Showing 1–25 of 25 results for all: aghil alaee

Boundary Behavior of Compact Manifolds With Scalar Curvature Lower Bounds and Static Quasi-Local Mass of Tori

Authors: Aghil Alaee, Pei-Ken Hung, Marcus Khuri

Abstract: A classic result of Shi and Tam states that a 2-sphere of positive Gauss and mean curvature bounding a compact 3-manifold with nonnegative scalar curvature, must have total mean curvature not greater than that of the isometric embedding into Euclidean 3-space, with equality only for domains in this reference manifold. We generalize this result to 2-tori of Guass curvature greater than $-1$, which… ▽ More A classic result of Shi and Tam states that a 2-sphere of positive Gauss and mean curvature bounding a compact 3-manifold with nonnegative scalar curvature, must have total mean curvature not greater than that of the isometric embedding into Euclidean 3-space, with equality only for domains in this reference manifold. We generalize this result to 2-tori of Guass curvature greater than $-1$, which bound a compact 3-manifold having scalar curvature not less than $-6$ and at least one other boundary component satisfying a 'trapping condition'. The conclusion is that the total weighted mean curvature is not greater than that of an isometric embedding into the Kottler manifold, with equality only for domains in this space. Examples are given to show that the assumption of a secondary boundary component cannot be removed. The result gives a positive mass theorem for the static Brown-York mass of tori, in analogy to the Shi-Tam positivity of the standard Brown-York mass, and represents the first such quasi-local mass positivity result for non-spherical surfaces. Furthermore, we prove a Penrose-type inequality in this setting. △ Less

Submitted 4 March, 2024; originally announced March 2024.

Comments: 14 pages

A Quasi-Local Mass

Authors: Aghil Alaee, Marcus Khuri, Shing-Tung Yau

Abstract: We define a new gauge independent quasi-local mass and energy, and show its relation to the Brown-York Hamilton-Jacobi analysis. A quasi-local proof of the positivity, based on spacetime harmonic functions, is given for admissible closed spacelike 2-surfaces which enclose an initial data set satisfying the dominant energy condition. Like the Wang-Yau mass, the new definition relies on isometric em… ▽ More We define a new gauge independent quasi-local mass and energy, and show its relation to the Brown-York Hamilton-Jacobi analysis. A quasi-local proof of the positivity, based on spacetime harmonic functions, is given for admissible closed spacelike 2-surfaces which enclose an initial data set satisfying the dominant energy condition. Like the Wang-Yau mass, the new definition relies on isometric embeddings into Minkowski space, although our notion of admissibility is different from that of Wang-Yau. Rigidity is also established, in that vanishing energy implies that the 2-surface arises from an embedding into Minkowski space, and conversely the mass vanishes for any such surface. Furthermore, we show convergence to the ADM mass at spatial infinity, and provide the equation associated with optimal isometric embedding. △ Less

Submitted 6 September, 2023; originally announced September 2023.

Comments: 21 pages

A Penrose-type inequality with angular momenta for black holes with 3-sphere horizon topology

Authors: Aghil Alaee, Hari K. Kunduri

Abstract: We establish a Penrose-type inequality with angular momenta for four dimensional, biaxially symmetric, maximal, asymptotically flat initial data sets $(M,g,k)$ for the Einstein equations with fixed angular momenta and horizon inner boundary associated to a 3-sphere outermost minimal surface. Moreover, equality holds if and only if the initial data set is isometric to a canonical time slice of a st… ▽ More We establish a Penrose-type inequality with angular momenta for four dimensional, biaxially symmetric, maximal, asymptotically flat initial data sets $(M,g,k)$ for the Einstein equations with fixed angular momenta and horizon inner boundary associated to a 3-sphere outermost minimal surface. Moreover, equality holds if and only if the initial data set is isometric to a canonical time slice of a stationary Myers-Perry black hole. △ Less

Submitted 4 November, 2023; v1 submitted 19 May, 2022; originally announced May 2022.

