Fractional Yamabe problem on locally flat conformal infinities of Poincare-Einstein manifolds

We study in this paper the fractional Yamabe problem first considered by Gonzalez-Qing on the conformal infinity $(M^n , [h])$ of a Poincar\'e-Einstein manifold $(X^{n+1} , g^+ )$ with either $n = 2$ or $n \geq 3$ and $(M^n , [h])$ is locally flat - namely $(M, h)$ is locally conformally flat. However, as for the classical Yamabe problem, because of the involved quantization phenomena, the variational analysis of the fractional one exhibits also a local situation and a global one. Furthermore the latter global situation includes the case of conformal infinities of Poincar\'e-Einstein manifolds of dimension either 2 or of dimension greater than $2$ and which are locally flat, and hence the minimizing technique of Aubin- Schoen in that case clearly requires an analogue of the positive mass theorem of Schoen-Yau which is not known to hold. Using the algebraic topological argument of Bahri-Coron, we bypass the latter positive mass issue and show that any conformal infinity of a Poincar\'e-Einstein manifold of dimension either $n = 2$ or of dimension $n \geq 3$ and which is locally flat admits a Riemannian metric of constant fractional scalar curvature.Comment: The current version - as of July 2021 - corresponds to sections 5,6,7 of the previous one. We have split out the others to a separate paper 'Asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian

Morse theory and the resonant Q-curvature problem

In this paper, we study the prescribed $Q$-curvature problem on closed four-dimensional Riemannian manifolds when the total integral of the $Q$-curvature is a positive integer multiple of the one of the four-dimensional round sphere. This problem has a variational structure with a lack of compactness. Using some topological tools of the theory of "critical points at infinity" combined with a refined blow-up analysis and some dynamical arguments, we identify the accumulations points of all noncompact flow lines of a pseudogradient flow, the so called critical points at infinity of the associated variational problem, and associate to them a natural Morse index. We then prove strong Morse type inequalities, extending the full Morse theory to this noncompact variational problem. Finally, we derive from our results Poincar\'e-Hopf index type criteria for existence, extending known results in the literature and deriving new ones

Earnings and Percentage Female: A Longitudinal Study

Comparable worth is designed to raise the earnings of women assumed to be penalized for working in female-dominated occupations. Comparable worth advocates assume that the relation between earnings and percentage female in an occupation is due to crowding or other forms of discrimination. An alternative explanation is that the relation stems from women freely choosing different occupations. In other words, preferences are an omitted variable. In our study, we first replicate previous research that has used cross-sectional data to find a negative relation between earnings and percentage female (in an occupation) for both men and women. However, using longitudinal data to control for time-invariant omitted variables, we find that while men\u27s estimated penalty is not reduced, the percentage female penalty falls substantially for women and is not statistically significant. These results imply that estimates of the percentage female effect based on cross-sectional data may be inflated for women. An exception to this general finding is that women with intermittent labor force participation do experience a sizeable penalty for working in female-dominated occupations. Hence, this pattern of results suggests that a comparable worth policy would most likely benefit women with discontinuous employment--perhaps an unintended outcome