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-dimensional space-periodic solutions of the static vacuum Einstein equations
A bstract An affirmative answer is given to a conjecture of Myers concerning the existence of 5-dimensional regular static vacuum solutions that balance an infinite number of black holes, which have Kasner asymptotics. A variety of examples are constructed, having different combinations of ring S 1 × S 2 and sphere S 3 cross-sectional horizon topologies. Furthermore, we show the existence of 5-dimensional vacuum solitons with Kasner asymptotics. These are regular static space-periodic vacuum spacetimes devoid of black holes. Consequently, we also obtain new examples of complete Riemannian manifolds of nonnegative Ricci curvature in dimension 4, and zero Ricci curvature in dimension 5, having arbitrarily large as well as infinite second Betti number.
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- Award ID(s):
- 1708798
- NSF-PAR ID:
- 10217349
- Date Published:
- Journal Name:
- Journal of High Energy Physics
- Volume:
- 2020
- Issue:
- 12
- ISSN:
- 1029-8479
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract We address the one‐parameter minmax construction for the Allen–Cahn energy that has recently lead to a new proof of the existence of a closed minimal hypersurface in an arbitrary compact Riemannian manifold with (Guaraco's work, relying on works by Hutchinson, Tonegawa, and Wickramasekera when sending the Allen–Cahn parameter to 0). We obtain the following result: if the Ricci curvature of
N is positive then the minmax Allen–Cahn solutions concentrate around amultiplicity‐1 minimal hypersurface (possibly having a singular set of dimension ). This multiplicity result is new for (for it is also implied by the recent work by Chodosh–Mantoulidis). We exploit directly the minmax characterization of the solutions and the analytic simplicity of semilinear (elliptic and parabolic) theory in . While geometric in flavour, our argument takes advantage of the flexibility afforded by the analytic Allen–Cahn framework, where hypersurfaces are replaced by diffused interfaces; more precisely, they are replaced by sufficiently regular functions (fromN to ), whose weighted level sets give rise to diffused interfaces. We capitalise on the fact that (unlike a hypersurface) a function can be deformed both in the domainN (deforming the level sets) and in the target (varying the values). We induce different geometric effects on the diffused interface by using these two types of deformations; this enables us to implement in a continuous way certain operations, whose analogues on a hypersurface would be discontinuous. An immediate corollary of the multiplicity‐1 conclusion is that every compact Riemannian manifold with and positive Ricci curvature admits a two‐sided closed minimal hypersurface, possibly with a singular set of dimension at most . (This geometric corollary also follows from results obtained by different ideas in an Almgren–Pitts minmax framework.) -
A bstract Perturbations of massless fields in the Kerr-Newman black hole background enjoy a (“Love”) SL(2 , ℝ) symmetry in the suitably defined near zone approximation. We present a detailed study of this symmetry and show how the intricate behavior of black hole responses in four and higher dimensions can be understood from the SL(2 , ℝ) representation theory. In particular, static perturbations of four-dimensional black holes belong to highest weight SL(2 , ℝ) representations. It is this highest weight properety that forces the static Love numbers to vanish. We find that the Love symmetry is tightly connected to the enhanced isometries of extremal black holes. This relation is simplest for extremal charged spherically symmetric (Reissner-Nordström) solutions, where the Love symmetry exactly reduces to the isometry of the near horizon AdS 2 throat. For rotating (Kerr-Newman) black holes one is lead to consider an infinite-dimensional SL(2 , ℝ) ⋉ $$ \hat{\textrm{U}}{(1)}_{\mathcal{V}} $$ U ̂ 1 V extension of the Love symmetry. It contains three physically distinct subalgebras: the Love algebra, the Starobinsky near zone algebra, and the near horizon algebra that becomes the Bardeen-Horowitz isometry in the extremal limit. We also discuss other aspects of the Love symmetry, such as the geometric meaning of its generators for spin weighted fields, connection to the no-hair theorems, non-renormalization of Love numbers, its relation to (non-extremal) Kerr/CFT correspondence and prospects of its existence in modified theories of gravity.more » « less
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Abstract In general relativity (without matter), there is typically a one parameter family of static, maximally symmetric black hole solutions labeled by their mass. We show that there are situations with many more black holes. We study asymptotically anti-de Sitter solutions in six and seven dimensions having a conformal boundary which is a product of spheres cross time. We show that the number of families of static, maximally symmetric black holes depends on the ratio, λ , of the radii of the boundary spheres. As λ approaches a critical value, λ c , the number of such families becomes infinite. In each family, we can take the size of the black hole to zero, obtaining an infinite number of static, maximally symmetric non-black hole solutions. We discuss several applications of these results, including Hawking–Page phase transitions and the phase diagram of dual field theories on a product of spheres, new positive energy conjectures, and more.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/JHEP12(2020)002
