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Journal articles on the topic 'Kähler- Einstein equation'

Author: Grafiati

Published: 6 September 2023

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1

Zhang, Xi, and Xiangwen Zhang. "Generalized Kähler–Einstein Metrics and Energy Functionals." Canadian Journal of Mathematics 66, no. 6 (December 1, 2014): 1413–35. http://dx.doi.org/10.4153/cjm-2013-034-3.

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Abstract.In this paper, we consider a generalized Kähler–Einstein equation on a Kähler manifold M. Using the twisted 𝒦–energy introduced by Song and Tian, we show that the existence of generalized Kähler–Einstein metrics with semi–positive twisting (1, 1)–form θ is also closely related to the properness of the twisted 𝒦-energy functional. Under the condition that the twisting form θ is strictly positive at a point or M admits no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of generalized Kähler–Einstein metric implies a Moser–Trudinger type inequality.

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2

PAN, LISHUANG, AN WANG, and LIYOU ZHANG. "ON THE KÄHLER–EINSTEIN METRIC OF BERGMAN–HARTOGS DOMAINS." Nagoya Mathematical Journal 221, no. 1 (March 2016): 184–206. http://dx.doi.org/10.1017/nmj.2016.4.

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We study the complete Kähler–Einstein metric of certain Hartogs domains ${\rm\Omega}_{s}$ over bounded homogeneous domains in $\mathbb{C}^{n}$. The generating function of the Kähler–Einstein metric satisfies a complex Monge–Ampère equation with Dirichlet boundary condition. We reduce the Monge–Ampère equation to an ordinary differential equation and solve it explicitly when we take the parameter $s$ for some critical value. This generalizes previous results when the base is either the Euclidean unit ball or a bounded symmetric domain.

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3

Zhang, Xi. "Hermitian Yang–Mills–Higgs Metrics on Complete Kähler Manifolds." Canadian Journal of Mathematics 57, no. 4 (August 1, 2005): 871–96. http://dx.doi.org/10.4153/cjm-2005-034-3.

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AbstractIn this paper, first, we will investigate the Dirichlet problem for one type of vortex equation, which generalizes the well-known Hermitian Einstein equation. Secondly, we will give existence results for solutions of these vortex equations over various complete noncompact Kähler manifolds.

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4

Visinescu, Mihai. "Sasaki–Ricci flow equation on five-dimensional Sasaki–Einstein space Yp,q." Modern Physics Letters A 35, no. 14 (March 20, 2020): 2050114. http://dx.doi.org/10.1142/s021773232050114x.

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We analyze the transverse Kähler–Ricci flow equation on Sasaki-Einstein space [Formula: see text]. Explicit solutions are produced representing new five-dimensional Sasaki structures. Solutions which do not modify the transverse metric preserve the Sasaki–Einstein feature of the contact structure. If the transverse metric is altered, the deformed metrics remain Sasaki, but not Einstein.

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5

Alekseevsky, Dmitri V., and Fabio Podestà. "Homogeneous almost-Kähler manifolds and the Chern–Einstein equation." Mathematische Zeitschrift 296, no. 1-2 (December 4, 2019): 831–46. http://dx.doi.org/10.1007/s00209-019-02446-y.

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6

ARVANITOYEORGOS, ANDREAS, IOANNIS CHRYSIKOS, and YUSUKE SAKANE. "HOMOGENEOUS EINSTEIN METRICS ON GENERALIZED FLAG MANIFOLDS WITH FIVE ISOTROPY SUMMANDS." International Journal of Mathematics 24, no. 10 (September 2013): 1350077. http://dx.doi.org/10.1142/s0129167x13500778.

