# Research

Submitted
Alaee, A., Lesourd, M. & Yau, S.-T. Stable surfaces and free boundary marginally outer trapped surfaces. (Submitted). Publisher's VersionAbstract
We explore various notions of stability for surfaces embedded and immersed in spacetimes and initial data sets. The interest in such surfaces lies in their potential to go beyond the variational techniques which often underlie the study of minimal and CMC surfaces. We prove two versions of Christodoulou-Yau estimate for H-stable surfaces, a Cohn-Vossen type inequality for non-compact stable marginally outer trapped surface (MOTS), and a global theorem on the topology of H-stable surfaces. Moreover, we give a definition of capillary stability for MOTS with boundary. This notion of stability leads to an area inequality and a local splitting theorem for free boundary stable MOTS. Finally, we establish an index estimate and a diameter estimate for free boundary MOTS. These are straightforward generalizations of Chen-Fraser-Pang and Carlotto-Franz results for free boundary minimal surfaces, respectively.
Alaee, A., Lesourd, M. & Yau, S.-T. A localized spacetime Penrose inequality and horizon detection with quasi-local mass. (Submitted). Publisher's VersionAbstract
Our setting is a simply connected bounded domain with a smooth connected boundary, which arises as an initial data set for the general relativistic constraint equations satisfying the dominant energy condition. Assuming the domain to be admissible in a certain precise sense, we prove a localized spacetime Penrose inequality for the Liu-Yau and Wang-Yau quasi-local masses and the area of an outermost marginally outer trapped surface (MOTS). On the basis of this inequality, we obtain sufficient conditions for the existence and non-existence of a MOTS (along with outer trapped surfaces) in the domain, and for the existence of a minimal surface in its Jang graph, expressed in terms of various quasi-local mass quantities and the boundary geometry of the domain.

Alaee, A., Khuri, M. & Kunduri, H. Existence and Uniqueness of Stationary Solutions in 5-Dimensional Minimal Supergravity. (Submitted). Publisher's VersionAbstract
We study the problem of stationary bi-axially symmetric solutions of the 5-dimensional minimal supergravity equations. Essentially all possible solutions with nondegenerate horizons are produced, having the allowed horizon cross-sectional topologies of the sphere $$S^3$$, ring $$S^1\times S^2$$, and lens $$L(p,q)$$, as well as the three different types of asymptotics. The solutions are smooth apart from possible conical singularities at the fixed point sets of the axial symmetry. This analysis also includes the solutions known as solitons in which horizons are not present but are rather replaced by nontrivial topology called bubbles which are sustained by dipole fluxes. Uniqueness results are also presented which show that the solutions are completely determined by their angular momenta, electric and dipole charges, and rod structure which fixes the topology. Consequently we are able to identify the finite number of parameters that govern a solution. In addition, a generalization of these results is given where the spacetime is allowed to have orbifold singularities.
Forthcoming
Alaee, A., Pacheco, A.J.C. & McCormick, S. Stability of a quasi-local positive mass theorem for graphical hypersurfaces of Euclidean space. To appear in Transactions of the American Mathematical Society (Forthcoming). Publisher's VersionAbstract
We present a quasi-local version of the stability of the positive mass theorem. We work with the Brown--York quasi-local mass as it possesses positivity and rigidity properties, and therefore the stability of this rigidity statement can be studied. Specifically, we ask if the Brown--York mass of the boundary of some compact manifold is close to zero, must the manifold be close to a Euclidean domain in some sense?
Here we consider a class of compact n-manifolds with boundary that can be realized as graphs in $\mathbb{R}^{n+1}$, and establish the following. If the Brown--York mass of the boundary of such a compact manifold is small, then the manifold is close to a Euclidean hyperplane with respect to the Federer--Fleming flat distance.
