We prove uniqueness and existence theorems for four-dimensional asymptotically flat, Ricci-flat, gravitational instantons with a torus symmetry. In particular, we prove that such instantons are uniquely characterised by their rod structure, which is
data that encodes the fixed point sets of the torus action. Furthermore, we establish that for every admissible rod structure there exists an instanton that is smooth up to possible conical singularities at the axes of symmetry. The proofs involve adapting the methods that are used to establish black hole uniqueness theorems, to a harmonic map formulation of Ricci-flat metrics with torus symmetry, where the target space is directly related to the metric (rather than auxiliary potentials). We also give an elementary proof of the nonexistence of asymptotically flat toric half-flat instantons. Finally, we derive a general set of identities that relate asymptotic invariants such as the mass to the rod structure.
We classify finite energy harmonic 2-forms on the asymptotically flat gravitational instanton constructed by Chen and Teo. We prove that every $U(1)$-bundle admits a unique anti-self-dual Yang-Mills instanton (up to gauge equivalence) which we descri
be explicitly in coordinates. As an application, we compute the classical partition function for Maxwell theory with theta term.
Within the framework of the complexity equals action and complexity equals volume conjectures, we study the properties of holographic complexity for rotating black holes. We focus on a class of odd-dimensional equal-spinning black holes for which con
siderable simplification occurs. We study the complexity of formation, uncovering a direct connection between complexity of formation and thermodynamic volume for large black holes. We consider also the growth-rate of complexity, finding that at late-times the rate of growth approaches a constant, but that Lloyds bound is generically violated.
We study the holographic complexity conjectures for rotating black holes, uncovering a relationship between the complexity of formation and the thermodynamic volume of the black hole. We suggest that it is the thermodynamic volume and not the entropy
that controls the complexity of formation of large black holes in both the Complexity Equals Action and Complexity Equals Volume proposals in general. Our proposal reduces to known results involving the entropy in settings where the thermodynamic volume and entropy are not independent, but has broader scope. Assuming a conjectured inequality is obeyed by the thermodynamic volume, we establish that the complexity of formation is bounded from below by the entropy for large black holes.
We consider a family of globally stationary (horizonless), asymptotically flat solutions of five-dimensional supergravity. We prove that massless linear scalar waves in such soliton spacetimes cannot have a uniform decay rate faster than inverse loga
rithmically in time. This slow decay can be attributed to the stable trapping of null geodesics. Our proof uses the construction of quasimodes which are time periodic approximate solutions to the wave equation. The proof is based on previous work to prove an analogous result in Kerr-AdS black holes cite{holzegel:2013kna}. We remark that this slow decay is suggestive of an instability at the nonlinear level.
Minimal $D=5$ supergravity admits asymptotically globally AdS$_5$ gravitational solitons (strictly stationary, geodesically complete spacetimes with positive mass). We show that, like asymptotically flat gravitational solitons, these solutions satisf
y mass and mass variation formulas analogous to those satisfied by AdS black holes. A thermodynamic volume associated to the non-trivial topology of the spacetime plays an important role in this construction. We then consider these solitons within the holographic ``complexity equals action and ``complexity equals volume conjectures as simple examples of spacetimes with nontrivial rotation and topology. We find distinct behaviours for the volume and action, with the counterterm for null boundaries playing a significant role in the latter case. For large solitons we find that both proposals yield a complexity of formation proportional to a power of the thermodynamic volume, $V^{3/4}$. In fact, up to numerical prefactors, the result coincides with the analogous one for large black holes.
We derive new identities for the thermodynamic variables of five-dimensional, asymptotically flat, stationary and biaxisymmetric vacuum black holes. These identities depend on the topology of the solution and include contributions arising from certai
n topological charges. The proof employs the harmonic map formulation of the vacuum Einstein equations for solutions with these symmetries.
We prove that any asymptotically flat static spacetime in higher dimensional Einstein-Maxwell theory must have no magnetic field. This implies that there are no static soliton spacetimes and completes the classification of static non-extremal black h
oles in this theory. In particular, these results establish that there are no asymptotically flat static spacetimes with non-trivial topology, with or without a black hole, in Einstein-Maxwell theory.
We examine potential deformations of inner black hole and cosmological horizons in Reissner-Nordstrom de-Sitter spacetimes. While the rigidity of the outer black hole horizon is guaranteed by theorem, that theorem applies to neither the inner black h
ole nor past cosmological horizon. Further for pure deSitter spacetime it is clear that the cosmological horizon can be deformed (by translation). For specific parameter choices, it is shown that both inner black hole and cosmological horizons can be infinitesimally deformed. However these do not extend to finite deformations. The corresponding results for general spherically symmetric spacetimes are considered.
In five-dimensional minimal supergravity, there are spherical black holes with nontrivial topology outside the horizon which have the same conserved charges at infinity as the BMPV solution. We show that some of these black holes have greater entropy
than the BMPV solution. These spacetimes are all asymptotically flat, stationary, and supersymmetric. We also show that there is a limit in which the black hole shrinks to zero size and the solution becomes a nonsingular bubbling geometry. Thus, these solutions provide explicit analytic examples of placing black holes inside solitons.
