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 $d$-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