Let (M, g) be a compact Riemannian manifold on which a tracefree and divergence-free σ ∈ W 1,p and a positive function τ ∈ W 1,p , p > n, are fixed. In this paper, we study the vacuum Einstein constraint equations using the well known conformal method with data σ and τ . We show that if no solution exists then there is a non-trivial solution of another non-linear limit equation on 1-forms. This last equation can be shown to be without solutions no solution in many situations. As a corollary, we get existence of solutions of the vacuum Einstein constraint equation under explicit assumptions which in particular hold on a dense set of metrics g for the C 0 -topology.
Let (M, g) be a compact Riemannian manifold of dimension n 3. We define the second Yamabe invariant as the infimum of the second eigenvalue of the Yamabe operator over the metrics conformal to g and of volume 1. We study when it is attained. As an application, we find nodal solutions of the Yamabe equation.
Let M be a compact spin manifold with a chosen spin structure. The Atiyah-Singer index theorem implies that for any Riemannian metric on M the dimension of the kernel of the Dirac operator is bounded from below by a topological quantity depending only on M and the spin structure. We show that for generic metrics on M this bound is attained.
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}.
Let (M, g, σ ) be a compact Riemannian spin manifold of dimension ≥ 2. For any metricg conformal to g, we denote byλ the first positive eigenvalue of the Dirac operator on (M,g, σ ). We show thatThis inequality is a spinorial analogue of Aubin's inequality, an important inequality in the solution of the Yamabe problem. The inequality is already known in the case n ≥ 3 and in the case n = 2, ker D = {0}. Our proof also works in the remaining case n = 2, ker D = {0}. With the same method we also prove that any conformal class on a Riemann surface contains a metric with 2λ 2 ≤μ, whereμ denotes the first positive eigenvalue of the Laplace operator.
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