arXiv subjects
Publications and source records attributed to Cipriana Anghel.
We prove a limited asymptotic expansion up to order $t^4\log t$ of the logarithmic derivative of the Selberg zeta function for the spin Dirac operator on compact surfaces for families of hyperbolic metrics degenerating towards complete hyperbolic metrics with cusps in a pinching process where the lengths $l_j(t)$ of certain disjoint simple closed geodesics converge smoothly to $0$ as $t\to 0$.
We study the behavior of the spectrum of the Dirac operator on degenerating families of compact Riemannian surfaces, when the length $t$ of a simple closed geodesic shrinks to zero, under the hypothesis that the spin structure along the pinched geodesic is non-trivial. The difficulty of the problem stems from the non-compactness of the limit surface, which has finite area and two cusps. The main idea in this investigation is to construct an adapted pseudodifferential calculus, in the spirit of the celebrated b-algebra of Melrose, which includes both the family of Dirac operators on the family of compact surfaces and the Dirac operator on the limit non-compact surface, together with their resolvents. We obtain smoothness of the spectral projectors, and $t^2 \log t$ regularity for the cusp-surgery trace of the relative resolvent in the degeneracy process as $t \searrow 0$.
We describe the small-time heat kernel asymptotics of real powers $Δ^r$, $r \in (0,1)$ of a non-negative self-adjoint generalized Laplacian $Δ$ acting on the sections of a hermitian vector bundle $\mathcal E$ over a closed oriented manifold $M$. First we treat separately the asymptotic on the diagonal of $M \times M$ and in a compact set away from it. Logarithmic terms appear only if $n$ is odd and $r$ is rational with even denominator. We prove the non-triviality of the coefficients appearing in the diagonal asymptotics, and also the non-locality of some of the coefficients. In the special case $r=1/2$, we give a simultaneous formula by proving that the heat kernel of $Δ^{1/2}$ is a polyhomogeneous conormal section in $\mathcal E \boxtimes {\mathcal E}^* $ on the standard blow-up space $M_{heat}$ of the diagonal at time $t=0$ inside $[0,\infty)\times M \times M$.
We give an elementary and self-contained proof of the uniformization theorem for non-compact simply-connected Riemann surfaces.