215 research outputs found
Isoperimetric Inequalities for Minimal Submanifolds in Riemannian Manifolds: A Counterexample in Higher Codimension
For compact Riemannian manifolds with convex boundary, B.White proved the
following alternative: Either there is an isoperimetric inequality for minimal
hypersurfaces or there exists a closed minimal hypersurface, possibly with a
small singular set. There is the natural question if a similar result is true
for submanifolds of higher codimension. Specifically, B.White asked if the
non-existence of an isoperimetric inequality for k-varifolds implies the
existence of a nonzero, stationary, integral k-varifold. We present examples
showing that this is not true in codimension greater than two. The key step is
the construction of a Riemannian metric on the closed four-dimensional ball B
with the following properties: (1) B has strictly convex boundary. (2) There
exists a complete nonconstant geodesic. (3) There does not exist a closed
geodesic in B.Comment: 11 pages, We changed the title and added a section that exhibits the
relation between our example and the question posed by Brian White concerning
isoperimetric inequalities for minimal submanifold
On uniqueness of tangent cones for Einstein manifolds
We show that for any Ricci-flat manifold with Euclidean volume growth the
tangent cone at infinity is unique if one tangent cone has a smooth
cross-section. Similarly, for any noncollapsing limit of Einstein manifolds
with uniformly bounded Einstein constants, we show that local tangent cones are
unique if one tangent cone has a smooth cross-section
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
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
Mean curvature flow in a Ricci flow background
Following work of Ecker, we consider a weighted Gibbons-Hawking-York
functional on a Riemannian manifold-with-boundary. We compute its variational
properties and its time derivative under Perelman's modified Ricci flow. The
answer has a boundary term which involves an extension of Hamilton's Harnack
expression for the mean curvature flow in Euclidean space. We also derive the
evolution equations for the second fundamental form and the mean curvature,
under a mean curvature flow in a Ricci flow background. In the case of a
gradient Ricci soliton background, we discuss mean curvature solitons and
Huisken monotonicity.Comment: final versio
The area of horizons and the trapped region
This paper considers some fundamental questions concerning marginally trapped
surfaces, or apparent horizons, in Cauchy data sets for the Einstein equation.
An area estimate for outermost marginally trapped surfaces is proved. The proof
makes use of an existence result for marginal surfaces, in the presence of
barriers, curvature estimates, together with a novel surgery construction for
marginal surfaces. These results are applied to characterize the boundary of
the trapped region.Comment: 44 pages, v3: small changes in presentatio
-minimal surface and manifold with positive -Bakry-\'{E}mery Ricci curvature
In this paper, we first prove a compactness theorem for the space of closed
embedded -minimal surfaces of fixed topology in a closed three-manifold with
positive Bakry-\'{E}mery Ricci curvature. Then we give a Lichnerowicz type
lower bound of the first eigenvalue of the -Laplacian on compact manifold
with positive -Bakry-\'{E}mery Ricci curvature, and prove that the lower
bound is achieved only if the manifold is isometric to the -shpere, or the
-dimensional hemisphere. Finally, for compact manifold with positive
-Bakry-\'{E}mery Ricci curvature and -mean convex boundary, we prove an
upper bound for the distance function to the boundary, and the upper bound is
achieved if only if the manifold is isometric to an Euclidean ball.Comment: 15 page
Complete Classification of the String-like Solutions of the Gravitating Abelian Higgs Model
The static cylindrically symmetric solutions of the gravitating Abelian Higgs
model form a two parameter family. In this paper we give a complete
classification of the string-like solutions of this system. We show that the
parameter plane is composed of two different regions with the following
characteristics: One region contains the standard asymptotically conic cosmic
string solutions together with a second kind of solutions with Melvin-like
asymptotic behavior. The other region contains two types of solutions with
bounded radial extension. The border between the two regions is the curve of
maximal angular deficit of .Comment: 12 pages, 4 figure
Cosmic Strings in the Abelian Higgs Model with Conformal Coupling to Gravity
Cosmic string solutions of the abelian Higgs model with conformal coupling to
gravity are shown to exist. The main characteristics of the solutions are
presented and the differences with respect to the minimally coupled case are
studied. An important difference is the absence of Bogomolnyi cosmic string
solutions for conformal coupling. Several new features of the abelian Higgs
cosmic strings of both types are discussed. The most interesting is perhaps a
relation between the angular deficit and the central magnetic field which is
bounded by a critical value.Comment: 22 pages, 10 figures; to appear in Phys. Rev.
Instability and `Sausage-String' Appearance in Blood Vessels during High Blood Pressure
A new Rayleigh-type instability is proposed to explain the `sausage-string'
pattern of alternating constrictions and dilatations formed in blood vessels
under influence of a vasoconstricting agent. Our theory involves the nonlinear
elasticity characteristics of the vessel wall, and provides predictions for the
conditions under which the cylindrical form of a blood vessel becomes unstable.Comment: 4 pages, 4 figures submitted to Physical Review Letter
- …