Study locally conformally balanced metrics on specific Lie algebras.
problem Characterize and classify locally conformally balanced metrics on almost abelian Lie algebras.
method Characterizations and classifications based on specific properties of Lie algebras.
result Classification of six-dimensional almost abelian Lie algebras with locally conformally balanced metrics.
Study non-Kähler metrics on complex nilmanifolds, proving torus structure under certain conditions.
problem Understanding special non-Kähler metrics on complex nilmanifolds.
method Analyzing locally conformally Kähler, k-Gauduchon, balanced, and locally conformally balanced metrics on compact complex manifolds. result Compact complex nilmanifolds with balanced or k-Gauduchon metrics are tori, extending previous results. Flow stabilizes on non-Kähler metrics near Calabi-Yau.
problem Stability of conformally balanced metrics flow near Calabi-Yau manifolds.
method Proving stability of the anomaly flow around Calabi-Yau metrics.
result The flow can converge on non-Kähler metrics near Calabi-Yau.
The paper solves a problem related to curvature in complex geometry.
problem Resolving the prescribed Chern scalar curvature problem.
method Divided into three cases based on the sign of the Gauduchon degree, analyzed separately.
result Proves that certain functions are Chern scalar curvatures of conformal metrics.
Characterizes complex Finsler metrics and their properties.
problem Characterize complex Finsler metrics and their geometric properties.
method Defined the canonical connection and investigated holomorphic sectional curvature tensors and Ricci curvatures.
result Characterizes balanced complex Finsler metrics and provides sufficient and necessary conditions.
While the Anomaly flow was originally motivated by string theory, its zero slope case is potentially of considerable interest in non-Kahler geometry, as it is a flow of conformally balanced metrics whose stationary points are precisely Kahler metrics. We establish its convergence on Kahler manifolds for suitable initia…
The abstract discusses conjectures about metrics on complex manifolds.
problem The abstract tackles the conjectures about metrics on complex manifolds, specifically balanced, SKT, and LCK.
method The abstract uses complex Hermitian manifolds, closed 1-forms, and conjectures to explore these metrics.
result The abstract verifies a conjecture about the Bott--Chern homology for all known classes of LCK manifolds.
Study balanced Hermitian structures on almost abelian Lie algebras, classifying six-dimensional cases.
problem Classify balanced Hermitian structures on almost abelian Lie algebras.
method Classify six-dimensional almost abelian Lie algebras with balanced structures, investigate flow of balanced metrics and anomaly flow.
result Prove conjecture for compact almost abelian solvmanifolds with left-invariant complex structures.
We will consider locally conformally balanced manifolds. We prove that a locally conformally balanced condition is not stable under a small deformation. We prove that locally conformally balanced condition is stable under any proper modification. We prove that symmetric products of the Kodaira surface can be resolve to…
Study shows compact Vaisman manifolds cannot have certain special Hermitian metrics.
problem Compact Vaisman manifolds and their compatibility with special Hermitian structures.
method Proof of non-existence of specific Hermitian metrics on compact Vaisman manifolds.
result Compact Vaisman manifolds cannot admit special Hermitian metrics like special k-Gauduchon metrics or pluriclosed metrics. The paper analyzes systoles of complex projective spaces under various metrics.
problem Behavior of systoles in complex projective spaces for different metrics.
method Integral geometric techniques and careful analysis of systole functional.
result Balanced metrics locally minimize the systole on volume-normalized metrics.
A Hermitian metric on a complex manifold of complex dimension n is called {\em astheno-Kähler} if its fundamental 2-form F satisfies the condition ∂∂Fn−2=0. If n=3, then the metric is {\em strong KT}, i.e. F is ∂∂-closed. By using blow-ups and the …
We prove that if a compact smooth polarized complex manifold admits in the corresponding Hodge Kähler class a conformally Kähler, Einstein--Maxwell metric, or more generally, a Kähler metric of constant (ξ,a,p)-scalar curvature, then this metric minimizes the (ξ,a,p)-Mabuchi functional. Our method of proof extend…
Unified flow approach to curvature problem on specific manifolds.
problem Prescribed Chern scalar curvature problem on compact Hermitian manifolds with negative Gauduchon degree.
method Unified flow approach with conditions on curvature function f. result Flow converges to a conformal Hermitian metric with specified curvature.
The study explores metrics with constant curvature on compact manifolds.
problem Finding Hermitian metrics with constant second scalar curvature on compact manifolds.
method Analyzes Yamabe-type and elliptic equations, derives geometric consequences, and proves existence under specific curvature conditions.
result Under certain curvature conditions, a pluriclosed Gauduchon Hermitian metric has constant second Chern scalar curvature, leading to the existence of Kähler-Einstein metrics.
