Yamabe Invariants and Spin Structures

Yamabe Invariants and Spin Stru turesMatthew

Yamabe Invariants and Spinc Structures

The Yamabe invariant of a smooth compact manifold is by definition the supremum of the scalar curvatures of unit-volume Yamabe metrics on the manifold. For an explicit infinite class of 4-manifolds, we show that this invariant is positive but strictly less than that of the 4-sphere. This is done by using spinc Dirac operators to control the lowest eigenvalue of a perturbation of the Yamabe Lapl...

Connected Sums of Closed Riemannian Manifolds and Fourth Order Conformal Invariants

In this note we take some initial steps in the investigation of a fourth order analogue of the Yamabe problem in conformal geometry. The Paneitz constants and the Paneitz invariants considered are believed to be very helpful to understand the topology of the underlined manifolds. We calculate how those quantities change, analogous to how the Yamabe constants and the Yamabe invariants do, under ...

Low-dimensional surgery and the Yamabe invariant

Assume that M is a compact n-dimensional manifold and that N is obtained by surgery along a k-dimensional sphere, k ≤ n − 3. The smooth Yamabe invariants σ(M) and σ(N) satisfy σ(N) ≥ min(σ(M),Λ) for Λ > 0. We derive explicit lower bounds for Λ in dimensions where previous methods failed, namely for (n, k) ∈ {(4, 1), (5, 1), (5, 2), (9, 1), (10, 1)}. With methods from surgery theory and bordism ...

Dependence on the Spin Structure of the Eta and Rokhlin Invariants

We study the dependence of the eta invariant ηD on the spin structure, where D is a twisted Dirac operator on a (4k+ 3)-dimensional spin manifold. The difference between the eta invariants for two spin structures related by a cohomolgy class which is the reduction of a H(M,Z)-class is shown to be a half integer. As an application of the technique of proof the generalized Rokhlin invariant is sh...