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

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

Weyl law for the volume spectrum

Given $M$ a Riemannian manifold with (possibly empty) boundary, we show that its volume spectrum $\{\omega_p(M)\}_{p\in\mathbb{N}}$ satisfies a Weyl law that was conjectured by Gromov.Comment: Revised version. To appear in Annals of Mathematic

Density of minimal hypersurfaces for generic metrics

For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a closed manifold $M^{n+1}$, $3\leq (n+1)\leq 7$, we prove that the union of all closed, smooth, embedded minimal hypersurfaces is dense. This implies there are infinitely many minimal hypersurfaces thus proving a conjecture of Yau (1982) for generic metrics.Comment: Revised version. To appear in Annals of Mathematic

Designing incenttives in local public utilities, an international comparison of the drinking water sector

Direct and indirect standardization procedures aim at comparing differences in health or differences in health care expenditures between subgroups of the population after controlling for observable morbidity differences. There is a close analogy between this problem and the issue of risk adjustment in health insurance. We analyse this analogy within the theoretical framework proposed in the recent social choice literature on responsibility and compensation. Traditional methods of risk adjustment are analogous to indirect standardization. They are equivalent to the so-called conditional egalitarian mechanism in social choice. In general, they do not remove incentives for risk selection, even if the effect of non-morbidity variables is correctly taken into account. A method of risk adjustment based on direct standardization (as proposed for Ireland) does remove the incentives for risk selection, but at the cost of violating a neutrality condition, stating that insurers should receive the same premium subsidy for all members of the same risk group. Direct standardization is equivalent to the egalitarianequivalent (or proportional) mechanism in social choice. The conflict between removing incentives for risk selection and neutrality is unavoidable if the health expenditure function is not additively separable in the morbidity and efficiency variables.

Spin-orbit coupling and magnetic spin states in cylindrical quantum dots

We make detailed analysis of each possible spin-orbit coupling of zincblende narrow-gap cylindrical quantum dots built in two-dimensional electron gas. These couplings are related to both bulk (Dresselhaus) and structure (Rashba) inversion asymmetries. We study the competition between electron-electron and spin-orbit interactions on electronic properties of 2-electron quantum dots.Comment: 6 pages, 6 figures, submitted to MR