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We construct the homogeneous Einstein equation for generalized flag manifolds G/K of a compact simple Lie group G whose isotropy representation decomposes into five inequivalent irreducible Ad (K)-submodules. To this end, we apply a new technique which is based on a fibration of a flag manifold over another such space and the theory of Riemannian submersions. We classify all generalized flag manifolds with five isotropy summands, and we use Gröbner bases to study the corresponding polynomial systems for the Einstein equation. For the generalized flag manifolds E6/(SU(4) × SU(2) × U(1) × U(1)) and E7/(U(1) × U(6)) we find explicitly all invariant Einstein metrics up to isometry. For the generalized flag manifolds SO (2ℓ + 1)/( U (1) × U (p) × SO (2(ℓ - p - 1) + 1)) and SO (2ℓ)/( U (1) × U (p) × SO (2(ℓ - p - 1))) we prove existence of at least two non-Kähler–Einstein metrics. For small values of ℓ and p we give the precise number of invariant Einstein metrics.

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7

Li, Chi. "On the limit behavior of metrics in the continuity method for the Kähler–Einstein problem on a toric Fano manifold." Compositio Mathematica 148, no. 6 (October 12, 2012): 1985–2003. http://dx.doi.org/10.1112/s0010437x12000334.

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AbstractThis work is a continuation of the author’s previous paper [Greatest lower bounds on the Ricci curvature of toric Fano manifolds, Adv. Math. 226 (2011), 4921–4932]. On any toric Fano manifold, we discuss the behavior of the limit metric of a sequence of metrics which are solutions to a continuity family of complex Monge–Ampère equations in the Kähler–Einstein problem. We show that the limit metric satisfies a singular complex Monge–Ampère equation. This gives a conic-type singularity for the limit metric. Information on conic-type singularities can be read off from the geometry of the moment polytope.

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8

SAKAGUCHI, MAKOTO. "FOUR-DIMENSIONAL N=2 SUPERSTRING BACKGROUNDS AND THE REAL HEAVENS." International Journal of Modern Physics A 11, no. 07 (March 20, 1996): 1279–97. http://dx.doi.org/10.1142/s0217751x96000572.

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We study N=2 superstring backgrounds which are four-dimensional, non-Kählerian and contain nontrivial dilaton and torsion fields. In particular we consider the case where the backgrounds possess at least one U(1) isometry and are characterized by the continual Toda equation and the Laplace equation. We obtain a string background associated with a nontrivial solution of the continual Toda equation, which is mapped, under the T duality transformation, to the Taub-NUT instanton background. It is shown that the integrable property of the non-Kählerian spaces has a direct origin in the real heavens: real, self-dual, Euclidean, Einstein spaces. The Laplace equation and the continual Toda equation imposed on quasi-Kähler geometry for consistent string propagation are related to the self-duality conditions of the real heavens with “translational” and “rotational” Killing symmetry respectively.

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9

Biswas, Indranil. "Yang–Mills connections on compact complex tori." Journal of Topology and Analysis 07, no. 02 (March 26, 2015): 293–307. http://dx.doi.org/10.1142/s1793525315500107.

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Let G be a connected reductive complex affine algebraic group and K ⊂ G a maximal compact subgroup. Let M be a compact complex torus equipped with a flat Kähler structure and (EG, θ) a polystable Higgs G-bundle on M. Take any C∞ reduction of structure group EK ⊂ EG to the subgroup K that solves the Yang–Mills equation for (EG, θ). We prove that the principal G-bundle EG is polystable and the above reduction EK solves the Einstein–Hermitian equation for EG. We also prove that for a semistable (respectively, polystable) Higgs G-bundle (EG, θ) on a compact connected Calabi–Yau manifold, the underlying principal G-bundle EG is semistable (respectively, polystable).

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10

ARVANITOYEORGOS, ANDREAS. "GEOMETRY OF FLAG MANIFOLDS." International Journal of Geometric Methods in Modern Physics 03, no. 05n06 (September 2006): 957–74. http://dx.doi.org/10.1142/s0219887806001399.

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A flag manifold is a homogeneous space M = G/K, where G is a compact semisimple Lie group, and K the centralizer of a torus in G. Equivalently, M can be identified with the adjoint orbit Ad (G)w of an element w in the Lie algebra of G. We present several aspects of flag manifolds, such as their classification in terms of painted Dynkin diagrams, T-roots and G-invariant metrics, and Kähler metrics. We give a Lie-theoretic expression of the Ricci tensor in M, hence reducing the Einstein equation on flag manifolds into an algebraic system of equations, which can be solved in several cases. A flag manifold is also a complex manifold, and this dual representation as a real and a complex manifold is related to a similar property of an infinite-dimensional manifold, the loop space, which in fact can be viewed as a "universal" flag manifold.