Alaee, A. & Yau, S.-T. Positive mass theorem for initial data sets with corners along a hypersurface. To appear in Communications in Analysis and Geometry (Forthcoming). Publisher's VersionAbstract
We prove positive mass theorem with angular momentum and charges for axially symmetric, simply connected, maximal, complete initial data sets with two ends, one designated asymptotically flat and the other either (Kaluza-Klein) asymptotically flat or asymptotically cylindrical, for 4-dimensional Einstein-Maxwell theory and 5-dimensional minimal supergravity theory which metrics fail to be $$C^1$$ and second fundamental forms and electromagnetic fields fail to be $$C^0$$ across an axially symmetric hypersurface $$\Sigma$$. Furthermore, we remove the completeness and simple connectivity assumptions in this result and prove it for manifold with boundary such that the mean curvature of the boundary is non-positive.
Alaee, A. & Woolgar, E. Formal power series for asymptotically hyperbolic Bach-flat metrics. To appear in Letters in Mathematical Physics (Forthcoming). Publisher's VersionAbstract
It has been observed by Maldacena that one can extract asymptotically anti-de Sitter Einstein 4-metrics from Bach-flat spacetimes by imposing simple principles and data choices. We cast this problem in a conformally compact Riemannian setting. Following an approach pioneered by Fefferman and Graham for the Einstein equation, we find formal power series for conformally compactifiable, asymptotically hyperbolic Bach-flat 4-metrics expanded about conformal infinity. We also consider Bach-flat metrics in the special case of constant scalar curvature and in the special case of constant Q-curvature. This allows us to determine the free data at conformal infinity, and to select those choices that lead to Einstein metrics. Interestingly, the mass is part of that free data, in contrast to the pure Einstein case. We then choose a convenient generalization of the Bach tensor to (bulk) dimensions n>4 and consider the higher dimensional problem. We find that the free data for the expansions split into low-order and high-order pairs. The former pair consists of the metric on the conformal boundary and its first radial derivative, while the latter pair consists of the radial derivatives of order n−2 and n−1. Higher dimensional generalizations of the Bach tensor lack some of the geometrical meaning of the 4-dimensional case. This is reflected in the relative complexity of the higher dimensional problem, but we are able to obtain a relatively complete result if conformal infinity is not scalar flat.
2020
Alaee, A., Khuri, M. & Yau, S.-T. Geometric Inequalities for Quasi-Local Masses. Communications in Mathematical Physics 378, 467–505 (2020). Publisher's VersionAbstract
In this paper lower bounds are obtained for quasi-local masses in terms of charge, angular momentum, and horizon area. In particular we treat three quasi-local masses based on a Hamiltonian approach, namely the Brown-York, Liu-Yau, and Wang-Yau masses. The geometric inequalities are motivated by analogous results for the ADM mass. They may be interpreted as localized versions of these inequalities, and are also closely tied to the conjectured Bekenstein bounds for entropy of macroscopic bodies. In addition, we give a new proof of the positivity property for the Wang-Yau mass which is used to remove the spin condition in higher dimensions. Furthermore, we generalize a recent result of Lu and Miao to obtain a localized version of the Penrose inequality for the static Wang-Yau mass.
2019
Alaee, A., Pacheco, A.C. & Cederbaum, C. Asymptotically flat extensions with charge. Advances in Theoretical and Mathematical Physics 23, 9, 1951-1980 (2019). Publisher's VersionAbstract
The Bartnik mass is a notion of quasi-local mass which is remarkably difficult to compute. Mantoulidis and Schoen [2016] developed a novel technique to construct asymptotically flat extensions of minimal Bartnik data in such a way that the ADM mass of these extensions is well-controlled, and thus, they were able to compute the Bartnik mass for minimal spheres satisfying a stability condition. In this work, we develop extensions and gluing tools, à la Mantoulidis and Schoen, for time-symmetric initial data sets for the Einstein-Maxwell equations that allow us to compute the value of an ad-hoc notion of charged Barnik mass for suitable charged minimal Bartnik data.
