Manifolds with cylindrical ends having a finite and positive number of embedded eigenvalues

Three-Manifolds and Manifolds with Cylindrical Ends

Contents Introduction. 1 Chapter 1. Four-dimensional Theory. 4 1. Setup. 4 2. A slice theorem for singular connections. 9 3. Laplacians on forms. 17 Chapter 2. Three-dimensional Theory. 22 1. Setup. 22 2. The Chern-Simons functional. 24 3. The deformation complex. 27 Chapter 3. Singular connections on manifolds with cylindrical ends. 33 1. Setup. 33 2. The translation-invariant case. 35 3. Glob...

Forward and Inverse Scattering on Manifolds with Asymptotically Cylindrical Ends

We study an inverse problem for a non-compact Riemannian manifold whose ends have the following properties : On each end, the Riemannian metric is assumed to be a short-range perturbation of the metric of the form (dy) + h(x, dx), h(x, dx) being the metric of some compact manifold of codimension 1. Moreover one end is exactly cylindrical, i.e. the metric is equal to (dy) +h(x, dx). Given two su...

Some Upper Bounds on the Number of Resonances for Manifolds with Infinite Cylindrical Ends

We prove some sharp upper bounds on the number of resonances associated with the Laplacian, or Laplacian plus potential, on a manifold with infinite cylidrical ends. The purpose of this note is to bound the number of resonances, poles of the meromorphic continuation of the resolvent, associated to a manifold with infinite cylindrical ends. These manifolds have an infinity which is in some sense...

Lefschetz Fixed Point Formula for Manifolds with Cylindrical Ends

Embedded Minimal Ends of Finite Type

We prove that the end of a complete embedded minimal surface in R3 with infinite total curvature and finite type has an explicit Weierstrass representation that only depends on a holomorphic function that vanishes at the puncture. Reciprocally, any choice of such an analytic function gives rise to a properly embedded minimal end E provided that it solves the corresponding period problem. Furthe...