Paper constructs solutions to a system using Aeppli class without auxiliary gauge connection.
problem Constructing solutions to the Hull-Strominger system without auxiliary gauge connection.
method Deforming conformally balanced metric and tuning by Aeppli class to satisfy anomaly cancellation condition.
result Existence of family of solutions obtained via implicit function theorem.
We classify both local and global Kähler structures admitting totally geodesic homothetic foliations with complex leaves. The main building blocks are related to Swann's twists and are obtained by applying Weinstein's method of constructing symplectic bundles to Kähler data. As a byproduct we obtain new classes of: hol…
Study on balanced Hermitian structures on Lie algebras twisted by representations.
problem Conditions for balanced and locally conformally balanced Hermitian structures on Lie algebras.
method Analysis of Hermitian structures on twisted cartesian products of Lie algebras.
result Classification of six-dimensional balanced Hermitian twisted cartesian products Lie algebras.
Study of complex structures on specific solvmanifolds, proving existence and non-existence results.
problem Classification and properties of complex structures on six-dimensional solvmanifolds.
method Classification of Lie algebras, analysis of complex structures, study of Hermitian metrics.
result Determination of new balanced solvmanifolds and confirmation of conjectures.
Let (J,g) be a Hermitian structure on a compact nilmanifold M with invariant complex structure J and compatible metric g, which is not required to be invariant. We give classifications of 6-dimensional nilmanifolds M admitting strong Kähler with torsion, balanced or locally conformal Kähler structures (J,g).
New boundary and point constraints for controlling conformal surfaces.
problem Controlling the geometry of surfaces defined by minimizers of conformal variational problems.
method Introducing new boundary conditions, point constraints, and flux constraints to control the metric and conformal scale factor.
result Introduces intuitive controls for exploring a subspace of conformal immersions.
The study classifies metrics with vanishing curvature on complex manifolds.
problem Understanding metrics with vanishing curvature on complex manifolds.
method Analyzing Hermitian metrics, pluriclosed metrics, and metrics with real bisectional curvature.
result Hermitian metrics with vanishing curvature are Kähler and conformally balanced.
The paper explores Hermitian structures on tangent bundles of affine manifolds with Riemannian metrics.
problem Understanding Hermitian structures on tangent bundles of affine manifolds.
method Analyzes the conditions for various generalized Kähler conditions on (TM,J1,g1) and (TkM,Jk,gk). result Develops a machinery to construct generalized Kähler manifolds.
Characterizes pluriclosed metrics on Oeljeklaus-Toma manifolds.
problem Characterizing existence of pluriclosed metrics on OT manifolds.
method Purely number-theoretical conditions.
result Explicit examples of pluriclosed OT manifolds in arbitrary complex dimension.
We give a procedure for constructing an 8n-dimensional HKT Lie algebra starting from a 4n-dimensional one by using a quaternionic representation of the latter. The strong (respectively, weak, hyper-Kähler, balanced) condition is preserved by our construction. As an application of our results we obtain a new compact…
Study of special Kato manifolds derived from toric geometry.
problem Characterize and study properties of Kato manifolds.
method Construction from toric geometry, topological and analytical properties, combinatorial data, flat degenerations, Hermitian geometry.
result No Kato manifold supports balanced or pluriclosed metrics.
Proposes SCD-split for CP to balance interpretability and efficiency.
problem Difficult interpretation of disconnected subintervals in CP prediction sets.
method Incorporates smoothing operations into CP framework.
result SCD-split balances interval length and subinterval number, theoretically provable.
Locally conformal SKT structures are introduced and studied on Lie groups and their compact quotients.
problem Existence and classification of LCSKT structures on Lie groups and their quotients.
method Introducing LCSKT structures and studying their properties on Lie groups and their quotients.
result Existence of non-trivial LCSKT structures on 6-dimensional nilpotent Lie algebras and almost abelian Lie algebras.
Consider E a holomorphic vector bundle over a projective manifold X polarized by an ample line bundle L. Fix k large enough, the holomorphic sections H0(E⊗Lk) provide embeddings of X in a Grassmanian space. We define the \textit{balancing flow for bundles} as a flow on the space of projectively equ…
Characterizes metrics on Lie groups, proving non-simultaneous existence of balanced and pluriclosed metrics.
problem Existence of balanced and pluriclosed metrics on real semisimple Lie groups.
method Characterization using Vogan diagrams and revisiting complex structure classification.
result Complex manifolds cannot simultaneously admit balanced and pluriclosed metrics.
Study on special Hermitian metrics on cohomogeneity one manifolds.
problem Characterizing and constructing Hermitian metrics on cohomogeneity one manifolds.
method Investigation of geometry of Hermitian manifolds with compact Lie group action by holomorphic isometries.
result Construction of new examples of cohomogeneity one Hermitian metrics solving specific equations.