Comments: 23 pages, v2: minor corrections, agrees with published version

Journal ref: J.Geom.Anal. 33 (2023) 7, 231

The Positive Energy Theorem for Asymptotically Hyperboloidal Initial Data Sets With Toroidal Infinity and Related Rigidity Results

Authors: Aghil Alaee, Pei-Ken Hung, Marcus Khuri

Abstract: We establish the positive energy theorem and a Penrose-type inequality for 3-dimensional asymptotically hyperboloidal initial data sets with toroidal infinity, weakly trapped boundary, and satisfying the dominant energy condition. In the umbilic case, a rigidity statement is proven showing that the total energy vanishes precisely when the initial data manifold is isometric to a portion of the cano… ▽ More We establish the positive energy theorem and a Penrose-type inequality for 3-dimensional asymptotically hyperboloidal initial data sets with toroidal infinity, weakly trapped boundary, and satisfying the dominant energy condition. In the umbilic case, a rigidity statement is proven showing that the total energy vanishes precisely when the initial data manifold is isometric to a portion of the canonical slice of the associated Kottler spacetime. Furthermore, we provide a new proof of the recent rigidity theorems of Eichmair-Galloway-Mendes [10] in dimension 3, with weakened hypotheses in certain cases. These results are obtained through an analysis of the level sets of spacetime harmonic functions. △ Less

Submitted 4 October, 2022; v1 submitted 12 January, 2022; originally announced January 2022.

Comments: 28 pages, final version

SURENA IV: Towards A Cost-effective Full-size Humanoid Robot for Real-world Scenarios

Authors: Aghil Yousefi-Koma, Behnam Maleki, Hessam Maleki, Amin Amani, Mohammad Ali Bazrafshani, Hossein Keshavarz, Ala Iranmanesh, Alireza Yazdanpanah, Hamidreza Alai, Sahel Salehi, Mahyar Ashkvari, Milad Mousavi, Milad Shafiee Ashtiani

Abstract: This paper describes the hardware, software framework, and experimental testing of SURENA IV humanoid robotics platform. SURENA IV has 43 degrees of freedom (DoFs), including seven DoFs for each arm, six DoFs for each hand, and six DoFs for each leg, with a height of 170 cm and a mass of 68 kg and morphological and mass properties similar to an average adult human. SURENA IV aims to realize a cost… ▽ More This paper describes the hardware, software framework, and experimental testing of SURENA IV humanoid robotics platform. SURENA IV has 43 degrees of freedom (DoFs), including seven DoFs for each arm, six DoFs for each hand, and six DoFs for each leg, with a height of 170 cm and a mass of 68 kg and morphological and mass properties similar to an average adult human. SURENA IV aims to realize a cost-effective and anthropomorphic humanoid robot for real-world scenarios. In this way, we demonstrate a locomotion framework based on a novel and inexpensive predictive foot sensor that enables walking with 7cm foot position error because of accumulative error of links and connections' deflection(that has been manufactured by the tools which are available in the Universities). Thanks to this sensor, the robot can walk on unknown obstacles without any force feedback, by online adaptation of foot height and orientation. Moreover, the arm and hand of the robot have been designed to grasp the objects with different stiffness and geometries that enable the robot to do drilling, visual servoing of a moving object, and writing his name on the white-board. △ Less

Submitted 30 August, 2021; originally announced August 2021.

Journal ref: 2020 IEEE-RAS International Conference on Humanoid Robots (HUMANOIDS)

A Minkowski inequality for Horowitz-Myers geon

Authors: Aghil Alaee, Pei-Ken Hung

Abstract: We prove a sharp inequality for toroidal hypersurfaces in three and four dimensional Horowitz-Myers geon. This extend previous results on Minkowski inequality in the static spacetime to toroidal surfaces in asymptotically hyperbolic manifold with flat toroidal conformal infinity. We prove a sharp inequality for toroidal hypersurfaces in three and four dimensional Horowitz-Myers geon. This extend previous results on Minkowski inequality in the static spacetime to toroidal surfaces in asymptotically hyperbolic manifold with flat toroidal conformal infinity. △ Less

Submitted 15 March, 2021; v1 submitted 12 March, 2021; originally announced March 2021.

