Following Matveev, a k-normal surface in a triangulated 3-manifold is a generalization of both normal and (octagonal) al- most normal surfaces. Using spines, complexity, and Turaev-Viro invariants of 3-manifolds, we prove the following results: a minimal triangulation of a closed irreducible or a boundedhyperbolic 3-manifold contains no non-trivial k-normal sphere; every triangulation of a closed manifold with at least 2 tetra-hedra contains some non-trivial normal surface; every manifold with boundary has only finitely many triangulations without non-trivial normal surfaces. Here, triangulations of bounded manifolds are actually ideal triangulations. We also calculate the number of normal surfaces of nonnegative Euler characteristics which are contained in the conjecturally minimal triangulations of all lens spaces Lp,q.

Let w be an Abelian differential on a compact Riemann surface of genus g ≥ 1. Then |w|2 defines a flat metric with conical singularities and trivial holonomy on the Riemann surface. We obtain an explicit holomorphic factorization formula for the ζ-regularized determinant of the Laplacian in the metric |w|2, generalizing the classical Ray-Singer result in g = 1.

The chains studied in this paper generalize Chern–Moser chains for CR structures. They form a distinguished family of one dimensional submanifolds in manifolds endowed with a parabolic contact structure. Both the parabolic contact structure and the system of chains can be equivalently encoded as Cartan geometries (of different types). The aim of this paper is to study the relation between these two Cartan geometries for Lagrangean contact structures and partially integrable almost CR structures. We develop a general method for extending Cartan geometries which generalizes the Cartan geometry interpretation of Fefferman’s construction of a conformal structure associated to a CR structure. For the two structures in question, we show that the Cartan geometry associated to the family of chains can be obtained in that way if and only if the original parabolic contact structure is torsion free. In particular, the procedure works exactly on the subclass of (integrable) CR structures. This tight relation between the two Cartan geometries leads to an explicit description of the Cartan curvature associated to the family of chains. On the one hand, this shows that the homogeneous models for the two parabolic contact structures give rise to examples of non–flat path geometries with large automorphism groups. On the other hand, we show that one may (almost) reconstruct the underlying torsion free parabolic contact structure from the Cartan curvature associated to the chains. In particular, this leads to a very conceptual proof of the fact that chain preserving contact diffeomorphisms are either isomorphisms or antiisomorphisms of parabolic contact structures.

We consider the zeros on the boundary @­ of a Neumann eigen- function '¸j of a real analytic plane domain ­. We prove that the number of its boundary zeros is O(¸j) where ¡¢'¸j = ¸2j '¸j . We also prove that the number of boundary critical points of either a Neumann or Dirichlet eigenfunction is O(¸j ). It follows that the number of nodal lines of '¸j (components of the nodal set) which touch the boundary is of order ¸j . This upper bound is of the same order of magnitude as the length of the total nodal line, but is the square root of the Courant bound on the number of nodal components in the interior. More generally, the results are proved for piecewise analytic domains.

The limiting behavior of the normalized KÄahler-Ricci °ow for manifolds with positive ¯rst Chern class is examined under certain stability conditions. First, it is shown that if the Mabuchi Kenergy is bounded from below, then the scalar curvature converges uniformly to a constant. Second, it is shown that if the Mabuchi Kenergy is bounded from below and if the lowest positive eigenvalue of the ¹@y ¹@ operator on smooth vector ¯elds is bounded away from0 along the °ow, then the metrics converge exponentially fast in C1 to a KÄahler-Einstein metric.