On an almost Hermitian manifold, we have two Hermitian scalar curvatures with respect to any canonical Hermitian connection defined by P. Gauduchon. Explicit formulas of these two Hermitian scalar curvatures are obtained in terms of Riemannian scalar curvature, norms of decompositions of covariant derivative of the fun…
Constructs many black hole spacetimes in de Sitter space.
problem Creating well-controlled many black hole spacetimes in de Sitter space.
method Gluing Schwarzschild-de Sitter or Kerr-de Sitter black hole metrics into neighborhoods of points on the future conformal boundary of de Sitter space, under certain balance conditions.
result Solves the Einstein equation directly for the metric, given scattering data at the future conformal boundary.
The paper finds conditions for smooth curves of balanced metrics in Hermitian non-Kähler settings.
problem Balanced metrics under deformations of complex structures.
method Necessary conditions for smooth curves of balanced metrics.
result Existence of smooth curves of balanced metrics starting from a fixed balanced metric.
No radial balanced metrics found on Kepler manifold unit ball with mild boundary conditions.
problem Finding radial balanced metrics on the unit ball of the Kepler manifold.
method Analyzing boundary behavior and weights of metrics.
result Explicit weights for radial metrics satisfying balanced condition identified.
Blowing up flat metrics yields balanced ones with constant curvature.
problem Constructing balanced metrics with constant curvature on orbifolds.
method Blowing up a compact orbifold with balanced Chern-Ricci flat metrics.
result Blown-up orbifolds admit balanced metrics with constant Chern scalar curvature.
Balanced metrics found on Lie groups and their quotients.
problem Existence of balanced metrics on Lie groups and quotients.
method Proved existence of invariant complex structures and Hermitian balanced metrics on Lie groups and quotients.
result Existence of balanced metrics on Lie groups and quotients, and no pluriclosed metrics.
Study confirms asymptotic behavior of logarithmic balanced metric near infinity.
problem Asymptotic behavior of logarithmic balanced metric near infinity.
method Non-trivial refinement of tools from previous work.
result Partial confirmation of conjecture on asymptotic behavior.
Study shows convergence of anticanonically balanced metrics to Kähler-Einstein metrics on Fano manifolds.
problem Finding anticanonically balanced metrics on Fano manifolds.
method Simplification of Donaldson's proof using Berezin-Toeplitz quantization.
result Sequence of anticanonically balanced metrics converges to Kähler-Einstein metric.
Paper proves non-existence of certain balanced metrics on six-manifolds.
problem Existence of balanced metrics on six-manifolds with specific properties.
method Analysis of cohomogeneity one actions and properties of balanced structures.
result Non-existence of balanced non-Kähler SU(3)-structures on six-manifolds.
The paper constructs flat metrics on orbifolds and resolutions.
problem Finding flat metrics on orbifolds and their resolutions.
method Gluing construction and analysis of singularities.
result All crepant resolutions of non-Kähler Calabi-Yau orbifolds with Chern-Ricci flat balanced metrics admit such metrics.
This paper consists of two results dealing with balanced metrics (in S. Donaldson terminology) on nonconpact complex manifolds. In the first one we describe all balanced metrics on Cartan domains. In the second one we show that the only Cartan-Hartogs domain which admits a balanced metric is the complex hyperbolic spac…
Symmetric Positive Definite (SPD) matrices have been used in many fields of medical data analysis. Many Riemannian metrics have been defined on this manifold but the choice of the Riemannian structure lacks a set of principles that could lead one to choose properly the metric. This drives us to introduce the principle …
We give a moment map interpretation of some relatively balanced metrics. As an application, we extend a result of S. K. Donaldson on constant scalar curvature Kähler metrics to the case of extremal metrics. Namely, we show that a given extremal metric is the limit of some specific relatively balanced metrics. As a coro…
The Anomaly flow is a flow which implements the Green-Schwarz anomaly cancellation mechanism originating from superstring theory, while preserving the conformally balanced condition of Hermitian metrics. There are several versions of the flow, depending on whether the gauge field also varies, or is assumed known. A dis…
Paper surveys balanced metrics and proves a geodesic convexity result.
problem Understanding balanced metrics and stability in algebraic geometry.
method Survey and proof of geodesic convexity result.
result Geodesically convex function on a complete Riemannian manifold admits a critical point if and only if its asymptotic slope at infinity is positive.
On a complex manifold an Hermitian metric which is simultaneously SKT and balanced has to be necessarily Kähler. It has been conjectured that if a compact complex manifold (M,J) has an SKT metric and a balanced metric both compatible with J, then (M, J) is necessarily Kähler. We show that the conjecture is true for nil…
The paper broadens a mathematical correspondence to include more balanced metrics.
problem Extending a mathematical correspondence to a broader class of metrics.
method Using key observations and known theorems to apply results to a new class of metrics.
result The known results can be applied to a larger class of metrics, including those arising from multipolarizations.