Alaee, A., Khuri, M. & Kunduri, H. Existence and uniqueness of near horizon geometries for five dimensional black holes. Journal of Geometry and Physics 144, 370-387 (2019). Publisher's VersionAbstract
We prove existence of all possible bi-axisymmetric near-horizon geometries of 5-dimensional minimal supergravity. These solutions possess the cross-sectional horizon topology $$S^3, S^1\times S^2,$$ or $$L(p,q)$$and come with prescribed electric charge, two angular momenta, and a dipole charge (in the ring case). Moreover, we establish uniqueness of these solutions up to an isometry of the symmetric space $$G_{2,2}/SO(4)$$.
2018
Alaee, A., Khuri, M. & Kunduri, H. Bounding horizon area by angular momentum, charge, and cosmological constant in five dimensional minimal supergravity. Annales Henri Poincaré 20, 2, 481-525 (2018). Publisher's VersionAbstract
We establish a class of area–angular momentum–charge inequalities satisfied by stable marginally outer trapped surfaces in 5-dimensional minimal supergravity which admit a $$U(1)^2$$symmetry. A novel feature is the fact that such surfaces can have the non-trivial topologies $$S^1\times S^2$$ and $$L(p,q)$$. In addition to two angular momenta, they may be characterized by ‘dipole charge’ as well as electric charge. We show that the unique geometries which saturate the inequalities are the horizon geometries corresponding to extreme black hole solutions. Analogous inequalities which also include contributions from a positive cosmological constant are also presented.
2017
Alaee, A., Khuri, M. & Kunduri, H. Mass-angular momentum inequality for black rings. Physical Review Letters 119, 7, 071101 (2017). Publisher's VersionAbstract
The inequality $$m^3\geq \frac{27\pi}{4}|\mathcal{J}_2||\mathcal{J}_1-\mathcal{J}_2|$$ relating total mass and angular momenta is established for (possibly dynamical) spacetimes admitting black holes of ring $$(S^1\times S^2)$$ topology. This inequality is shown to be sharp in the sense that it is saturated precisely for the extreme Pomeransky-Sen’kov black ring solutions. The physical significance of this inequality and its relation to new evidence of black ring instability, as well as the standard picture of gravitational collapse, are discussed.
Alaee, A., Khuri, M. & Kunduri, H. Relating mass to angular momentum and charge in 5-dimensional Minimal Supergravity. Annales Henri Poincaré 18, 5, 1703-1753 (2017). Publisher's VersionAbstract
We prove a mass-angular momentum-charge inequality for a broad class of maximal, asymptotically flat, bi-axisymmetric initial data within the context of five-dimensional minimal supergravity. We further show that the charged Myers–Perry black hole initial data are the unique minimizers. Also, we establish a rigidity statement for the relevant BPS bound, and give a variational characterization of BMPV black holes.
Alaee, A., Khuri, M. & Kunduri, H. Proof of the mass-angular momenta inequality for bi-Axisymmetric black holes with spherical topology. Advances in Theoretical and Mathematical Physics 20, 6, 1397-1441 (2017). Publisher's VersionAbstract
We show that extreme Myers–Perry initial data realize the unique absolute minimum of the total mass in a physically relevant (Brill) class of maximal, asymptotically flat, bi-axisymmetric initial data for the Einstein equations with fixed angular momenta. As a consequence, we prove the relevant mass-angular momentum inequality in this setting for 55-dimensional spacetimes. That is, all data in this class satisfy the inequality $$m^3\geq \frac{27\pi}{32}\left(|\mathcal{J}_1|+|\mathcal{J}_2|\right)^2$$, where mm and $$\mathcal{J}_i, i=1,2,$$ are the total mass and angular momenta of the spacetime. Moreover, equality holds if and only if the initial data set is isometric to the canonical slice of an extreme Myers–Perry black hole.
