We prove a surgery formula for the smooth Yamabe invariant σ(M ) of a compact manifold M . Assume that N is obtained from M by surgery of codimension at least 3. We prove the existence of a positive constant Λn, depending only on the dimension n of M , such that σ(N ) ≥ min{σ(M ), Λn}.
We introduce a differential topological invariant for compact differentiable manifolds by counting the small eigenvalues of the Conformal Laplace operator. This invariant vanishes if and only if the manifold has a metric of positive scalar curvature. We show that the invariant does not increase under surgery of codimension at least three and we give lower and upper bounds in terms of the α-genus.
Let (V,g) and (W,h) be compact Riemannian manifolds of dimension at least 3.
We derive a lower bound for the conformal Yamabe constant of the product
manifold (V x W, g+h) in terms of the conformal Yamabe constants of (V,g) and
(W,h).Comment: 12 pages, to appear in Proc. AMS; v3: small changes, very last
preprint version, close to published versio
Abstract. We show that on every compact spin manifold admitting a Riemannian metric of positive scalar curvature Friedrich's eigenvalue estimate for the Dirac operator can be made sharp up to an arbitrarily small given error by choosing the metric suitably.
We associate to a compact spin manifold M a real-valued invariant τ (M ) by taking the supremum over all conformal classes of the infimum inside each conformal class of the first positive Dirac eigenvalue, when the metrics are normalized to unit volume. This invariant is a spinorial analogue of Schoen's σ-constant, also known as the smooth Yamabe invariant.We prove that if N is obtained from M by surgery of codimension at least 2 then τ (N ) ≥ min{τ (M ), Λn}, where Λn is a positive constant depending only on n = dim M . Various topological conclusions can be drawn, in particular that τ is a spin-bordism invariant below Λn. Also, below Λn the values of τ cannot accumulate from above when varied over all manifolds of dimension n.
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