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Award ID contains: 2105478

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  1. ; (, Journal of the London Mathematical Society)
    Abstract There are two known classes of gravitational instantons with quadratic volume growth at infinity, known as type and . Gravitational instantons of type were previously classified by Chen–Chen. In this paper, we prove a classification theorem for gravitational instantons. We determine the topology and prove existence of “uniform” coordinates at infinity for both ALG and gravitational instantons. We also prove a result regarding the relationship between ALG gravitational instantons of order and those of order 2. 
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  2. ; (, Communications in Mathematical Physics)
    Abstract We study the prescribed scalar curvature problem in a conformal class on orbifolds with isolated singularities. We prove a compactness theorem in dimension 4, and an existence theorem which holds in dimensions$$n \ge 4$$ n ≥ 4 . This problem is more subtle than the manifold case since the positive mass theorem does not hold for ALE metrics in general. We also determine the$$\textrm{U}(2)$$ U ( 2 ) -invariant Leray–Schauder degree for a family of negative-mass orbifolds found by LeBrun. 
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  3. ; ; (, Duke Mathematical Journal)
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  4. ; ; (, Journal für die reine und angewandte Mathematik (Crelles Journal))
    Abstract We develop a Fredholm theory for the Hodge Laplacian in weighted spaces on ALG ∗ manifolds in dimension four.We then give several applications of this theory.First, we show the existence of harmonic functions with prescribed asymptotics at infinity.A corollary of this is a non-existence result for ALG ∗ manifolds with non-negative Ricci curvature having group Γ = { e } \Gamma=\{e\} at infinity.Next, we prove a Hodge decomposition for the first de Rham cohomology group of an ALG ∗ manifold.A corollary of this is vanishing of the first Betti number for any ALG ∗ manifold with non-negative Ricci curvature.Another application of our analysis is to determine the optimal order of ALG ∗ gravitational instantons. 
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  5. ; ; ; (, Journal of the American Mathematical Society)
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  6. ; ; (, Communications in Analysis and Geometry)
    null (Ed.)
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