A characterization of shortest geodesics on surfaces

Any finite configuration of curves with minimal intersections on a surface is a configuration of shortest geodesics for some Riemannian metric on the surface. The metric can be chosen to make the lengths of these geodesics equal to the number of intersections along them.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-17.abs.htm

Least area tori in 3-manifolds.

We obtain two results about the singularities of area-minimizing maps from the torus into a closed, orientable and irreducible 3-manifold. First we give three examples of manifolds of this kind that admit \pi\sb1-injective embeddings of the torus, but where the \pi\sb1-injective maps of minimum area may or may not be embeddings depending on the metric given to the manifold. Then we show that for any other such manifold, the area-minimizing maps are always embeddings or double coverings of embedded Klein bottles. The second result is about maps of the torus that minimize area in their homotopy classes. We prove that if the image of one of these maps has self-intersections, then the preimages of the curves of intersection are simple in the torus. As a corollary, we give a characterization of all the closed, orientable and irreducible manifolds which admit incompressible maps of the torus with one double curve, but do not admit any incompressible embeddings (this can be used to give an alternative proof of most cases of P. Scott's result that there are no fake Seifert fibre spaces with infinite fundamental group).Ph.D.MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/103281/1/9308409.pdfDescription of 9308409.pdf : Restricted to UM users only

Equilibria of pairs of nonlinear maps associated with cones

[[abstract]]Let K1, K2 be closed, full, pointed convex cones in finite-dimensional real vector spaces of the same dimension, and let F : K1 → span K2 be a homogeneous, continuous, K2-convex map that satisfies F(∂K1) ∩ int K2=∅ and FK1 ∩ int K2 ≠ ∅. Using an equivalent formulation of the Borsuk-Ulam theorem in algebraic topology, we show that we have F(K1∖{0})∩(−K2)=∅F(K1∖{0})∩(−K2)=∅ and K2⊆FK1.K2⊆FK1. We also prove that if, in addition, G : K1 → span K2 is any homogeneous, continuous map which is (K1, K2)-positive and K2-concave, then there exist a unique real scalar ω0 and a (up to scalar multiples) unique nonzero vector x0 ∈ K1 such that Gx0 = ω0Fx0, and moreover we have ω0 > 0 and x0 ∈ int K1 and we also have a characterization of the scalar ω0. Then, we reformulate the above result in the setting when K1 is replaced by a compact convex set and recapture a classical result of Ky Fan on the equilibrium value of a finite system of convex and concave functions.[[notice]]補正完畢[[journaltype]]國外[[incitationindex]]SC