Determinante regularizado do Laplaciano e conjuntos isoespectrais em superfícies

Abstract

On compact surfaces with boundary, with some conditions in a conformal class, we study the problem about to find a metric with constant Gaussian curvature with boundary of constant geodesic curvature for which the regularized determinant of Laplacian has a maximum. From this, we present applications to the problem to obtain a compact Riemannian manifold from its spectrum. Finally, we use the regularized determinant of Laplacian and the invariants of the heat kernel to study the compactness for isospectral sets of simply connected planar domains in a natural $C^{\infty}$ topology.