Renormalized volumes with boundary

We develop a general regulated volume expansion for the volume of a manifold with boundary whose measure is suitably singular along a separating hypersurface. The expansion is shown to have a regulator independent anomaly term and a renormalized volume term given by the primitive of an associated anomaly operator. These results apply to a wide range of structures. We detail applications in the setting of measures derived from a conformally singular metric. In particular, we show that the anomaly generates invariant ([Formula: see text]-curvature, transgression)-type pairs for hypersurfaces with boundary. For the special case of anomalies coming from the volume enclosed by a minimal hypersurface ending on the boundary of a Poincaré–Einstein structure, this result recovers Branson’s [Formula: see text]-curvature and corresponding transgression. When the singular metric solves a boundary version of the constant scalar curvature Yamabe problem, the anomaly gives generalized Willmore energy functionals for hypersurfaces with boundary. Our approach yields computational algorithms for all the above quantities, and we give explicit results for surfaces embedded in 3-manifolds.

New Perspectives On Automatic And Morphophonological Alternations: Harmonic Processes In Two Peninsular Varieties Of Spanish

The present study investigated vowel harmony (VH) in two varieties of Peninsular Spanish - Eastern Andalusian and Montañes. Despite both varieties exhibiting VH, the triggers and targets for each variety result in metaphonic alternations that are quite distinct. Although previous research has extensively documented the VH of Andalusia and Montañes, no study has yet systematically compared the two using a singular metric for determining automatic (i.e., phonological) and morphophonological alternations.To address these questions, VH in each variety is described in detail and then classified as either an automatic or morphophonological alternation according to the following eight criteria indicated in Haspelmath and Sims (2010): phonological versus morphological or lexical conditioning, phonetic coherency, phonetic distance, restriction to derived environments, extension to loanwords, sensitivity to speech-style, creation of new segments, and restriction to the word level. In order to gain a more compete understanding of the morphology-phonology interface in Spanish, we explore similarities and differences in the VH of Eastern Andalusia and of the north of Spain. We seek to determine if VH in each region is more characteristic of automatic or morphophonological alternations.An in-depth analysis of the VH in each variety is revealed that a binary classification was less appropriate than viewing these alternations on a continuum. The nuanced representation of these alternations on a continuum is a unique contribution to the literature on Spanish VH and provides a fresh perspective on the nature of VH alternations in Peninsular Spanish.

Gradient Trajectories For Plane Singular Metrics I: Oscillating Trajectories

In this short note, we construct an example of a real plane analytic singular metric, degenerating only at the origin, such that any gradient trajectory (respectively to this singular metric) of some well chosen function spirals around the origin. The inversion mapping carries this example into an example of a gradient spiraling dynamics at infinity

A LOCAL OPTIMAL DIASTOLIC INEQUALITY ON THE TWO-SPHERE

We prove a local optimal inequality on the two-sphere between the area and the diastole — defined by a minimax process on the one-cycle space — in a neighborhood of the singular metric made of two equilateral triangles glued along their boundaries.

DIMENSIONAL REDUCTION OF QUANTUM FIELDS ON A BRANE

A singular metric arising in some brane models can substantially influence the behavior of Green's functions. We show that the quantum scalar field defined by a restriction of the Green's function to the brane can behave as a canonical quantum field living on the lower dimensional submanifold.

NEW POLARIZATIONS ON THE MODULI SPACES AND THE THURSTON COMPACTIFICATION OF TEICHMÜLLER SPACE

Given a foliation F with closed leaves and with certain kinds of singularities on an oriented closed surface Σ, we construct in this paper an isotropic foliation on ℳ(Σ), the moduli space of flat G-connections, for G any compact simple simply connected Lie-group. We describe the infinitesimal structure of this isotropic foliation in terms of the basic cohomology with twisted coefficients of F. For any pair (F, g), where g is a singular metric on Σ compatible with F, we construct a new polarization on the symplectic manifold ℳ′(Σ), the open dense subset of smooth points of ℳ(Σ). We construct a sequence of complex structures on Σ, such that the corresponding complex structures on ℳ′(Σ) converges to the polarization associated to (F, g). In particular we see that the Jeffrey–Weitzman polarization on the SU(2)-moduli space is the limit of a sequence of complex structures induced from a degenerating family of complex structures on Σ, which converges to a point in the Thurston boundary of Teichmüller space of Σ. As a corollary of the above constructions, we establish a certain discontinuiuty at the Thurston boundary of Teichmüller space for the map from Teichmüller space to the space of polarizations on ℳ′(Σ). For any reducible finite order diffeomorphism of the surface, our constuction produces an invariant polarization on the moduli space.

Cone Preserving Mappings for Quadratic Cones Over Arbitrary Fields

Let F be a non-singular metric vector space, that is, a vector space over a field F not of characteristic two, upon which is defined a non-singular symmetric bilinear form ( , ). For any a ϵ V, we define the cone with vertex a to be the set