Comments: 20 pages

Cosmic Cloaking of Rich Extra Dimensions

Authors: Aghil Alaee, Marcus Khuri, Hari Kunduri

Abstract: We present arguments that show why it is difficult to see \emph{rich} extra dimensions in the Universe. More precisely, we study the conditions under which significant size and variation of the extra dimensions in a Kaluza-Klein compactification lead to a black hole in the lower dimensional theory. The idea is based on the hoop (or trapped surface) conjecture concerning black hole existence, as we… ▽ More We present arguments that show why it is difficult to see \emph{rich} extra dimensions in the Universe. More precisely, we study the conditions under which significant size and variation of the extra dimensions in a Kaluza-Klein compactification lead to a black hole in the lower dimensional theory. The idea is based on the hoop (or trapped surface) conjecture concerning black hole existence, as well as on the observation that dimensional reduction on macroscopically large, twisted, or highly dynamical extra dimensions contributes positively to the energy density in the lower dimensional theory and can induce gravitational collapse. We analyze these conditions and find that in an idealized scenario a threshold for the size exists, on the order of $10^{-19}m$, such that extra dimensions of length above this level must lie inside black holes, thus shielding them from the view of outside observers. The threshold is highly dependent on the size of the Universe, leading to the speculation that in the early stages of evolution truly macroscopic and large extra dimensions would have been visible. △ Less

Submitted 25 May, 2021; v1 submitted 5 March, 2021; originally announced March 2021.

Comments: A slightly modified version of this article received - Honorable Mention - in the Gravity Research Foundation 2021 Awards for Essays on Gravitation; 5 pages

Stable Surfaces and Free Boundary Marginally Outer Trapped Surfaces

Authors: Aghil Alaee, Martin Lesourd, Shing-Tung Yau

Abstract: 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 $\mathbf{H}$-stable surfaces, a Cohn-Vossen type inequality for non-compact st… ▽ More 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 $\mathbf{H}$-stable surfaces, a Cohn-Vossen type inequality for non-compact stable marginally outer trapped surface (MOTS), and a global theorem on the topology of $\mathbf{H}$-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. △ Less

Submitted 16 September, 2020; originally announced September 2020.

Comments: 25 pages, 1 figure. Comments welcome!

A localized spacetime Penrose inequality and horizon detection with quasi-local mass

Authors: Aghil Alaee, Martin Lesourd, Shing-Tung Yau

Abstract: For an admissible class of smooth compact initial data sets with boundary, we prove a comparison theorem between the Wang/Liu-Yau quasi-local mass of the boundary and the Hawking mass of strictly minimizing hulls in the Jang graphs of the domain. Using this, we prove a quasi-local Penrose inequality that involves these quasi-local masses of the boundary and the area of an outermost marginally oute… ▽ More For an admissible class of smooth compact initial data sets with boundary, we prove a comparison theorem between the Wang/Liu-Yau quasi-local mass of the boundary and the Hawking mass of strictly minimizing hulls in the Jang graphs of the domain. Using this, we prove a quasi-local Penrose inequality that involves these quasi-local masses of the boundary and the area of an outermost marginally outer trapped surface (MOTS) in the domain or the area of minimizing minimal surface within the Jang graphs. Moreover, we obtain sufficient conditions for the (non)existence of a MOTS within a domain, in the spirit of the folklore hoop conjecture. △ Less

Submitted 24 September, 2021; v1 submitted 3 December, 2019; originally announced December 2019.

Comments: 22 pages. J. Differential Geom., to appear

Stability of a quasi-local positive mass theorem for graphical hypersurfaces of Euclidean space

Authors: Aghil Alaee, Armando J. Cabrera Pacheco, Stephen McCormick

Abstract: 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 domai… ▽ More 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. △ Less

Submitted 21 January, 2020; v1 submitted 27 November, 2019; originally announced November 2019.

Comments: 24 pages, 3 figures. Comments welcome! v2: Some references added

Geometric Inequalities for Quasi-Local Masses

Authors: Aghil Alaee, Marcus Khuri, Shing-Tung Yau

Abstract: 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… ▽ More 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. △ Less

Submitted 15 October, 2019; originally announced October 2019.