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11

Mazzeo, Rafe, and Yanir A. Rubinstein. "The Ricci continuity method for the complex Monge–Ampère equation, with applications to Kähler–Einstein edge metrics." Comptes Rendus Mathematique 350, no. 13-14 (July 2012): 693–97. http://dx.doi.org/10.1016/j.crma.2012.07.001.

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12

DEMAILLY, JEAN-PIERRE, and NEFTON PALI. "DEGENERATE COMPLEX MONGE–AMPÈRE EQUATIONS OVER COMPACT KÄHLER MANIFOLDS." International Journal of Mathematics 21, no. 03 (March 2010): 357–405. http://dx.doi.org/10.1142/s0129167x10006070.

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We prove the existence and uniqueness of the solutions of some very general type of degenerate complex Monge–Ampère equations, and investigate their regularity. These types of equations are precisely what is needed in order to construct Kähler–Einstein metrics over irreducible singular Kähler spaces with ample or trivial canonical sheaf and singular Kähler–Einstein metrics over varieties of general type.

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13

ITOH, MITSUHIRO. "ALMOST KÄHLER 4-MANIFOLDS, L2-SCALAR CURVATURE FUNCTIONAL AND SEIBERG–WITTEN EQUATIONS." International Journal of Mathematics 15, no. 06 (August 2004): 573–80. http://dx.doi.org/10.1142/s0129167x04002478.

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We show in this paper by applying the Seiberg–Witten theory developed by Taubes and LeBrun that a compact almost Kähler–Einstein 4-manifold of negative scalar curvature s is Kähler–Einstein if and only if the L2-norm satisfies ∫Ms2dv=32π2(2χ+3τ)(M). The Einstein condition can be weakened by the topological condition (2χ+3τ)(M)>0.

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14

YANG, BO. "A CHARACTERIZATION OF NONCOMPACT KOISO-TYPE SOLITONS." International Journal of Mathematics 23, no. 05 (May 2012): 1250054. http://dx.doi.org/10.1142/s0129167x12500541.

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We construct complete gradient Kähler–Ricci solitons of various types on the total spaces of certain holomorphic line bundles over compact Kähler–Einstein manifolds with positive scalar curvature. Those are noncompact analogues of the compact examples found by Koiso [On rotationally symmetric Hamilton's equations for Kähler–Einstein metrics, in Recent Topics in Differential and Analytic Geometry, Advanced Studies in Pure Mathematics, Vol. 18-I (Academic Press, Boston, MA, 1990), pp. 327–337]. Our examples can be viewed a generalization of previous examples by Cao [Existense of gradient Kähler–Ricci solitons, in Elliptic and Parabolic Methods in Geometry (Minneapolis, MN, 1994), pp. 1–16], Chave and Valent [On a class of compact and non-compact quasi-Einstein metrics and their renormalizability properties, Nuclear Phys. B 478 (1996) 758–778], Pedersen, Tønnesen-Friedman, and Valent [Quasi-Einstein Kähler metrics, Lett. Math. Phys. 50(3) (1999) 229–241], and Feldman, Ilmanen and Knopf [Rotationally symmetric shrinking and expanding gradient Kähler–Ricci solitons, J. Differential Geom. 65 (2003) 169–209]. We also prove a uniformization result on complete steady gradient Kähler–Ricci solitons with non-negative Ricci curvature under additional assumptions.

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15

Fu, Xin, Bin Guo, and Jian Song. "Geometric estimates for complex Monge–Ampère equations." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 765 (August 1, 2020): 69–99. http://dx.doi.org/10.1515/crelle-2019-0020.

