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See detailA Polyakov formula for sectors
Aldana Dominguez, Clara Lucia UL; Rowlett, Julie

E-print/Working paper (2015)

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 ... [more ▼]

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 corner. As an application of the method, we obtain an analogues Polyakov formula for a surface with one conical singularity. We compute the zeta-regularized determinant of rectangular domains of fixed area and prove that it is uniquely maximized by the square. [less ▲]

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See detailCompactness of Relatively Isospectral Sets of Surfaces Via Conformal Surgeries
Aldana Dominguez, Clara Lucia UL; Albin, Pierre; Rochon, Frédéric

in Journal of Geometric Analysis (The) (2015), 25(2), 1185-1210

We introduce a notion of relative isospectrality for surfaces with boundary having possibly non-compact ends either conformally compact or asymptotic to cusps. We obtain a compactness result for such ... [more ▼]

We introduce a notion of relative isospectrality for surfaces with boundary having possibly non-compact ends either conformally compact or asymptotic to cusps. We obtain a compactness result for such families via a conformal surgery that allows us to reduce to the case of surfaces hyperbolic near infinity recently studied by Borthwick and Perry, or to the closed case by Osgood, Phillips, and Sarnak if there are only cusps. [less ▲]

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