Comments: 37 pages

Journal ref: Comm. Math. Phys., 378 (2020), no. 1, 467-505

Positive mass theorem for initial data sets with corners along a hypersurface

Authors: Aghil Alaee, Shing-Tung Yau

Abstract: 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 $C^1$… ▽ More 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 $Σ$. 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. △ Less

Submitted 12 February, 2020; v1 submitted 20 June, 2019; originally announced June 2019.

Comments: 26 pages, some references added, to appear in Communications in Analysis and Geometry

Existence and Uniqueness of Stationary Solutions in 5-Dimensional Minimal Supergravity

Authors: Aghil Alaee, Marcus Khuri, Hari Kunduri

Abstract: 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 fr… ▽ More 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. △ Less

Submitted 5 January, 2023; v1 submitted 28 April, 2019; originally announced April 2019.

Comments: 43 pages, final version

Asymptotically flat extensions with charge

Authors: Aghil Alaee, Armando J. Cabrera Pacheco, Carla Cederbaum

Abstract: 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 t… ▽ More 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. △ Less

Submitted 29 March, 2019; v1 submitted 21 March, 2019; originally announced March 2019.

Comments: We performed minor corrections in the statement of the main theorem related to the bound on the first eigenvalue, see Corollary 5.1 and Theorem 5.1. Moreover, we added remark on page 3 concerning the time-evolution of the initial data sets we construct. Comments are very welcome

MSC Class: 53C21; 83C22

Existence and Uniqueness of Near-Horizon Geometries for 5-Dimensional Black Holes

Authors: Aghil Alaee, Marcus Khuri, Hari Kunduri

Abstract: 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 symmetri… ▽ More 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 $G_{2(2)}/SO(4)$. △ Less

Submitted 19 January, 2021; v1 submitted 19 December, 2018; originally announced December 2018.

Journal ref: J. Geom. Phys., 144 (2019), 370-387

Formal power series for asymptotically hyperbolic Bach-flat metrics

Authors: Aghil Alaee, Eric Woolgar

Abstract: 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, asymptotica… ▽ More 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. △ Less

Submitted 12 October, 2020; v1 submitted 17 September, 2018; originally announced September 2018.

Comments: Section 5 shortened and affiliations updated to match version accepted for publication

Bounding Horizon Area by Angular Momentum, Charge, and Cosmological Constant in 5-Dimensional Minimal Supergravity

Authors: Aghil Alaee, Marcus Khuri, Hari Kunduri

Abstract: 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 nontrivial 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 elec… ▽ More 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 nontrivial 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. △ Less

Submitted 16 January, 2021; v1 submitted 5 December, 2017; originally announced December 2017.

Comments: 39 pages, final version

Journal ref: Ann. Henri Poincare, 20 (2019), no. 2, 481-525

Mass-Angular Momentum Inequality For Black Ring Spacetimes

Authors: Aghil Alaee, Marcus Khuri, Hari Kunduri

Abstract: The inequality $m^3\geq \frac{27π}{4} |\mathcal{J}_{2}||\mathcal{J}_{1}-\mathcal{J}_{2}|$ 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 significa… ▽ More The inequality $m^3\geq \frac{27π}{4} |\mathcal{J}_{2}||\mathcal{J}_{1}-\mathcal{J}_{2}|$ 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. △ Less

Submitted 8 September, 2017; v1 submitted 24 May, 2017; originally announced May 2017.

Comments: 5 pages; V2: minor improvements

Journal ref: Phys. Rev. Lett. 119, 071101 (2017)

Relating Mass to Angular Momentum and Charge in 5-Dimensional Minimal Supergravity

Authors: Aghil Alaee, Marcus Khuri, Hari Kunduri

Abstract: 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. In addition, we establish a rigidity statement for the relevant BPS bound, and give a variational characterizati… ▽ More 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. In addition, we establish a rigidity statement for the relevant BPS bound, and give a variational characterization of BMPV black holes. △ Less

Submitted 20 April, 2017; v1 submitted 23 August, 2016; originally announced August 2016.