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AbstractWe prove uniform gradient and diameter estimates for a family of geometric complex Monge–Ampère equations. Such estimates can be applied to study geometric regularity of singular solutions of complex Monge–Ampère equations. We also prove a uniform diameter estimate for collapsing families of twisted Kähler–Einstein metrics on Kähler manifolds of nonnegative Kodaira dimensions.

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16

Naderi, F., A. Rezaei-Aghdam, and F. Darabi. "Gravity and induced matter on nearly Kähler manifolds." International Journal of Modern Physics A 30, no. 03 (January 30, 2015): 1550015. http://dx.doi.org/10.1142/s0217751x15500153.

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We show that the conservation of energy–momentum tensor of a gravitational model with Einstein–Hilbert like action on a nearly Kähler manifold with the scalar curvature of a curvature-like tensor, is consistent with the nearly Kähler properties. In this way, the nearly Kähler structure is automatically manifested in the action as a induced matter field. As an example of nearly Kähler manifold, we consider the group manifold of R×II ×S3×S3 on which we identify a nearly Kähler structure and solve the Einstein equations to interpret the model. It is shown that the nearly Kähler structure in this example is capable of alleviating the well known fine tuning problem of the cosmological constant. Moreover, this structure may be considered as a potential candidate for dark energy.

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17

LEE, HYUK-JAE. "TOPOLOGICAL FIELD THEORY OF VORTICES OVER CLOSED KÄHLER MANIFOLD." International Journal of Modern Physics A 10, no. 30 (December 10, 1995): 4371–85. http://dx.doi.org/10.1142/s0217751x95002023.

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We show that the vortex equations of the n-dimensional closed Kähler manifolds can be derived from Einstein-Hermitian equations of the (n+1)-dimensional closed Kähler manifolds by setting invariance under translation in the (n+1)th component direction. We construct the topological theory about the vortex pair model through the dimensional reduction of the topological BRST structure.

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18

LeBrun, Claude. "The Einstein–Maxwell Equations and Conformally Kähler Geometry." Communications in Mathematical Physics 344, no. 2 (January 29, 2016): 621–53. http://dx.doi.org/10.1007/s00220-015-2568-5.

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19

LeBrun, Claude. "The Einstein–Maxwell equations, Kähler metrics, and Hermitian geometry." Journal of Geometry and Physics 91 (May 2015): 163–71. http://dx.doi.org/10.1016/j.geomphys.2015.01.009.

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20

Visinescu, Mihai. "Transverse Kähler–Ricci Solitons of Five-Dimensional Sasaki–Einstein Spaces Yp,q and T1,1." Symmetry 12, no. 3 (February 25, 2020): 330. http://dx.doi.org/10.3390/sym12030330.

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We investigate the deformations of the Sasaki–Einstein structures of the five-dimensional spaces T 1 , 1 and Y p , q by exploiting the transverse structure of the Sasaki manifolds. We consider local deformations of the Sasaki structures preserving the Reeb vector fields but modify the contact forms. In this class of deformations, we analyze the transverse Kähler–Ricci flow equations. We produce some particular explicit solutions representing families of new Sasakian structures.

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21

Araneda, Bernardo. "On self-dual Yang–Mills fields on special complex surfaces." Journal of Mathematical Physics 63, no. 5 (May 1, 2022): 052501. http://dx.doi.org/10.1063/5.0087276.

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We derive a generalization of the flat space equations of Yang and Newman for self-dual Yang–Mills fields to (locally) conformally Kähler Riemannian four-manifolds. The results also apply to Einstein metrics (whose full curvature is not necessarily self-dual). We analyze the possibility of hidden symmetries in the form of Bäcklund transformations, and we find a continuous group of hidden symmetries only for the case in which the geometry is conformally half-flat. No isometries are assumed.

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22

Talebaoui, Wathek. "Analytic solutions to the Einstein-fluid-Kähler field equations in conformal-time." Il Nuovo Cimento B 108, no. 8 (August 1993): 893–904. http://dx.doi.org/10.1007/bf02828736.

