodynamics. We analyze and clarify its algebraic and analytic properties, and we generalize it to non-selfadjoint partition operators χ and .