1,025 research outputs found
Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature
We prove convergence results for expanding curvature flows in the Euclidean
and hyperbolic space. The flow speeds have the form , where and
is a positive, strictly monotone and 1-homogeneous curvature function. In
particular this class includes the mean curvature . We prove that a
certain initial pinching condition is preserved and the properly rescaled
hypersurfaces converge smoothly to the unit sphere. We show that an example due
to Andrews-McCoy-Zheng can be used to construct strictly convex initial
hypersurfaces, for which the inverse mean curvature flow to the power
loses convexity, justifying the necessity to impose a certain pinching
condition on the initial hypersurface.Comment: 18 pages. We included an example for the loss of convexity and
pinching. In the third version we dropped the concavity assumption on F.
Comments are welcom
A simple expression for the ADM mass
We show by an almost elementary calculation that the ADM mass of an
asymptotically flat space can be computed as a limit involving a rate of change
of area of a closed 2-surface. The result is essentially the same as that given
by Brown and York. We will prove this result in two ways, first by direct
calculation from the original formula as given by Arnowitt, Deser and Misner
and second as a corollary of an earlier result by Brewin for the case of
simplicial spaces.Comment: 9 pages, 1 figur
Global embedding of the Kerr black hole event horizon into hyperbolic 3-space
An explicit global and unique isometric embedding into hyperbolic 3-space,
H^3, of an axi-symmetric 2-surface with Gaussian curvature bounded below is
given. In particular, this allows the embedding into H^3 of surfaces of
revolution having negative, but finite, Gaussian curvature at smooth fixed
points of the U(1) isometry. As an example, we exhibit the global embedding of
the Kerr-Newman event horizon into H^3, for arbitrary values of the angular
momentum. For this example, considering a quotient of H^3 by the Picard group,
we show that the hyperbolic embedding fits in a fundamental domain of the group
up to a slightly larger value of the angular momentum than the limit for which
a global embedding into Euclidean 3-space is possible. An embedding of the
double-Kerr event horizon is also presented, as an example of an embedding
which cannot be made global.Comment: 16 pages, 13 figure
Evaluating quasilocal energy and solving optimal embedding equation at null infinity
We study the limit of quasilocal energy defined in [7] and [8] for a family
of spacelike 2-surfaces approaching null infinity of an asymptotically flat
spacetime. It is shown that Lorentzian symmetry is recovered and an
energy-momentum 4-vector is obtained. In particular, the result is consistent
with the Bondi-Sachs energy-momentum at a retarded time. The quasilocal mass in
[7] and [8] is defined by minimizing quasilocal energy among admissible
isometric embeddings and observers. The solvability of the Euler-Lagrange
equation for this variational problem is also discussed in both the
asymptotically flat and asymptotically null cases. Assuming analyticity, the
equation can be solved and the solution is locally minimizing in all orders. In
particular, this produces an optimal reference hypersurface in the Minkowski
space for the spatial or null exterior region of an asymptotically flat
spacetime.Comment: 22 page
The regularized 3D Boussinesq equations with fractional Laplacian and no diffusion
In this paper, we study the 3D regularized Boussinesq equations. The velocity
equation is regularized \`a la Leray through a smoothing kernel of order
in the nonlinear term and a -fractional Laplacian; we consider
the critical case and we assume . The temperature equation is a pure transport equation, where
the transport velocity is regularized through the same smoothing kernel of
order . We prove global well posedness when the initial velocity is in
and the initial temperature is in for
. This regularity is enough to prove uniqueness of
solutions. We also prove a continuous dependence of the solutions on the
initial conditions.Comment: 28 pages; final version accepted for publication in Journal of
Differential Equation
Local and Global Analytic Solutions for a Class of Characteristic Problems of the Einstein Vacuum Equations in the "Double Null Foliation Gauge"
The main goal of this work consists in showing that the analytic solutions
for a class of characteristic problems for the Einstein vacuum equations have
an existence region larger than the one provided by the Cauchy-Kowalevski
theorem due to the intrinsic hyperbolicity of the Einstein equations. To prove
this result we first describe a geometric way of writing the vacuum Einstein
equations for the characteristic problems we are considering, in a gauge
characterized by the introduction of a double null cone foliation of the
spacetime. Then we prove that the existence region for the analytic solutions
can be extended to a larger region which depends only on the validity of the
apriori estimates for the Weyl equations, associated to the "Bel-Robinson
norms". In particular if the initial data are sufficiently small we show that
the analytic solution is global. Before showing how to extend the existence
region we describe the same result in the case of the Burger equation, which,
even if much simpler, nevertheless requires analogous logical steps required
for the general proof. Due to length of this work, in this paper we mainly
concentrate on the definition of the gauge we use and on writing in a
"geometric" way the Einstein equations, then we show how the Cauchy-Kowalevski
theorem is adapted to the characteristic problem for the Einstein equations and
we describe how the existence region can be extended in the case of the Burger
equation. Finally we describe the structure of the extension proof in the case
of the Einstein equations. The technical parts of this last result is the
content of a second paper.Comment: 68 page
Biogas as a fuel source for proton-exchange membrane fuel cell
Paper presented to the 10th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, Florida, 14-16 July 2014.This paper discusses the use of biogas in Proton exchange Membrane fuel cells. Biogas is a mixture of gases, mainly methane and carbon dioxide. This mixture must be purified and then reformed, producing syngas and separating hydrogen from its composition. The CO present in the reformed gas in contact with the surface of the catalyst particles (a platinum-based) of the anode of PEM fuel cell is adsorbed impairing the oxidation of hydrogen, which leads to a decrease in cell efficiency. The reformed gas may not contain in its composition carbon monoxide concentrations greater than 50 ppm. The operating conditions of the system and the cost of the process are the main challenges of using biogas in PEM fuel cells.cf201
Critical points of Wang-Yau quasi-local energy
In this paper, we prove the following theorem regarding the Wang-Yau
quasi-local energy of a spacelike two-surface in a spacetime: Let be a
boundary component of some compact, time-symmetric, spacelike hypersurface
in a time-oriented spacetime satisfying the dominant energy
condition. Suppose the induced metric on has positive Gaussian
curvature and all boundary components of have positive mean curvature.
Suppose where is the mean curvature of in and
is the mean curvature of when isometrically embedded in .
If is not isometric to a domain in , then 1. the Brown-York mass
of in is a strict local minimum of the Wang-Yau quasi-local
energy of , 2. on a small perturbation of in
, there exists a critical point of the Wang-Yau quasi-local energy of
.Comment: substantially revised, main theorem replaced, Section 3 adde
Differential Geometry of Quantum States, Observables and Evolution
The geometrical description of Quantum Mechanics is reviewed and proposed as
an alternative picture to the standard ones. The basic notions of observables,
states, evolution and composition of systems are analised from this
perspective, the relevant geometrical structures and their associated algebraic
properties are highlighted, and the Qubit example is thoroughly discussed.Comment: 20 pages, comments are welcome
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