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23

Talebaoui, Wathek. "Cosmological solutions of the Einstein-KÄhler field equations in Robertson-Walker backgrounds." General Relativity and Gravitation 26, no. 7 (July 1994): 663–79. http://dx.doi.org/10.1007/bf02116956.

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24

Berman, Robert J. "A thermodynamical formalism for Monge–Ampère equations, Moser–Trudinger inequalities and Kähler–Einstein metrics." Advances in Mathematics 248 (November 2013): 1254–97. http://dx.doi.org/10.1016/j.aim.2013.08.024.

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25

Angella, Daniele, and Francesco Pediconi. "On cohomogeneity one Hermitian non-Kähler metrics." Proceedings of the Royal Society of Edinburgh: Section A Mathematics, February 17, 2022, 1–43. http://dx.doi.org/10.1017/prm.2022.5.

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We investigate the geometry of Hermitian manifolds endowed with a compact Lie group action by holomorphic isometries with principal orbits of codimension one. In particular, we focus on a special class of these manifolds constructed by following Bérard-Bergery which includes, among the others, the holomorphic line bundles on $\mathbb {C}\mathbb {P}^{m-1}$ , the linear Hopf manifolds and the Hirzebruch surfaces. We characterize their invariant special Hermitian metrics, such as balanced, Kähler-like, pluriclosed, locally conformally Kähler, Vaisman and Gauduchon. Furthermore, we construct new examples of cohomogeneity one Hermitian metrics solving the second-Chern–Einstein equation and the constant Chern-scalar curvature equation.

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26

Cortés, Vicente, Calin Lazaroiu, and C. S. Shahbazi. "Spinors of real type as polyforms and the generalized Killing equation." Mathematische Zeitschrift, March 19, 2021. http://dx.doi.org/10.1007/s00209-021-02726-6.

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AbstractWe develop a new framework for the study of generalized Killing spinors, where every generalized Killing spinor equation, possibly with constraints, can be formulated equivalently as a system of partial differential equations for a polyform satisfying algebraic relations in the Kähler–Atiyah bundle constructed by quantizing the exterior algebra bundle of the underlying manifold. At the core of this framework lies the characterization, which we develop in detail, of the image of the spinor squaring map of an irreducible Clifford module $$\Sigma $$ Σ of real type as a real algebraic variety in the Kähler–Atiyah algebra, which gives necessary and sufficient conditions for a polyform to be the square of a real spinor. We apply these results to Lorentzian four-manifolds, obtaining a new description of a real spinor on such a manifold through a certain distribution of parabolic 2-planes in its cotangent bundle. We use this result to give global characterizations of real Killing spinors on Lorentzian four-manifolds and of four-dimensional supersymmetric configurations of heterotic supergravity. In particular, we find new families of Einstein and non-Einstein four-dimensional Lorentzian metrics admitting real Killing spinors, some of which are deformations of the metric of $$\text {AdS}_4$$ AdS 4 space-time.

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27

Dervan, Ruadhaí, and Lars Martin Sektnan. "Uniqueness of optimal symplectic connections." Forum of Mathematics, Sigma 9 (2021). http://dx.doi.org/10.1017/fms.2021.15.

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Abstract Consider a holomorphic submersion between compact Kähler manifolds, such that each fibre admits a constantscalar curvature Kähler metric. When the fibres admit continuous automorphisms, a choice of fibrewise constant scalarcurvature Kähler metric is not unique. An optimal symplectic connection is a choice of fibrewise constant scalar curvature Kähler metric satisfying a geometric partial differential equation. The condition generalises the Hermite-Einstein condition for a holomorphic vector bundle through the induced fibrewise Fubini-Study metric on the associated projectivisation. We prove various foundational analytic results concerning optimal symplectic connections. Our main result proves that optimal symplectic connections are unique, up to the action of the automorphism group of the submersion, when they exist. Thus optimal symplectic connections are canonical relatively Kähler metrics when they exist. In addition, we show that the existence of an optimal symplectic connection forces the automorphism group of the submersion to be reductive and that an optimal symplectic connection is automatically invariant under a maximal compact subgroup of this automorphism group. We also prove that when a submersion admits an optimal symplectic connection, it achieves the absolute minimum of a natural log norm functional, which we define.

