Solves a Dirichlet problem for flat metrics on Riemann surfaces with boundary.
problem Solving a Dirichlet problem for flat hermitian metrics on Hilbert bundles over compact Riemann surfaces with boundary.
method Proves solvability using flat hermitian metrics and factorization results.
result Solves the Dirichlet problem for flat metrics on Riemann surfaces with boundary.
Flat Hermitian Lie algebras are always Kähler.
problem Classifying Lie groups with Hermitian structures that are flat.
method Analysis on the Hermitian geometry of 2-step solvable Lie groups.
result Flat Hermitian Lie algebras are Kähler.
Study on Gauduchon manifolds finds metrics for projectively flat bundles.
problem Existence of Hermitian-Poisson metrics on projectively flat bundles.
method Heat flow techniques and continuity methods.
result Established a correspondence between Hermitian-Poisson metrics and semi-simplicity.
New metrics found on non-Kähler Calabi-Yau manifolds.
problem Constructing Ricci-flat metrics on non-Kähler Calabi-Yau manifolds.
method Using t-Gauduchon metrics on principal torus bundles over rational homogeneous varieties. result Examples of new metrics on non-Kähler Calabi-Yau manifolds.
Hermitian metrics with zero second Chern Ricci curvature are rigid and exist on specific manifolds.
problem Characterizing Hermitian metrics with vanishing second Chern Ricci curvature.
method Analyzing the rigidity of the second Chern Ricci curvature on compact complex manifolds.
result Characterization of second Chern Ricci-flat Hermitian metrics and non-existence results.
New insights prevent certain types of metrics on compact spaces.
problem Preventing the existence of specific types of metrics on compact spaces.
method Computing cohomology and analyzing stability of metrics for the pluriclosed flow.
result Prevents the existence of non-flat homogeneous Bismut Hermitian Einstein metrics on C-spaces.
Flow smooths Chern-Ricci-flat metrics on Hermitian manifolds.
problem Smooth Chern-Ricci-flat metrics on Hermitian manifolds.
method An analogue of the Calabi flow for compact Hermitian manifolds with vanishing first Bott-Chern class.
result The flow converges to the unique Chern-Ricci-flat metric under certain conditions.
The study finds conditions for scalar-flat metrics on ruled surfaces.
problem Conditions for the existence of scalar-flat metrics on ruled surfaces.
method Analyzes intrinsic number and complex structure of ruled surfaces.
result Scalar-flat metrics exist only for ruled surfaces with genus g≥2 and m(X)>2−2g. New findings on Chern flat metrics and their criticality.
problem Understanding critical Hermitian metrics on Chern flat manifolds.
method Analyzing Chern flat manifolds as compact quotients of complex Lie groups and studying their criticality.
result Chern flat metrics on semi-simple Lie groups are torsion-critical and vice versa.
A local classification of the Hermitian manifolds with flat associated connection is given. Hermitian manifolds admitting locally a conformal metric with flat associated connection are characterized by a curvature identity. Locally conformal Kaehler manifolds as well as Hermitian surfaces with vanishing associated conf…
The study examines deformations of Ricci-flat ALF spaces, showing they must be Hermitian.
problem Classifying and understanding deformations of Ricci-flat ALF spaces.
method Assuming suitable fall-off conditions, the authors show that deformations must be Hermitian and carry a non-trivial Killing vector field.
result The new Ricci-flat metric must belong to the family of previously classified metrics under mild additional hypotheses.
The paper classifies 4D Ricci-flat ALE manifolds and their properties.
problem Characterizing 4D Ricci-flat ALE manifolds and their properties.
method Analyzing the correspondence between ALE gravitational instantons and Kähler orbifolds, and proving properties of these structures.
result There is a one-to-one correspondence between Hermitian non-Kähler ALE gravitational instantons and Bach-flat Kähler orbifolds in 2D.
Uniform estimates for complex Monge-Ampère equations on hermitian varieties.
problem Uniform boundedness of Chern-Ricci flat potentials in conifold transitions.
method Proving uniform a priori estimates for degenerate complex Monge-Ampère equations.
result Generalization of a theorem to hermitian contexts.
Study on Lie groups with flat Gauduchon connections, focusing on Kähler structures.
problem Classifying Hermitian manifolds with flat Gauduchon connections.
method Investigating left-invariant Hermitian structures on Lie groups, focusing on flat connections.
result If either dimension is 2 or there exists a parallel frame, the metric must be Kähler.
New metrics found on non-Kähler Calabi-Yau manifolds.
problem Finding Levi-Civita Ricci-flat metrics on non-Kähler Calabi-Yau manifolds.
method Constructing new metrics on specific types of non-Kähler Calabi-Yau manifolds.
result Examples of Levi-Civita Ricci-flat metrics on various non-Kähler Calabi-Yau manifolds.
