Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · dg-ga/9703022

Separated nets in Euclidean space and Jacobians of biLipschitz maps

Dmitri Burago·Bruce Kleiner
View original source ↗Read source document ↗Export metadata ↓

Abstract

We show that there are separated nets in the Euclidean plane which are not biLipschitz equivalent to the integer lattice. The argument is based on the construction of a continuous function which is not the Jacobian of a biLipschitz map.

Explore related subjects

dg-gamath.DGmath.MG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Dmitri Burago, Bruce Kleiner. 1997-03-30. Separated nets in Euclidean space and Jacobians of biLipschitz maps. https://arxiv.org/abs/dg-ga/9703022

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2018

A Simple Geometric Representative for $μ$ of a Point

For $SU(2)$ (or $SO(3)$) Donaldson theory on a 4-manifold $X$, we construct a simple geometric representative for $μ$ of a point. Let $p$ be a generic point in $X$. Then the set $\{ [A] | F_A^-(p) $ is reducible $\}$, with coefficient -1/4 and appropriate orientation, is our desired geometric representative.

dg-ga↗
arXiv2013

Moduli spaces of PU(2)-monopoles

The goal of this article was the S^1-equivariant transversality-problem and the compactification-problem for the moduli spaces of (perturbed) PU(2)-monopoles. A substantially improved version entitled "Moduli spaces of PU(2)-monopoles (revised version)" which gives simpler, clearer proofs of the transversality results, has been published on arxiv in June 99 and appeared in Asian J. Math, see Moduli spaces of PU(2)-Monopoles, Asian J. Math. Vol. 4, No. 2 (2000), 391-436.

dg-ga↗
arXiv2008

L^2-torsion of hyperbolic manifolds of finite volume

Suppose $\bar{M}$ is a compact connected odd-dimensional manifold with boundary, whose interior $M$ comes with a complete hyperbolic metric of finite volume. We will show that the $L^2$-topological torsion of $\bar{M}$ and the $L^2$-analytic torsion of the Riemannian manifold $M$ are equal. In particular, the $L^2$-topological torsion of $\bar{M}$ is proportional to the hyperbolic volume of $M$, with a constant of proportionality which depends only on the dimension and which is known to be nonzero in dimension 3, 5 and 7. In dimension 3 this proves the conjecture Of Lott and Lueck which gives a complete calculation of the $L^2$-topological torsion of compact $L^2$-acyclic 3-manifolds which admit a geometric torus-decomposition. In an appendix we give a counterexample to an extension of the Cheeger-Mueller theorem to manifolds with boundary: if the metric is not a product near the boundary, in general analytic and topological torsion are not equal, even if the Euler characteristic of the boundary vanishes. Keywords: L^2-torsion, hyperbolic manifolds, 3-manifolds

dg-ga↗