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28

Apostolov, Vestislav, David M. J. Calderbank, and Paul Gauduchon. "Ambitoric geometry I: Einstein metrics and extremal ambikähler structures." Journal für die reine und angewandte Mathematik (Crelles Journal) 2016, no. 721 (January 1, 2016). http://dx.doi.org/10.1515/crelle-2014-0060.

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AbstractWe present a local classification of conformally equivalent but oppositely oriented 4-dimensional Kähler metrics which are toric with respect to a common 2-torus action. In the generic case, these “ambitoric” structures have an intriguing local geometry depending on a quadratic polynomialWe use this description to classify 4-dimensional Einstein metrics which are hermitian with respect to both orientations, as well as a class of solutions to the Einstein–Maxwell equations including riemannian analogues of the Plebański–Demiański metrics. Our classification can be viewed as a riemannian analogue of a result in relativity due to R. Debever, N. Kamran, and R. McLenaghan, and is a natural extension of the classification of selfdual Einstein hermitian 4-manifolds, obtained independently by R. Bryant and the first and third authors.These Einstein metrics are precisely the ambitoric structures with vanishing Bach tensor, and thus have the property that the associated toric Kähler metrics are extremal (in the sense of E. Calabi). Our main results also classify the latter, providing new examples of explicit extremal Kähler metrics. For both the Einstein–Maxwell and the extremal ambitoric structures,

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29

Dang, Quang-Tuan. "Continuity of Monge–Ampère Potentials in Big Cohomology Classes." International Mathematics Research Notices, July 10, 2021. http://dx.doi.org/10.1093/imrn/rnab183.

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Abstract Extending Di Nezza–Lu’s approach [16] to the setting of big cohomology classes, we prove that solutions of degenerate complex Monge–Ampère equations on compact Kähler manifolds are continuous on a Zariski open set. This allows us to show that singular Kähler–Einstein metrics on log canonical varieties of general type have continuous potentials on the ample locus outside of the non-klt part.

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30

Chi, Hanci, Ioannis Chrysikos, and Eivind Schneider. "Decomposable (5, 6)-solutions in eleven-dimensional supergravity." Journal of Mathematical Physics 64, no. 6 (June 1, 2023). http://dx.doi.org/10.1063/5.0142572.

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We present decomposable (5, 6)-solutions M̃1,4×M6 in eleven-dimensional supergravity by solving the bosonic supergravity equations for a variety of non-trivial flux forms. Many of the bosonic backgrounds presented here are induced by various types of null flux forms on products of certain totally Ricci-isotropic Lorentzian Walker manifolds and Ricci-flat Riemannian manifolds. These constructions provide an analogy of the work performed by Chrysikos and Galaev [Classical Quantum Gravity 37, 125004 (2020)], who made similar computations for decomposable (6, 5)-solutions. We also present bosonic backgrounds that are products of Lorentzian Einstein manifolds with a negative Einstein constant (in the “mostly plus” convention) and Riemannian Kähler–Einstein manifolds with a positive Einstein constant. This conclusion generalizes a result of Pope and van Nieuwenhuizen [Commun. Math. Phys. 122, 281–292 (1989)] concerning the appearance of six-dimensional Kähler–Einstein manifolds in eleven-dimensional supergravity. In this setting, we construct infinitely many non-symmetric decomposable (5, 6)-supergravity backgrounds by using the infinitely many Lorentzian Einstein–Sasakian structures with a negative Einstein constant on the 5-sphere, known from the work of Boyer et al. [Commun. Math. Phys. 262, 177–208 (2006)].

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Ghosh, Kartick. "Coupled Kähler-Einstein and Hermitian-Yang-Mills equations." Bulletin des Sciences Mathématiques, January 2023, 103232. http://dx.doi.org/10.1016/j.bulsci.2023.103232.

