We present the first examples of formally asymptotically flat black hole solutions with horizons ... more We present the first examples of formally asymptotically flat black hole solutions with horizons of general lens space topology Lðp; qÞ. These five-dimensional static or stationary spacetimes are regular on and outside the event horizon for any choice of relatively prime integers 1 ≤ q < p; in particular, conical singularities are absent. They are supported by Kaluza-Klein matter fields arising from higher dimensional vacuum solutions through reduction on tori. The technique is sufficiently robust that it leads to the explicit construction of regular solutions, in any dimension, realizing the full range of possible topologies for the horizon as well as the domain of outer communication, that are allowable with multi-axisymmetry. Lastly, as a by-product, we obtain new examples of regular gravitational instantons in higher dimensions.
We present the first examples of formally asymptotically flat black hole solutions with horizons ... more We present the first examples of formally asymptotically flat black hole solutions with horizons of general lens space topology L(p, q). These 5-dimensional static/stationary spacetimes are regular on and outside the event horizon for any choice of relatively prime integers 1 ≤ q < p, in particular conical singularities are absent. They are supported by Kaluza-Klein matter fields arising from higher dimensional vacuum solutions through reduction on tori. The technique is sufficiently robust that it leads to the explicit construction of regular solutions, in any dimension, realising the full range of possible topologies for the horizon as well as the domain of outer communication, that are allowable with multi-axisymmetry. Lastly, as a by product, we obtain new examples of regular gravitational instantons in higher dimensions.
Extending recent work in 5 dimensions, we prove the existence and uniqueness of solutions to the ... more Extending recent work in 5 dimensions, we prove the existence and uniqueness of solutions to the reduced Einstein equations for vacuum black holes in $(n+3)$-dimensional spacetimes admitting the isometry group $\mathbb{R}\times U(1)^{n}$, with Kaluza-Klein asymptotics for $n\geq3$. This is equivalent to establishing existence and uniqueness for singular harmonic maps $\varphi: \mathbb{R}^3\setminus\Gamma\rightarrow SL(n+1,\mathbb{R})/SO(n+1)$ with prescribed blow-up along $\Gamma$, a subset of the $z$-axis in $\mathbb{R}^3$. We also analyze the topology of the domain of outer communication for these spacetimes, by developing an appropriate generalization of the plumbing construction used in the lower dimensional case. Furthermore, we provide a counterexample to a conjecture of Hollands-Ishibashi concerning the topological classification of the domain of outer communication. A refined version of the conjecture is then presented and established in spacetime dimensions less than 8.
The question of whether a closed Riemannian manifold has infinitely many geometrically distinct c... more The question of whether a closed Riemannian manifold has infinitely many geometrically distinct closed geodesics has a long history. Though unsolved in general, it is well understood in the case of surfaces. For surfaces of revolution diffeomorphic to the sphere, a refinement of this problem was introduced by Borzellino, Jordan-Squire, Petrics, and Sullivan. In this article, we quantify their result by counting distinct geodesics of bounded length. In addition, we reframe these results to obtain a couple of characterizations of the round two-sphere.
We present the first examples of formally asymptotically flat black hole solutions with horizons ... more We present the first examples of formally asymptotically flat black hole solutions with horizons of general lens space topology Lðp; qÞ. These five-dimensional static or stationary spacetimes are regular on and outside the event horizon for any choice of relatively prime integers 1 ≤ q < p; in particular, conical singularities are absent. They are supported by Kaluza-Klein matter fields arising from higher dimensional vacuum solutions through reduction on tori. The technique is sufficiently robust that it leads to the explicit construction of regular solutions, in any dimension, realizing the full range of possible topologies for the horizon as well as the domain of outer communication, that are allowable with multi-axisymmetry. Lastly, as a by-product, we obtain new examples of regular gravitational instantons in higher dimensions.
We present the first examples of formally asymptotically flat black hole solutions with horizons ... more We present the first examples of formally asymptotically flat black hole solutions with horizons of general lens space topology L(p, q). These 5-dimensional static/stationary spacetimes are regular on and outside the event horizon for any choice of relatively prime integers 1 ≤ q < p, in particular conical singularities are absent. They are supported by Kaluza-Klein matter fields arising from higher dimensional vacuum solutions through reduction on tori. The technique is sufficiently robust that it leads to the explicit construction of regular solutions, in any dimension, realising the full range of possible topologies for the horizon as well as the domain of outer communication, that are allowable with multi-axisymmetry. Lastly, as a by product, we obtain new examples of regular gravitational instantons in higher dimensions.
Extending recent work in 5 dimensions, we prove the existence and uniqueness of solutions to the ... more Extending recent work in 5 dimensions, we prove the existence and uniqueness of solutions to the reduced Einstein equations for vacuum black holes in $(n+3)$-dimensional spacetimes admitting the isometry group $\mathbb{R}\times U(1)^{n}$, with Kaluza-Klein asymptotics for $n\geq3$. This is equivalent to establishing existence and uniqueness for singular harmonic maps $\varphi: \mathbb{R}^3\setminus\Gamma\rightarrow SL(n+1,\mathbb{R})/SO(n+1)$ with prescribed blow-up along $\Gamma$, a subset of the $z$-axis in $\mathbb{R}^3$. We also analyze the topology of the domain of outer communication for these spacetimes, by developing an appropriate generalization of the plumbing construction used in the lower dimensional case. Furthermore, we provide a counterexample to a conjecture of Hollands-Ishibashi concerning the topological classification of the domain of outer communication. A refined version of the conjecture is then presented and established in spacetime dimensions less than 8.
The question of whether a closed Riemannian manifold has infinitely many geometrically distinct c... more The question of whether a closed Riemannian manifold has infinitely many geometrically distinct closed geodesics has a long history. Though unsolved in general, it is well understood in the case of surfaces. For surfaces of revolution diffeomorphic to the sphere, a refinement of this problem was introduced by Borzellino, Jordan-Squire, Petrics, and Sullivan. In this article, we quantify their result by counting distinct geodesics of bounded length. In addition, we reframe these results to obtain a couple of characterizations of the round two-sphere.
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Papers by Jordan Rainone