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
=
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“…In this subsection, we review the Gibbons–Hawking construction of the model metric. See [10, Section 2] for more details. Let be any positive integer.…”
Section: The Model Hyperkähler Structuresmentioning
confidence: 99%
“…Proof Since is invariant under , by [10, Theorem 1.3], there exists a harmonic function on such that is asymptotic to on . Since we have for all …”
Section: Classification Of Alg∗${\rm Alg}^*$ Gravitational Instantonsmentioning
confidence: 99%
“…By Subsection 2.2, the hyperkähler structure is according to Definition 1.3. Finally, the order can be improved to 2 by [10, Theorem 1.10].…”
Section: Classification Of Alg∗${\rm Alg}^*$ Gravitational Instantonsmentioning
confidence: 99%
“…Remark We note that gravitational instantons satisfy the following properties. (1) as .(2)The tangent cone at infinity is .(3) as .(4)There exist coordinates on so that the order satisfies ; see [10, Theorem 1.10].…”
Section: Introductionmentioning
confidence: 99%
“…We first show that the elliptic surface constructed in Proposition 3.1 is a rational elliptic surface. To see this, it follows from [10,Corollary 1.6] that the first betti number 𝑏 1 (𝑋) = 0. Since 𝐷 is a configuration of 2-spheres corresponding to an extended Dynkin diagram of dihedral type, we have 𝑏 1 (𝐷) = 0.…”
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.
“…In this subsection, we review the Gibbons–Hawking construction of the model metric. See [10, Section 2] for more details. Let be any positive integer.…”
Section: The Model Hyperkähler Structuresmentioning
confidence: 99%
“…Proof Since is invariant under , by [10, Theorem 1.3], there exists a harmonic function on such that is asymptotic to on . Since we have for all …”
Section: Classification Of Alg∗${\rm Alg}^*$ Gravitational Instantonsmentioning
confidence: 99%
“…By Subsection 2.2, the hyperkähler structure is according to Definition 1.3. Finally, the order can be improved to 2 by [10, Theorem 1.10].…”
Section: Classification Of Alg∗${\rm Alg}^*$ Gravitational Instantonsmentioning
confidence: 99%
“…Remark We note that gravitational instantons satisfy the following properties. (1) as .(2)The tangent cone at infinity is .(3) as .(4)There exist coordinates on so that the order satisfies ; see [10, Theorem 1.10].…”
Section: Introductionmentioning
confidence: 99%
“…We first show that the elliptic surface constructed in Proposition 3.1 is a rational elliptic surface. To see this, it follows from [10,Corollary 1.6] that the first betti number 𝑏 1 (𝑋) = 0. Since 𝐷 is a configuration of 2-spheres corresponding to an extended Dynkin diagram of dihedral type, we have 𝑏 1 (𝐷) = 0.…”
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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