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32

Bor, Gil, Omid Makhmali, and Paweł Nurowski. "Para-Kähler-Einstein 4-manifolds and non-integrable twistor distributions." Geometriae Dedicata 216, no. 1 (January 4, 2022). http://dx.doi.org/10.1007/s10711-021-00665-4.

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AbstractWe study the local geometry of 4-manifolds equipped with a para-Kähler-Einstein (pKE) metric, a special type of split-signature pseudo-Riemannian metric, and their associated twistor distribution, a rank 2 distribution on the 5-dimensional total space of the circle bundle of self-dual null 2-planes. For pKE metrics with non-zero scalar curvature this twistor distribution has exactly two integral leaves and is ‘maximally non-integrable’ on their complement, a so-called (2,3,5)-distribution. Our main result establishes a simple correspondence between the anti-self-dual Weyl tensor of a pKE metric with non-zero scalar curvature and the Cartan quartic of the associated twistor distribution. This will be followed by a discussion of this correspondence for general split-signature metrics which is shown to be much more involved. We use Cartan’s method of equivalence to produce a large number of explicit examples of pKE metrics with non-zero scalar curvature whose anti-self-dual Weyl tensor have special real Petrov type. In the case of real Petrov type D, we obtain a complete local classification. Combined with the main result, this produces twistor distributions whose Cartan quartic has the same algebraic type as the Petrov type of the constructed pKE metrics. In a similar manner, one can obtain twistor distributions with Cartan quartic of arbitrary algebraic type. As a byproduct of our pKE examples we naturally obtain para-Sasaki-Einstein metrics in five dimensions. Furthermore, we study various Cartan geometries naturally associated to certain classes of pKE 4-dimensional metrics. We observe that in some geometrically distinguished cases the corresponding Cartan connections satisfy the Yang-Mills equations. We then provide explicit examples of such Yang-Mills Cartan connections.

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33

Bruzzo, Ugo, Pietro Fré, Umar Shahzad, and Mario Trigiante. "D3-brane supergravity solutions from Ricci-flat metrics on canonical bundles of Kähler–Einstein surfaces." Letters in Mathematical Physics 113, no. 3 (June 4, 2023). http://dx.doi.org/10.1007/s11005-023-01683-x.

