A Polyakov formula for sectors

We consider finite area convex Euclidean circular sectors. We prove a variational Polyakov formula which shows how the zeta-regularized determinant of the Laplacian varies with respect to the opening angle. Varying the angle corresponds to a conformal deformation in the direction of a conformal factor with a logarithmic singularity at the origin. We compute explicitly all the contributions to this formula coming from the different parts of the sector. In the process, we obtain an explicit expression for the heat kernel on an infinite area sector using Carslaw-Sommerfeld's heat kernel. We also compute the zeta-regularized determinant of rectangular domains of unit area and prove that it is uniquely maximized by the square.Comment: 51 pages, 2 figures. Major modification of Lemma 4, it was revised and corrected. Other small misprints were corrected. Accepted for publication in The Journal of Geometric Analysi

Polyakov formulas for conical singularities in two dimensions

We investigate the zeta-regularized determinant and its variation in the presence of conical singularities, boundaries, and corners. For surfaces with isolated conical singularities which may also have one or more smooth boundary components, we demonstrate both a variational Polyakov formula as well as an integrated Polyakov formula for the conformal variation of the Riemannian metric with conformal factors which are smooth up to all singular points and boundary components. We demonstrate the analogous result for curvilinear polygonal domains in surfaces. We then specialize to finite circular sectors and cones and via two independent methods obtain variational Polyakov formulas for the dependence of the determinant on the opening angle. Notably, this requires the conformal factor to be logarithmically singular at the vertex. Although these formulas look quite different, we prove that they are indeed equal. We further obtain explicit formulas for the determinant for finite circular sectors and cones

Asymptotics of relative heat traces and determinants on open surfaces of finite area

The goal of this paper is to prove that on surfaces with asymptotically cusp ends the relative determinant of pairs of Laplace operators is well defined. We consider a surface with cusps (M,g) and a metric h on the surface that is a conformal transformation of the initial metric g. We prove the existence of the relative determinant of the pair $(\Delta_{h},\Delta_{g})$ under suitable conditions on the conformal factor. The core of the paper is the proof of the existence of an asymptotic expansion of the relative heat trace for small times. We find the decay of the conformal factor at infinity for which this asymptotic expansion exists and the relative determinant is defined. Following the paper by B. Osgood, R. Phillips and P. Sarnak about extremal of determinants on compact surfaces, we prove Polyakov's formula for the relative determinant and discuss the extremal problem inside a conformal class. We discuss necessary conditions for the existence of a maximizer.Comment: This is the final version of the article before it gets published. 51 page

Reconstructing the Deep Population History of Central and South America

We report genome-wide ancient DNA from 49 individuals forming four parallel time transects in Belize, Brazil, the Central Andes, and the Southern Cone, each dating to at least 9,000 years ago. The common ancestral population radiated rapidly from just one of the two early branches that contributed to Native Americans today. We document two previously unappreciated streams of gene flow between North and South America. One affected the Central Andes by 4,200 years ago, while the other explains an affinity between the oldest North American genome associated with the Clovis culture and the oldest Central and South Americans from Chile, Brazil, and Belize. However, this was not the primary source for later South Americans, as the other ancient individuals derive from lineages without specific affinity to the Clovis-associated genome, suggesting a population replacement that began at least 9,000 years ago and was followed by substantial population continuity in multiple regions

A∞A_\infty weights and compactness of conformal metrics under Ln/2L^{n/2} curvature bounds

International audienceWe study sequences of conformal deformations of a smooth closed Riemannian manifold of dimension $n$, assuming uniform volume bounds and $L^{n/2}$ bounds on their scalar curvatures. Singularities may appear in the limit. Nevertheless, we show that under such bounds the underlying metric spaces are pre-compact in the Gromov-Hausdorff topology. Our study is based on the use of $A_\infty$-weights from harmonic analysis, and provides geometric controls on the limit spaces thus obtained. Our techniques also show that any conformal deformation of the Euclidean metric on $R^n$ with infinite volume and finite $L^{n/2}$ norm of the scalar curvature satisfies the Euclidean isoperimetric inequality