317 research outputs found
Minimality of planes in normed spaces
We prove that a region in a two-dimensional affine subspace of a normed space
has the least 2-dimensional Hausdorff measure among all compact surfaces
with the same boundary. Furthermore, the 2-dimensional Hausdorff area density
admits a convex extension to . The proof is based on a (probably)
new inequality for the Euclidean area of a convex centrally-symmetric polygon.Comment: 10 pages, v2: minor changes according to referees' comments, to
appear in GAF
Profiles of inflated surfaces
We study the shape of inflated surfaces introduced in \cite{B1} and
\cite{P1}. More precisely, we analyze profiles of surfaces obtained by
inflating a convex polyhedron, or more generally an almost everywhere flat
surface, with a symmetry plane. We show that such profiles are in a
one-parameter family of curves which we describe explicitly as the solutions of
a certain differential equation.Comment: 13 pages, 2 figure
The connected components of the space of Alexandrov surfaces
Denote by the set of all compact Alexandrov surfaces
with curvature bounded below by without boundary, endowed with the
topology induced by the Gromov-Hausdorff metric. We determine the connected
components of and of its closure
Spectral stability of metric-measure Laplacians
We consider a "convolution mm-Laplacian" operator on metric-measure spaces and study its spectral properties. The definition is based on averaging over small metric balls. For reasonably nice metric-measure spaces we prove stability of convolution Laplacian's spectrum with respect to metric-measure perturbations and obtain Weyl-type estimates on the number of eigenvalues
Filling minimality of Finslerian 2-discs
We prove that every Riemannian metric on the 2-disc such that all its
geodesics are minimal, is a minimal filling of its boundary (within the class
of fillings homeomorphic to the disc). This improves an earlier result of the
author by removing the assumption that the boundary is convex. More generally,
we prove this result for Finsler metrics with area defined as the
two-dimensional Holmes-Thompson volume. This implies a generalization of Pu's
isosystolic inequality to Finsler metrics, both for Holmes-Thompson and
Busemann definitions of Finsler area.Comment: 16 pages, v2: improved introduction and formattin
Relative entropy as a measure of inhomogeneity in general relativity
We introduce the notion of relative volume entropy for two spacetimes with
preferred compact spacelike foliations. This is accomplished by applying the
notion of Kullback-Leibler divergence to the volume elements induced on
spacelike slices. The resulting quantity gives a lower bound on the number of
bits which are necessary to describe one metric given the other. For
illustration, we study some examples, in particular gravitational waves, and
conclude that the relative volume entropy is a suitable device for quantitative
comparison of the inhomogeneity of two spacetimes.Comment: 15 pages, 7 figure
A Non-Riemannian Metric on Space-Time Emergent From Scalar Quantum Field Theory
We show that the two-point function
\sigma(x,x')=\sqrt{} of a scalar quantum field theory
is a metric (i.e., a symmetric positive function satisfying the triangle
inequality) on space-time (with imaginary time). It is very different from the
Euclidean metric |x-x'| at large distances, yet agrees with it at short
distances. For example, space-time has finite diameter which is not universal.
The Lipschitz equivalence class of the metric is independent of the cutoff.
\sigma(x,x') is not the length of the geodesic in any Riemannian metric.
Nevertheless, it is possible to embed space-time in a higher dimensional space
so that \sigma(x,x') is the length of the geodesic in the ambient space.
\sigma(x,x') should be useful in constructing the continuum limit of quantum
field theory with fundamental scalar particles
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