arXiv research

Embeds Higson compactification into adelic solenoids.

A topology on a set $X$ is the same as a projection (i.e. an idempotent linear operator) $cl:2^X\to 2^X$ satisfying $A\subset cl(A)$ for all $A\subset X$. That's a good way to summarize Kuratowski's closure operator. Basic geometry on a set $X$ is a dot product $\cdot:2^X\times 2^X\to 2^Y$. Its equivalent form is an or…

In this paper, we give a framework for the study of the extremal length geometry of Teichmüller space after S. Kerckhoff, F. Gardiner and H. Masur. There is a natural compactification using extremal length geometry introduced by Gardiner and Masur. The compactification is realized in a certain projective space. We deve…

Paper tackles conditional expectation estimation using compactification operators.

We produce some explicit examples of conformally compact Einstein manifolds, whose conformal compactifications are foliated by Riemannian products of a closed Einstein manifold with the total space of a principal circle bundle over products of Kahler-Einstein manifolds. We compute the associated conformal invariants, i…

The paper solves a specific Dirichlet problem for constant mean curvature surfaces in a particular manifold.

This is Part II of a series on noncompact isometry groups of Lorentz manifolds. We have introduced in Part I, a compactification of these isometry groups, and called ``bi-polarized'' those Lorentz manifolds having a ``trivial '' compactification. Here we show a geometric rigidity of non-bi-polarized Lorentz manifolds; …

We show that for $X$ a proper $\mathrm{CAT}(-1)$ space there is a maximal open subset of the horofunction compactification of $X\times X$ with respect to the maximum metric that compactifies the diagonal action of an infinite quasi-convex group of the isometries of $X$. We also consider the product action of two quasi-…

Defines products for fibered corners manifolds, generalizing resolutions.

We discuss `hd-compactifications' of $\SL(2,\bbK)$ for $\bbK=\bbC$ or $\bbR.$ These are compact manifolds with boundary on which both the Schwartz and the Harish-Chandra Schwartz spaces are shown to be relatively standard spaces of conormal functions relative to the boundary. Closure under convolution and other module …

Study compactifies representations space of hyperbolic surfaces.

Y-algebroids generalize symmetry structures for string/M-theory compactifications.

Geometrically describes Satake compactifications without root data.

Let $G$ be an almost simple, simply connected algebraic group defined over a number field $k$, and let $S$ be a finite set of places of $k$ including all infinite places. Let $X$ be the product over $v\in S$ of the symmetric spaces associated to $G(k_v)$, when $v$ is an infinite place, and the Bruhat-Tits buildings ass…

The Planck mass can be derived from gravitational potential behavior in compactifications.

Finite rank median spaces are a simultaneous generalisation of finite dimensional ${\rm CAT}(0)$ cube complexes and real trees. If $Γ$ is an irreducible lattice in a product of rank one simple Lie groups, we show that every action of $Γ$ on a complete, finite rank median space has a global fixed point. This is in sharp…

The paper extends Huber's theorem to higher dimensions using n-Laplace equations.

We construct a triangulation of a compactification of the Moduli space of a surface with at least one puncture that is closely related to the Deligne-Mumford compactification. Specifically, there is a surjective map from the compactification we construct to the Deligne-Mumford compactification so that the inverse image…

Study on instantons over product manifolds with a codimension-4 form.

Extends harmonic maps compactification to punctured Riemann surfaces.

We interpret heterotic M-theory in terms of h-cobordism, that is the eleven-manifold is a product of the ten-manifold times an interval is translated into a statement that the former is a cobordism of the latter which is a homtopy equivalence. In the non-simply connected case, which is important for model building, the…

Paper relates new compactification to classical moduli space.

We consider horofunction compactifications of symmetric spaces with respect to invariant Finsler metrics. We show that any (generalized) Satake compactification can be realized as a horofunction compactification with respect to a polyhedral Finsler metric.

Existence of Kähler-Einstein metrics on compactifications of Lie groups.

The horoboundary of Teichmüller space is path connected and has non-dense Busemann points.

Characterizes toroidal and semi-toric compactifications as log minimal models and applies to weak K-moduli.

Satake has constructed compactifications of symmetric spaces D=G/K which (under a condition called geometric rationality by Casselman) yield compactifications of the corresponding locally symmetric spaces. The different compactifications depend on the choice of a representation of G. One example is the Baily-Borel-Sata…

The paper studies compactifications of SL(2,C) character varieties for punctured surfaces.

New coordinates for Teichmüller space compactification.

New compactification for character varieties with good topological properties.

We discuss the `hd-compactification' of a semi-simple Lie group to a manifold with corners; it is the real analog of the wonderful compactification of deConcini and Procesi. There is a 1-1 correspondence between the boundary faces of the compactification and conjugacy classes of parabolic subgroups with the boundary fa…

We show that the horofunction compactification of Teichmüller space with the Teichmüller metric is homeomorphic to the Gardiner-Masur compactification.

We define a compactification of symmetric spaces of noncompact type, seen as spaces of isometry classes of marked lattices, analogous to the Thurston compactification of the Teichmüller space, and we show that it is equivariantly isomorphic to a Satake compactification. We then use it to define a new compactification o…

We construct several examples of compactifications of Einstein metrics. We show that the Eguchi--Hanson instanton admits a projective compactification which is non--metric, and that a metric cone over any (pseudo)--Riemannian manifolds admits a metric projective compactification. We construct a para--$c$--projective co…

Researchers create a new compactification of character varieties using geometric and algebraic methods.

Paper studies compactifications of Higgs bundles and self-duality equations.

Guilbault and author prove sufficient conditions for $\mathcal{Z}$-compactifiability of certain manifolds.

No natural topological compactification for Fulton-MacPherson.

Paper shows geometric properties preserved by compactifications in relation to coarse structures and group actions.

The paper classifies and computes limits of equivariant compactifications of groups.

We consider the asymptotic behavior of properly embedded minimal surfaces in the product of the hyperbolic plane with the line, taking into account the fact that there is more than one natural compactification of this space. This provides a better setting in which to consider the general problem of determining which cu…

The study finds only finitely many Kähler-Einstein compactifications for semisimple groups.

The paper extends end concepts to arbitrary groups and spaces.

In this paper we present a topological way of building a compactification of a symmetric space from a compactification of a Weyl Chamber.

To study a noncompact Riemannian manifold, it is often useful to find a compactification. We discuss several common compactifications and survey some recent results.

The group of direct isometries of the real n-dimensional hyperbolic space is G=SOo(n,1). This isometric action admits many differentiable compactifications into an action on the closed ball. We prove that all such compactifications are topologically conjugate but not necessarily differentiably conjugate. We give the cl…

Consider a finite dimensional (generally reducible) polynomial representation ρof GL_n. A projective compactification of GL_n is the closure of ρ(GL_n) in the space of all operators defined up to a factor (this class of spaces can be characterized as equivariant projective normal compactifications of GL_n). We give an …

Schwartz functions smoothly extend to real projective spaces.