We study the behavior of spatially homogeneous brane-worlds close to the initial singularity in the presence of both local and nonlocal stresses. It is found that the singularity in these brane-worlds can be locally either isotropic or anisotropic. We then investigate the Weyl curvature conjecture, according to which some measure of the Weyl curvature is related to a gravitational entropy. In p...

We derive the redshift and the angular diameter distance in rotationless dust universes which are statistically homogeneous and isotropic, but have otherwise arbitrary geometry. The calculation from first principles shows that the Dyer-Roeder approximation does not correctly describe the effect of clumping. Instead, the redshift and the distance are determined by the average expansion rate, the...

We prove that Ricci flows with almost maximal extinction time must be nearly round, provided that they have positive isotropic curvature when crossed with R. As an application, we show that positively curved metrics on S and RP 3 with almost maximal width must be nearly round.

In this paper we study the properties of special (α, β)-metric α α−β + β, the Randers change of Matsumoto metric. We find a necessary and sufficient condition for this metric to be of locally projectively flat and we prove the conditions for this metric to be of Berwald and Douglas type.

Elasticity theory calculations predict the number N of depressions that appear at the surface of a spherical thin shell submitted to an external isotropic pressure. Using a model that mainly considers curvature deformations, we show that N depends on the relative volume variation and on an adimensional parameter that takes into account both the relative spontaneous curvature and the relative th...

The growth of nematic liquid crystal nuclei from an isotropic melt follows a power law behavior with exponent n found experimentally to vary between 1/2 for low quench depths, up to 1 for high quench depths. This behavior has been attributed to the competition between curvature and free energy. We show that curvature cannot account for the low quench depth behavior of the nucleus growth, and at...

The Dyson‐Maleev formalism is used to calculate the decay rate of antiferromagnetic spin waves at low temperatures and long wavelengths. Various regimes must be distinguished depending on the relation between the wavevector k, the temperature T, and the anisotropy energy. For the isotropic system the relevant parameters are (a) the incident energy, (b) the thermal energy, (c) the deviation from...

By using the geometric concept of PDEs with prescribed curvature representation, we prove that the 1+2 dimensional isotropic Landau-Lifshitz equation is gauge equivalent to a 1+2 dimensional nonlinear Schrödinger-type system. From the nonlinear Schrödinger-type system, we construct blowing up H3(R2)-solutions to the Landau-Lifshitz equation, which reveals the blow up phenomenon of the equation.

We study the formation of five-dimensional string solutions including the Gregory-Laflamme (GL) black string, the Kaluza-Klein (KK) bubble, and the geometry with a naked singularity from the gravitational collapse. The interior solutions of five-dimensional Einstein equations describe collapsing non-isotropic matter clouds. It is shown that the matter cloud always forms the GL black string solu...