Study classifies gravitational instantons based on their asymptotic geometry.
problem Classifying gravitational instantons based on their asymptotic properties.
method Investigation of asymptotic geometry of Hermitian non-Kähler Ricci-flat metrics.
result All Hermitian non-Kähler gravitational instantons can be compactified to log del Pezzo surfaces.
Regularities and stability shown for a specific type of complex parallelizable manifolds.
problem Stability and regularity of Chern-flat metrics on complex parallelizable manifolds.
method Study of Hermitian metrics governed by the second Chern-Ricci form on compact complex manifolds.
result Chern-flat metrics are dynamically stable on compact complex parallelizable manifolds.
Study numerically flat bundles on non-Kähler manifolds.
problem Characterize holomorphic vector bundles on non-Kähler manifolds.
method Analyze conditions for numerically flatness and effectiveness.
result Numerically flatness is equivalent to specific stability conditions.
Classifies 4D toric Hermitian ALF metrics with conical singularities.
problem Classifying specific types of 4D Riemannian metrics.
method Explicit formulas provided for classification.
result Examples of metrics with conical singularities have infinitely many distinct topologies.
We develop a theory of stable bundles and affine Hermitian-Einstein metrics for flat vector bundles over a special affine manifold (a manifold admitting an atlas whose gluing maps are all locally constant volume-preserving affine maps). Our paper presents a parallel to Donaldson-Uhlenbeck-Yau's proof of the existence o…
Proves unique ALE instanton with toric Hermitian structure.
problem Classify Ricci flat ALE instantons with toric Hermitian non-Kähler structure.
method Direct global analysis of Tod form in Weyl-Papapetrou coordinates, avoiding toric Kähler geometry.
result Eguchi-Hanson instanton is the only smooth, Ricci flat, ALE instanton with toric Hermitian non-Kähler structure.
The paper proves conditions for Kähler-Einstein metrics on certain bundles.
problem Conditions for the existence of Kähler-Einstein metrics on unit sphere bundles.
method Analyzes curvature conditions and Ricci eigenvalues of Kähler manifolds.
result Conditions for obstruction flatness and existence of Kähler-Einstein metrics.
New metrics found on non-Kähler complex manifolds.
problem Finding metrics on non-Kähler complex manifolds.
method Reinterpretation of Bismut Hermitian-Einstein condition and associated holomorphic Courant algebroid.
result Infinitely many non-Kähler manifolds without Bismut Hermitian-Einstein metrics.
A flat complex vector bundle (E,D) on a compact Riemannian manifold (X,g) is stable (resp. polystable) in the sense of Corlette [C] if it has no D-invariant subbundle (resp. if it is the D-invariant direct sum of stable subbundles). It has been shown in [C] that the polystability of (E,D) in this sense is equivalent to…
Study on Hermitian-Yang-Mills connections on collapsing K3 surfaces.
problem Analyzing Hermitian-Yang-Mills connections on elliptically fibered K3 surfaces.
method Examined a sequence of Ricci-flat metrics collapsing fibers, and stable holomorphic bundles.
result Proved convergence of restricted Hermitian-Yang-Mills connections to a flat connection.
Gravitational instantons are non-Kähler, providing a counterexample to Euclidean Black Hole Uniqueness.
problem Classical Euclidean Black Hole Uniqueness conjecture
method Analyzing the Chen-Teo gravitational instanton
result The Chen-Teo instanton is Hermitian and non-Kähler
The paper examines SKT and CYT metrics on Lie groups.
problem Existence and properties of SKT and CYT metrics on Lie groups.
method Study left-invariant metrics on compact semi-simple Lie groups.
result Left-invariant metrics that are both CYT and SKT must be Bismut flat.
Let M be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric g and a covariant constant volume form. Let G be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat princip…
We study two types of isotropic planes: weakly isotropic and strongly isotropic planes. We prove that a Riemannian manifold of indefinite metric is conformally flat if and only if its curvature tensor vanishes on all the strongly isotropic planes. We specialize the plane axiom for Riemannian manifolds of indefinite met…
In this paper, we introduce the first Aeppli-Chern class for complex manifolds and show that the (1,1)- component of the curvature 2-form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We systematically investigate the relationship between a variety of Ricci curvatures on Her…
Holomorphic functions on certain manifolds are isometric to Lie groups.
problem Characterizing holomorphic functions on specific Hermitian manifolds.
method Gradient estimate for holomorphic functions, sub-Riemannian geometry.
result Universal cover of complete Hermitian manifolds with flat Chern connection is holomorphically isometric to a complex Lie group.