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AbstractD3 brane solutions of type IIB supergravity can be obtained by means of a classical Ansatz involving a harmonic warp factor, $$H(\textbf{y},\bar{\textbf{y}})$$ H ( y , y ¯ ) multiplying at power $$-1/2$$ - 1 / 2 the first summand, i.e., the Minkowski metric of the D3 brane world-sheet, and at power 1/2 the second summand, i.e., the Ricci-flat metric on a six-dimensional transverse space $$\mathcal {M}_6$$ M 6 , whose complex coordinates y are the arguments of the warp factor. Of particular interest is the case where $$\mathcal {M}_6={\text {tot}}[ K\left[ \left( \mathcal {M}_B\right) \right] $$ M 6 = tot [ K M B is the total space of the canonical bundle over a complex Kähler surface $$\mathcal {M}_B$$ M B . This situation emerges in many cases while considering the resolution à la Kronheimer of singular manifolds of type $$\mathcal {M}_6=\mathbb {C}^3/\Gamma $$ M 6 = C 3 / Γ , where $$\Gamma \subset \mathrm {SU(3)} $$ Γ ⊂ SU ( 3 ) is a discrete subgroup. When $$\Gamma = \mathbb {Z}_4$$ Γ = Z 4 , the surface $$\mathcal {M}_B$$ M B is the second Hirzebruch surface endowed with a Kähler metric having $$\mathrm {SU(2)\times U(1)}$$ SU ( 2 ) × U ( 1 ) isometry. There is an entire class $${\text {Met}}(\mathcal{F}\mathcal{V})$$ Met ( F V ) of such cohomogeneity one Kähler metrics parameterized by a single function $$\mathcal{F}\mathcal{K}(\mathfrak {v})$$ F K ( v ) that are best described in the Abreu–Martelli–Sparks–Yau (AMSY) symplectic formalism. We study in detail a two-parameter subclass $${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{KE}}\subset {\text {Met}}(\mathcal{F}\mathcal{V})$$ Met ( F V ) KE ⊂ Met ( F V ) of Kähler–Einstein metrics of the aforementioned class, defined on manifolds that are homeomorphic to $$S^2\times S^2$$ S 2 × S 2 , but are singular as complex manifolds. Actually, $${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{KE}}\subset {\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{ext}}\subset {\text {Met}}(\mathcal{F}\mathcal{V})$$ Met ( F V ) KE ⊂ Met ( F V ) ext ⊂ Met ( F V ) is a subset of a four parameter subclass $${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{ext}}$$ Met ( F V ) ext of cohomogeneity one extremal Kähler metrics originally introduced by Calabi in 1983 and translated by Abreu into the AMSY action-angle formalism.$${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{ext}}$$ Met ( F V ) ext contains also a two-parameter subclass $${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{ext}\mathbb {F}_2}$$ Met ( F V ) ext F 2 disjoint from $${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{KE}}$$ Met ( F V ) KE of extremal smooth metrics on the second Hirzebruch surface that we rederive using constraints on period integrals of the Ricci 2-form. The Kähler–Einstein nature of the metrics in $${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{KE}}$$ Met ( F V ) KE allows the construction of the Ricci-flat metric on their canonical bundle via the Calabi Ansatz, which we recast in the AMSY formalism deriving some new elegant formulae. The metrics in $${\text {Met}}(\mathcal{F}\mathcal{V})_{\textrm{KE}}$$ Met ( F V ) KE are defined on the base manifolds of U(1) fibrations supporting the family of Sasaki–Einstein metrics $$\textrm{SEmet}_5$$ SEmet 5 introduced by Gauntlett et al. (Adv Theor Math Phys 8:711–734, 2004), and already appeared in Gibbons and Pope (Commun Math Phys 66:267–290, 1979). However, as we show in detail using Weyl tensor polynomial invariants, the six-dimensional Ricci-flat metric on the metric cone of $${\mathcal M}_5 \in {\text {Met}}(\textrm{SE})_5$$ M 5 ∈ Met ( SE ) 5 is different from the Ricci-flat metric on $${\text {tot}}[ K\left[ \left( \mathcal {M}_{\textrm{KE}}\right) \right] $$ tot [ K M KE constructed via Calabi Ansatz. This opens new research perspectives. We also show the full integrability of the differential system of geodesics equations on $$\mathcal {M}_B$$ M B thanks to a certain conserved quantity which is similar to the Carter constant in the case of the Kerr metric.

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Marconnet, Paul, and Dimitrios Tsimpis. "Universal accelerating cosmologies from 10d supergravity." Journal of High Energy Physics 2023, no. 1 (January 9, 2023). http://dx.doi.org/10.1007/jhep01(2023)033.

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Abstract We study 4d Friedmann-Lemaître-Robertson-Walker cosmologies obtained from time-dependent compactifications of Type IIA 10d supergravity on various classes of 6d manifolds (Calabi-Yau, Einstein, Einstein-Kähler). The cosmologies we present are universal in that they do not depend on the detailed features of the compactification manifold, but only on the properties which are common to all the manifolds belonging to that class. Once the equations of motion are rewritten as an appropriate dynamical system, the existence of solutions featuring a phase of accelerated expansion is made manifest. The fixed points of this dynamical system, as well as the trajectories on the boundary of the phase space, correspond to analytic solutions which we determine explicitly. Furthermore, some of the resulting cosmologies exhibit eternal or semi-eternal acceleration, whereas others allow for a parametric control on the number of e-foldings. At future infinity, one can achieve both large volume and weak string coupling. Moreover, we find several smooth accelerating cosmologies without Big Bang singularities: the universe is contracting in the cosmological past (T < 0), expanding in the future (T > 0), while in the vicinity of T = 0 it becomes de Sitter in hyperbolic slicing. We also obtain several cosmologies featuring an infinite number of cycles of alternating periods of accelerated and decelerated expansions.

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