157 research outputs found
A simple proof of Perelman's collapsing theorem for 3-manifolds
We will simplify earlier proofs of Perelman's collapsing theorem for
3-manifolds given by Shioya-Yamaguchi and Morgan-Tian. Among other things, we
use Perelman's critical point theory (e.g., multiple conic singularity theory
and his fibration theory) for Alexandrov spaces to construct the desired local
Seifert fibration structure on collapsed 3-manifolds. The verification of
Perelman's collapsing theorem is the last step of Perelman's proof of
Thurston's Geometrization Conjecture on the classification of 3-manifolds. Our
proof of Perelman's collapsing theorem is almost self-contained, accessible to
non-experts and advanced graduate students. Perelman's collapsing theorem for
3-manifolds can be viewed as an extension of implicit function theoremComment: v1: 9 Figures. In this version, we improve the exposition of our
arguments in the earlier arXiv version. v2: added one more grap
Polynomial Growth Harmonic Functions on Finitely Generated Abelian Groups
In the present paper, we develop geometric analytic techniques on Cayley
graphs of finitely generated abelian groups to study the polynomial growth
harmonic functions. We develop a geometric analytic proof of the classical
Heilbronn theorem and the recent Nayar theorem on polynomial growth harmonic
functions on lattices \mathds{Z}^n that does not use a representation formula
for harmonic functions. We also calculate the precise dimension of the space of
polynomial growth harmonic functions on finitely generated abelian groups.
While the Cayley graph not only depends on the abelian group, but also on the
choice of a generating set, we find that this dimension depends only on the
group itself.Comment: 15 pages, to appear in Ann. Global Anal. Geo
Doubly connected minimal surfaces and extremal harmonic mappings
The concept of a conformal deformation has two natural extensions:
quasiconformal and harmonic mappings. Both classes do not preserve the
conformal type of the domain, however they cannot change it in an arbitrary
way. Doubly connected domains are where one first observes nontrivial conformal
invariants. Herbert Groetzsch and Johannes C. C. Nitsche addressed this issue
for quasiconformal and harmonic mappings, respectively. Combining these
concepts we obtain sharp estimates for quasiconformal harmonic mappings between
doubly connected domains. We then apply our results to the Cauchy problem for
minimal surfaces, also known as the Bjorling problem. Specifically, we obtain a
sharp estimate of the modulus of a doubly connected minimal surface that
evolves from its inner boundary with a given initial slope.Comment: 35 pages, 2 figures. Minor edits, references adde
More about Birkhoff's Invariant and Thorne's Hoop Conjecture for Horizons
A recent precise formulation of the hoop conjecture in four spacetime
dimensions is that the Birkhoff invariant (the least maximal length of
any sweepout or foliation by circles) of an apparent horizon of energy and
area should satisfy . This conjecture together with the
Cosmic Censorship or Isoperimetric inequality implies that the length of
the shortest non-trivial closed geodesic satisfies . We have
tested these conjectures on the horizons of all four-charged rotating black
hole solutions of ungauged supergravity theories and find that they always
hold. They continue to hold in the the presence of a negative cosmological
constant, and for multi-charged rotating solutions in gauged supergravity.
Surprisingly, they also hold for the Ernst-Wild static black holes immersed in
a magnetic field, which are asymptotic to the Melvin solution. In five
spacetime dimensions we define as the least maximal area of all
sweepouts of the horizon by two-dimensional tori, and find in all cases
examined that , which we conjecture holds
quiet generally for apparent horizons. In even spacetime dimensions ,
we find that for sweepouts by the product , is
bounded from above by a certain dimension-dependent multiple of the energy .
We also find that is bounded from above by a certain
dimension-dependent multiple of the horizon area . Finally, we show that
is bounded from above by a certain dimension-dependent multiple of
the energy, for all Kerr-AdS black holes.Comment: 25 page
On the structure of phase transition maps for three or more coexisting phases
This paper is partly based on a lecture delivered by the author at the ERC
workshop "Geometric Partial Differential Equations" held in Pisa in September
2012. What is presented here is an expanded version of that lecture.Comment: 23 pages, 6 figure
The stability inequality for Ricci-flat cones
In this article, we thoroughly investigate the stability inequality for Ricci-flat cones. Perhaps most importantly, we prove that the Ricci-flat cone over CP^2 is stable, showing that the first stable non-flat Ricci-flat cone occurs in the smallest possible dimension. On the other hand, we prove that many other examples of Ricci-flat cones over 4-manifolds are unstable, and that Ricci-flat cones over products of Einstein manifolds and over Kähler-Einstein manifolds with h^{1,1}>1 are unstable in dimension less than 10. As results of independent interest, our computations indicate that the Page metric and the Chen-LeBrun-Weber metric are unstable Ricci shrinkers. As a final bonus, we give plenty of motivations, and partly confirm a conjecture of Tom Ilmanen relating the lambda-functional, the positive mass theorem and the nonuniqueness of Ricci flow with conical initial data
Non stationary Einstein-Maxwell fields interacting with a superconducting cosmic string
Non stationary cylindrically symmetric exact solutions of the
Einstein-Maxwell equations are derived as single soliton perturbations of a
Levi-Civita metric, by an application of Alekseev inverse scattering method. We
show that the metric derived by L. Witten, interpreted as describing the
electrogravitational field of a straight, stationary, conducting wire may be
recovered in the limit of a `wide' soliton. This leads to the possibility of
interpreting the solitonic solutions as representing a non stationary
electrogravitational field exterior to, and interacting with, a thin, straight,
superconducting cosmic string. We give a detailed discussion of the
restrictions that arise when appropiate energy and regularity conditions are
imposed on the matter and fields comprising the string, considered as `source',
the most important being that this `source' must necessarily have a non-
vanishing minimum radius. We show that as a consequence, it is not possible,
except in the stationary case, to assign uniquely a current to the source from
a knowledge of the electrogravitational fields outside the source. A discussion
of the asymptotic properties of the metrics, the physical meaning of their
curvature singularities, as well as that of some of the metric parameters, is
also included.Comment: 14 pages, no figures (RevTex
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