New Ricci flat Kähler metrics found on complex symmetric spaces.
problem Finding Ricci flat Kähler metrics on complex symmetric spaces.
method Using an explicit asymptotic model with interpretation of geometry at infinity in the wonderful compactification.
result Obtained new Ricci flat Kähler metrics on complex symmetric spaces of rank two.
The study classifies flat Gauduchon connections on compact Hermitian manifolds.
problem Understanding when compact Hermitian manifolds can have flat Gauduchon connections.
method Investigates the s-Gauduchon connections and their flatness conditions. result For certain values of s, the metric g is Kähler. We find a new class of invariant metrics existing on the tangent bundle of any given almost-Hermitian manifold. We focus here on the case of Riemannian surfaces, which yield new examples of Kählerian Ricci-flat manifolds in four real dimensions.
We discuss a technique to construct Ricci-flat hermitian metrics on complements of (some) anticanonical divisors of almost homogeneous manifolds and discuss when this metric is complete and Kähler. This construction has a strong interplay with invariance groups of the same dimension as the manifold acting with an open …
The paper studies Kähler-Einstein metrics on circle bundles and their obstruction flatness.
problem Understanding Kähler-Einstein metrics on circle bundles and their smoothness properties.
method Analyzing the obstruction flatness of hypersurfaces arising as unit circle bundles over Kähler manifolds.
result Complete Kähler-Einstein metrics on disk bundles are possible under certain conditions.
We generalize and study the Zermelo navigation problem on Hermitian manifolds in the presence of a perturbation W determined by a mild complex velocity vector field ∣∣W(z)∣∣h<∣∣u(z)∣∣h, with application of complex Finsler metric of complex Randers type. By admitting space-dependence of ship's relative speed $||u(…
A Hermitian metric on a complex manifold is called strong Kähler with torsion (SKT) if its fundamental 2-form ω is ∂∂ˉ-closed. We review some properties of strong KT metrics also in relation with symplectic forms taming complex structures. Starting from a 2n-dimensional SKT Lie algebra $\mathfr…
Almost hypercomplex manifolds with Hermitian and Norden metrics and more specially the corresponding quaternionic Kaehler manifolds are considered. Some necessary and sufficient conditions the investigated manifolds be isotropic hyper-Kaehlerian and flat are found. It is proved that the quaternionic Kaehler manifolds w…
The subject of investigations are the almost hypercomplex manifolds with Hermitian and anti-Hermitian (Norden) metrics. A linear connection D is introduced such that the structure of these manifolds is parallel with respect to D and its torsion is totally skew-symmetric. The class of the nearly Kaehler manifolds with r…
Anomaly flow studied on flat and non-flat nilmanifolds.
problem Analyzing the Anomaly flow on nilmanifolds.
method Examined with respect to Hermitian connections, focusing on flat and non-flat cases.
result General solutions and qualitative behavior of the Anomaly flow on nilmanifolds.
The study characterizes compact homogeneous manifolds with Bismut parallel torsion.
problem Characterizing compact homogeneous manifolds with specific geometric properties.
method Investigating Hermitian manifolds with Bismut parallel torsion, focusing on locally homogeneous manifolds.
result Characterization of compact Chern flat BTP manifolds and properties of BTP compact Hermitian locally homogeneous manifolds.
The class of the hypercomplex pseudo-Hermitian manifolds is considered. The flatness of the considered manifolds with the 3 parallel complex structures is proved. Conformal transformations of the metrics are introduced. The conformal invariance and the conformal equivalence of the basic types manifolds are studied. A k…
Introduces a new geometric structure for statistical manifolds with degenerate metrics.
problem Degenerate metrics in statistical manifolds affect geometric structures and applications.
method Introduces quasi-Codazzi structure for degenerate metrics and coherent tangent bundles.
result Generalizes geometric structures and relations for statistical models with degenerate metrics.
Ray Singer torsion is a numerical invariant associated with a compact Riemannian manifold equipped with a flat bundle and a Hermitian structure on this bundle. In this note we show how one can remove the dependence on the Riemannian metric and on the Hermitian structure with the help of a base point and of an Euler str…
Complete classification of Hermitian manifolds with flat Gauduchon connections.
problem Classifying compact Hermitian manifolds with flat Gauduchon connections.
method Analyzing properties of Hermitian manifolds and using Gauduchon connections.
result Established a conjecture about Kähler-like conditions and flatness.
We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is Kähler if and only if the Lie group is the direct p…
New result on Levi-flat hypersurfaces' normal bundles without positive curvature.
problem Understanding Levi-flat hypersurfaces' normal bundles and their curvature properties.
method Analyzing the normal bundle of Levi-flat real hypersurfaces in complex manifolds.
result The normal bundle to the Levi foliation does not admit a Hermitian metric with positive curvature.