Levi-Civita connections for conformally deformed metrics on tame differential calculi

Given a tame differential calculus over a noncommutative algebra [Formula: see text] and an [Formula: see text]-bilinear metric [Formula: see text] consider the conformal deformation [Formula: see text] [Formula: see text] being an invertible element of [Formula: see text] We prove that there exists a unique connection [Formula: see text] on the bimodule of one-forms of the differential calculus which is torsionless and compatible with [Formula: see text] We derive a concrete formula connecting [Formula: see text] and the Levi-Civita connection for the metric [Formula: see text] As an application, we compute the Ricci and scalar curvatures for a general conformal perturbation of the canonical metric on the noncommutative [Formula: see text]-torus as well as for a natural metric on the quantum Heisenberg manifold. For the latter, the scalar curvature turns out to be a negative constant.

A Polyakov Formula for Sectors

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 origin. We compute explicitly all the contributions to this formula coming from the different parts of the sector. In the process, we obtain an explicit expression for the heat kernel on an infinite area sector using Carslaw–Sommerfeld’s heat kernel. We also compute the zeta-regularized determinant of rectangular domains of unit area and prove that it is uniquely maximized by the square.

Compactness and non-compactness for the Yamabe problem on manifolds with boundary

We study the problem of conformal deformation of Riemannian structure to constant scalar curvature with zero mean curvature on the boundary. We prove compactness for the full set of solutions when the boundary is umbilic and the dimension