2016
Alaee, A. & Kunduri, H. Remarks on mass and angular momenta for U(1)^2-invariant initial data. Journal of Mathematical Physics 57, 3, 032502 (2016). Publisher's VersionAbstract
We extend Brill’s positive mass theorem to a large class of asymptotically flat, maximal, $$U(1)^2$$-invariant initial data sets on simply connected four dimensional manifolds $$\Sigma$$. Moreover, we extend the local mass angular momenta inequality result [A. Alaee and H. K. Kunduri, Classical Quantum Gravity 32(16), 165020 (2015)] for $$U(1)^2$$ invariant black holes to the case with nonzero stress energy tensor with positive matter density and energy-momentum current invariant under the above symmetries.
2015
Alaee, A. & Kunduri, H. Proof of the local mass-angular momenta inequality for U(1)^2 invariant black holes. Classical and Quantum Gravity 32, 16, 6845-6856 (2015). Publisher's VersionAbstract
We consider initial data for extreme vacuum asymptotically flat black holes with $$\mathbb{R}\times U(1)^2$$ symmetry. Such geometries are critical points of a mass functional defined for a wide class of asymptotically flat, $$t-\phi^i$$ symmetric maximal initial data for the vacuum Einstein equations. We prove that the above extreme geometries are local minima of mass among nearby initial data (with the same interval structure) with fixed angular momenta. Thus the ADM mass of nearby data $$m\geq f(J_1,J_2)$$ for some function $$f$$ depending on the interval structure. The proof requires that the initial data of the critical points satisfy certain conditions that are satisfied by the extreme Myers–Perry and extreme black ring data.
Alaee, A. & Kunduri, H. Small deformations of extreme 5D Myers-Perry black hole initial data. General Relativity and Gravitation 47, 2, (2015). Publisher's VersionAbstract
We demonstrate the existence of a one-parameter family of initial data for the vacuum Einstein equations in five dimensions representing small deformations of the extreme Myers–Perry black hole. This initial data set has $$t-\phi^i$$ symmetry and preserves the angular momenta and horizon geometry of the extreme solution. Our proof is based upon an earlier result of Dain and Gabach-Clement concerning the existence of $$U(1)^2$$-invariant initial data sets which preserve the geometry of extreme Kerr (at least for short times). In addition, we construct a general class of transverse, traceless symmetric rank 2 tensors in these geometries.
2014
Alaee, A. & Kunduri, H. Mass functional for initial data in 4 + 1 -dimensional spacetime. Physical Review D. 90, 12, 124078 (2014). Publisher's VersionAbstract
We consider a broad class of asymptotically flat, maximal initial data sets satisfying the vacuum constraint equations, admitting two commuting rotational symmetries. We construct a mass functional for $$t-\phi^i$$-symmetric data which evaluates to the Arnowitt-Deser-Misner mass. We then show that $$\mathbb{R}\times U(1)^2$$-invariant solutions of the vacuum Einstein equations are critical points of this functional amongst this class of data. We demonstrate the positivity of this functional for a class of rod structures which include the Myers-Perry initial data. The construction is a natural extension of Dain’s mass functional to $$D=5$$, although several new features arise.
Alaee, A., Kunduri, H. & Pedroza, E.M. Notes on maximal slices of five-dimensional black holes. Classical and Quantum Gravity 31, 5, 055004 (2014). Publisher's VersionAbstract
We consider maximal slices of the Myers–Perry black hole, the doubly spinning black ring, and the Black Saturn solution. These slices are complete, asymptotically flat Riemannian manifolds with inner boundaries corresponding to black hole horizons. Although these spaces are simply connected as a consequence of topological censorship, they have non-trivial topology. In this paper we investigate the question of whether the topology of spatial sections of the horizon uniquely determines the topology of the maximal slices. We show that the horizon determines the homological invariants of the slice under certain conditions. The homological analysis is extended to black holes for which explicit geometries are not yet known. We believe that these results could provide insights in the context of proving existence of deformations of this initial data. For the topological slices of the doubly spinning black ring and the Black Saturn we compute the homotopy groups up to dimension 3 and show that their four-dimensional homotopy group is not trivial.