Let (M, g) be an odd-dimensional incomplete compact Riemannian singular space with a simple edge singularity. We study the analytic torsion on M , and in particular consider how it depends on the metric g. If g is an admissible edge metric, we prove that the torsion zeta function is holomorphic near s = 0, hence the torsion is well-defined, but possibly depends on g. In general dimensions, we prove that the torsion depends only on the asymptotic structure of g near the singular stratum of M ; when the dimension of the edge is odd, we prove that the analytic torsion is independent of the choice of even admissible edge metrics. The main tool is the construction, via the methodology of geometric microlocal analysis, of the heat kernel for the Friedrichs extension of the Hodge Laplacian in all degrees. In this way we obtain detailed asymptotics of this heat kernel and its trace.
Abstract. This is a continuation of the first author's development [17] of the theory of elliptic differential operators with edge degeneracies. That first paper treated basic mapping theory, focusing on semiFredholm properties on weighted Sobolev and Hölder spaces and regularity in the form of asymptotic expansions of solutions. The present paper builds on this through the formulation of boundary conditions and the construction of parametrices for the associated boundary problems. As in [17], the emphasis is on the geometric microlocal structure of the Schwartz kernels of parametrices and generalized inverses.
Abstract. Let (M, g) be a compact oriented Riemannian manifold with an incomplete edge singularity. This article shows that it is possible to evolve g by the Yamabe flow within a class of singular edge metrics. As the main analytic step we establish parabolic Schaudertype estimates for the heat operator on certain Hölder spaces adapted to the singular edge geometry. We apply these estimates to obtain local existence for a variety of quasilinear equations, including the Yamabe flow. This provides a setup for a subsequent discussion of the Yamabe problem using flow techniques in the singular setting.
Torsion invariants for manifolds which are not simply connected were
introduced by K. Reidemeister and generalized to higher dimensions by W. Franz.
The Reidemeister torsion, was the first invariant of manifolds which was not a
homotopy invariant. The analytic counterpart of the combinatorial Reidemeister
torsion was introduced by D. B. Ray and I. M. Singer in form of a weighted
product of zeta-regularized determinants of Laplace operators on differential
forms. The celebrated Cheeger-Mueller Theorem, established independently by J.
Cheeger and W. Mueller, proved equality between the analytic Ray-Singer torsion
and the combinatorial Reidemeister torsion for any smooth closed manifold with
an orthogonal representation of its fundamental group. Motivated by the vision
of a Cheeger-Mueller type result on manifolds with conical singularities, we
compute the analytic torsion of a bounded generalized cone by generalizing the
computational methods of M. Spreafico and using the symmetry of the de Rham
complex, as established by M. Lesch.Comment: Some references were corrected. 41 pages, 2 figure
We discuss the refined analytic torsion, introduced by M Braverman and T Kappeler as a canonical refinement of analytic torsion on closed manifolds. Unfortunately there seems to be no canonical way to extend their construction to compact manifolds with boundary. We propose a different refinement of analytic torsion, similar to Braverman and Kappeler, which does apply to compact manifolds with and without boundary. In a subsequent publication we prove a surgery formula for our construction.
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