Degenerations of ideal hyperbolic triangulations

Let M be a cusped 3-manifold, and let T be an ideal triangulation of M. The deformation variety D(T), a subset of which parameterises (incomplete) hyperbolic structures obtained on M using T, is defined and compactified by adding certain projective classes of transversely measured singular codimension-one foliations of M. This leads to a combinatorial and geometric variant of well-known constructions by Culler, Morgan and Shalen concerning the character variety of a 3-manifold.Comment: 31 pages, 11 figures; minor changes; to appear in Mathematische Zeitschrif

Global classification of isolated singularities in dimensions (4,3)(4,3) and (8,5)(8,5)

We characterize those closed $2k$-manifolds admitting smooth maps into $(k+1)$-manifolds with only finitely many critical points, for $k\in\{2,4\}$. We compute then the minimal number of critical points of such smooth maps for $k=2$ and, under some fundamental group restrictions, also for $k=4$. The main ingredients are King's local classification of isolated singularities, decomposition theory, low dimensional cobordisms of spherical fibrations and 3-manifolds topology.Comment: 31p, revised version, Ann. Scuola Norm. Sup. Pisa Cl. Sci., to appea

Computing Periods of Hypersurfaces

We give an algorithm to compute the periods of smooth projective hypersurfaces of any dimension. This is an improvement over existing algorithms which could only compute the periods of plane curves. Our algorithm reduces the evaluation of period integrals to an initial value problem for ordinary differential equations of Picard-Fuchs type. In this way, the periods can be computed to extreme-precision in order to study their arithmetic properties. The initial conditions are obtained by an exact determination of the cohomology pairing on Fermat hypersurfaces with respect to a natural basis.Comment: 33 pages; Final version. Fixed typos, minor expository changes. Changed code repository lin

A discrete Laplace-Beltrami operator for simplicial surfaces

We define a discrete Laplace-Beltrami operator for simplicial surfaces. It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finite-elements Laplacian (the so called ``cotan formula'') except that it is based on the intrinsic Delaunay triangulation of the simplicial surface. This leads to new definitions of discrete harmonic functions, discrete mean curvature, and discrete minimal surfaces. The definition of the discrete Laplace-Beltrami operator depends on the existence and uniqueness of Delaunay tessellations in piecewise flat surfaces. While the existence is known, we prove the uniqueness. Using Rippa's Theorem we show that, as claimed, Musin's harmonic index provides an optimality criterion for Delaunay triangulations, and this can be used to prove that the edge flipping algorithm terminates also in the setting of piecewise flat surfaces.Comment: 18 pages, 6 vector graphics figures. v2: Section 2 on Delaunay triangulations of piecewise flat surfaces revised and expanded. References added. Some minor changes, typos corrected. v3: fixed inaccuracies in discussion of flip algorithm, corrected attributions, added references, some minor revision to improve expositio