Comments: 42 pages; final version

Journal ref: Ann. Henri Poincare, 18 (2017), no. 5, 1703-1753

Proof of the Mass-Angular Momentum Inequality for Bi-Axisymmetric Black Holes With Spherical Topology

Authors: Aghil Alaee, Marcus Khuri, Hari Kunduri

Abstract: 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 5-dimensional spacetimes. That is, all data in this… ▽ More 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 5-dimensional spacetimes. That is, all data in this class satisfy the inequality $m^3\geq \frac{27π}{32}\left(|\mathcal{J}_1|+|\mathcal{J}_2|\right)^2$, where $m$ 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. △ Less

Submitted 24 January, 2017; v1 submitted 23 October, 2015; originally announced October 2015.

Comments: 29 pages; final version

Journal ref: Adv. Theor. Math. Phys., 20 (2016), no. 6, 1397-1441

Remarks on mass and angular momenta for $U(1)^2$-invariant initial data

Authors: Aghil Alaee, Hari K. Kunduri

Abstract: 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 $Σ$. Moreover, we extend the local mass angular momenta inequality result Ref [1] for $U(1)^2$ invariant black holes to the case with nonzero stress energy tensor with positive matter density and energy-momentum current invari… ▽ More 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 $Σ$. Moreover, we extend the local mass angular momenta inequality result Ref [1] 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. △ Less

Submitted 6 December, 2015; v1 submitted 10 August, 2015; originally announced August 2015.

Comments: 17 pages, LaTeX; v2. assumptions on generalized Brill data weakened and the main theorem modified accordingly

Proof of the local mass-angular momenta inequality for $U(1)^2$ invariant black holes

Authors: Aghil Alaee, Hari K. Kunduri

Abstract: 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, `$(t-φ^i)$' symmetric maximal initial data for the vacuum Einstein equations. We prove that the above extreme geometries are local minima of mass amongst nearby initial data (w… ▽ More 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, `$(t-φ^i)$' symmetric maximal initial data for the vacuum Einstein equations. We prove that the above extreme geometries are local minima of mass amongst 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. △ Less

Submitted 12 August, 2015; v1 submitted 11 March, 2015; originally announced March 2015.

Comments: v2. statement of main theorem clarified, various minor improvements and clarifications

Journal ref: Class.Quant.Grav. 32 (2015) 16, 165020

On a mass functional for initial data in 4+1 dimensional spacetime

Authors: Aghil Alaee, Hari K. Kunduri

Abstract: 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 `$t-φ^i$' symmetric data which evaluates to the ADM mass. We then show that $\mathbb{R} \times U(1)^2$-invariant solutions of the vacuum Einstein equations are critical points of this functional amo… ▽ More 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 `$t-φ^i$' symmetric data which evaluates to the ADM 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 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. △ Less

Submitted 3 June, 2015; v1 submitted 3 November, 2014; originally announced November 2014.

Comments: V2: 31 pages-Section 2.2 revised and lengthened (differs from published version); minor typos corrected

Journal ref: Phys. Rev. D 90, 124078 (2014)

Small deformations of extreme five dimensional Myers-Perry black hole initial data

Authors: Aghil Alaee, Hari K Kunduri

Abstract: In this note 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 `$t-φ^i$' 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-Clemen… ▽ More In this note 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 `$t-φ^i$' 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 $U(1)$-invariant initial data sets which preserve the geometry of extreme Kerr (at least for short times) △ Less

Submitted 6 December, 2015; v1 submitted 3 July, 2014; originally announced July 2014.

Comments: LateX, 29 pages; v2: revised and streamlined to agree with published version

Journal ref: Gen.Rel.Grav. 47 (2015) 2, 13

Notes on maximal slices of five-dimensional black holes

Authors: Aghil Alaee, Hari K. Kunduri, Eduardo Martínez-Pedroza

Abstract: 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 note we investigate the… ▽ More 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 note 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 4-dimensional homotopy group is not trivial. △ Less

Submitted 13 February, 2014; v1 submitted 10 September, 2013; originally announced September 2013.

Comments: LateX, 26 pages; v2: added computation of homotopy groups up to dimension 3; minor improvements. to appear in Classical and Quantum Gravity

Journal ref: Class. Quantum Grav. 31 (2014) 055004