Period: 2021-2023.

Financed by: ANCS-UEFISCDI contract 117/2021

Budget: 1118032 lei

Grant director: Sergiu Moroianu

Team members:

Rezumat.

Spatiul de moduli de suprafete Riemann este intim legat de geometria hiperbolica. Metrica Riemann pe acest spatiu este definita de produsul scalar L^2 fata de unica metrica hiperbolica dintr-o clasa conforma. Mai mult, forma Kahler pe partea regulata a spatiului de moduli admite diverse potentiale Kahler pe acoperirea universala in sens orbifold, adica pe spatiul Teichmuller. Unul dintre aceste potentiale este determinantul Laplacianului fata de metrica hiperbolica. Un altul este volumul renormalizat al unei metrici hiperbolice pe o 3-varietate de volum infinit care corespunde unui cobordism hiperbolic in dimensiune 2+1. Vom extinde aceste obiecte - si rezultate - la frontiera spatiului Teichmuller. Vom investiga determinantul operatorului Dirac pe suprafete hiperbolice, limita sa adiabatica (colapsarea unei familii de geodezice simple disjuncte) si sa construim in acest fel un potential Kahler care se extinde continuu la unele din componentele de bord, adica la spatiilor Teichmuller ale suprafetelor punctate. Este cunoscut ca determinantul Laplacianului devine singular la toate fetele de bord ale spatiului Teichmuller space corespunzatoare limitei adiabatice. Anticipam ca formula de urma Selberg va juca un rol in analiza noastra, dar intentionam de asemenea sa folosim algebre de operatori pseudodiferentiali adaptate acestei limite geometrice, pentru a controla nucleul Schwartz al puterilor complexe ale operatorului Dirac.

Abstract.

The moduli space of Riemann surfaces is intimately related to hyperbolic geometry. The Riemann metric on this space is defined by the scalar product L^2 with respect to the only hyperbolic metric in a conformal class. Moreover, the Kahler form on the regular part of the moduli space admits various Kahler potentials on the universal covering in the orbifold sense, i.e. on the Teichmuller space. One of these potentials is the determinant of the Laplacian against the hyperbolic metric. Another is the renormalized volume of a hyperbolic metric on a 3-manifold of infinite volume that corresponds to a hyperbolic cobordism in dimension 2+1. We will extend these objects - and results - to the frontier of Teichmuller space. We will investigate the determinant of the Dirac operator on hyperbolic surfaces, its adiabatic limit (the collapse of a family of simple disjoint geodesics) and in this way construct a Kahler potential that extends continuously to some of the board components, i.e. to the Teichmuller spaces of the dotted surfaces. It is known that the determinant of the Laplacian becomes singular at all edges of the Teichmuller space corresponding to the adiabatic limit. We anticipate that the Selberg trace formula will play a role in our analysis, but we also intend to use algebras of pseudo-differential operators adapted to this geometric limit, to control the Schwartz kernel of complex powers of the Dirac operator.

Updated: March 16, 2023