Surfaces with boundary: their uniformizations, determinants of Laplacians, and isospectrality

Let \Sigma be a compact surface of type (g, n), n > 0, obtained by removing n disjoint disks from a closed surface of genus g. Assuming \chi(\Sigma)<0, we show that on \Sigma, the set of flat metrics which have the same Laplacian spectrum of Dirichlet boundary condition is compact in the C^\infty topology. This isospectral compactness extends the result of Osgood, Phillips, and Sarnak \cite{OPS3} for type (0,n) surfaces, whose examples include bounded plane domains. Our main ingredients are as following. We first show that the determinant of the Laplacian is a proper function on the moduli space of geodesically bordered hyperbolic metrics on \Sigma. Secondly, we show that the space of such metrics is homeomorphic (in the C^\infty-topology) to the space of flat metrics (on \Sigma) with constantly curved boundary. Because of this, we next reduce the complicated degenerations of flat metrics to the simpler and well-known degenerations of hyperbolic metrics, and we show that determinants of Laplacians of flat metrics on \Sigma, with fixed area and boundary of constant geodesic curvature, give a proper function on the corresponding moduli space. This is interesting because Khuri \cite{Kh} showed that if the boundary length (instead of the area) is fixed, the determinant is not a proper function when \Sigma is of type (g, n), g>0; while Osgood, Phillips, and Sarnak \cite{OPS3} showed the properness when g=0.Comment: Further Revised. A technical error is corrected; the sections devoted to the proof of the insertion lemma and the separation of variables method are completely rewritten. (Sections 4, 5, and 6 in this revised version.) A lot of changes, corrections, and improvements are made throughout the paper. No mathematical change in the main theorems listed in the introductio

Prohibiting isolated singularities in optimal transport

We give natural topological conditions on the support of the target measure under which solutions to the optimal transport problem with cost function satisfying the (weak) Ma, Trudinger, and Wang condition cannot have any isolated singular points.Comment: 10 pages, minor correction in proof of Lemma 3.

A Generalization of Caffarelli's Contraction Theorem via (reverse) Heat Flow

A theorem of L. Caffarelli implies the existence of a map pushing forward a source Gaussian measure to a target measure which is more log-concave than the source one, which contracts Euclidean distance (in fact, Caffarelli showed that the optimal-transport Brenier map $T_{opt}$ is a contraction in this case). We generalize this result to more general source and target measures, using a condition on the third derivative of the potential, using two different proofs. The first uses a map $T$, whose inverse is constructed as a flow along an advection field associated to an appropriate heat-diffusion process. The contraction property is then reduced to showing that log-concavity is preserved along the corresponding diffusion semi-group, by using a maximum principle for parabolic PDE. In particular, Caffarelli's original result immediately follows by using the Ornstein-Uhlenbeck process and the Pr\'ekopa--Leindler Theorem. The second uses the map $T_{opt}$ by generalizing Caffarelli's argument, employing in addition further results of Caffarelli. As applications, we obtain new correlation and isoperimetric inequalities.Comment: 33 pages; corrected typos, shortened Section 6 and some of the standard proofs. To appear in Math. Anna