22,117 research outputs found
Existence of infinitely many minimal hypersurfaces in positive Ricci curvature
In the early 1980s, S. T. Yau conjectured that any compact Riemannian
three-manifold admits an infinite number of closed immersed minimal surfaces.
We use min-max theory for the area functional to prove this conjecture in the
positive Ricci curvature setting. More precisely, we show that every compact
Riemannian manifold with positive Ricci curvature and dimension at most seven
contains infinitely many smooth, closed, embedded minimal hypersurfaces.
In the last section we mention some open problems related with the geometry
of these minimal hypersurfaces.Comment: 34 pages, to appear in Inventiones Mathematica
Morse index and multiplicity of min-max minimal hypersurfaces
The Min-max Theory for the area functional, started by Almgren in the early
1960s and greatly improved by Pitts in 1981, was left incomplete because it
gave no Morse index estimate for the min-max minimal hypersurface.
We advance the theory further and prove the first general Morse index bounds
for minimal hypersurfaces produced by it. We also settle the multiplicity
problem for the classical case of one-parameter sweepouts.Comment: Cambridge Journal of Mathematics, 4 (4), 463-511, 201
Local entropic effects of polymers grafted to soft interfaces
In this paper, we study the equilibrium properties of polymer chains
end-tethered to a fluid membrane. The loss of conformational entropy of the
polymer results in an inhomogeneous pressure field that we calculate for
gaussian chains. We estimate the effects of excluded volume through a relation
between pressure and concentration. Under the polymer pressure, a soft surface
will deform. We calculate the deformation profile for a fluid membrane and show
that close to the grafting point, this profile assumes a cone-like shape,
independently of the boundary conditions. Interactions between different
polymers are also mediated by the membrane deformation. This pair-additive
potential is attractive for chains grafted on the same side of the membrane and
repulsive otherwise.Comment: 10 pages, 9 figure
Depletion forces between two spheres in a rod solution
We study the depletion interaction between spherical particles of radius R
immersed in a dilute solution of rigid rods of length L. The computed
interaction potential is, within numerical accuracy, exact for any value of
L/R. In particular we find that for L of order R, the depth of the depletion
well is smaller than the prediction of the Derjaguin approximation. Our results
bring new light into the discussion on the lack of phase separation in
colloidal mixtures of spheres and rods.Comment: 12 pages including figures. 5 eps figures. LaTeX with REVTe
Hawking radiation for non asymptotically flat dilatonic black holes using gravitational anomaly
The -dimensional scalar field action may be reduced, in the background
geometry of a black hole, to a 2-dimensional effective action. In the near
horizon region, it appears a gravitational anomaly: the energy-momentum tensor
of the scalar field is not conserved anymore. This anomaly is removed by
introducing a term related to the Hawking temperature of the black hole. Even
if the temperature term introduced is not covariant, a gauge transformation may
restore the covariance. We apply this method to compute the temperature of the
black hole of the dilatonic non asymptotically flat black holes. We compare the
results with those obtained through other methods.Comment: Latex file, 22 pages. Some discussions enlarged. New references.
Accepted for publication in the European Physical Journal
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