The study proves conditions for Hermitian metrics on compact almost complex manifolds.
problem Conditions for Hermitian metrics on compact almost complex manifolds.
method Analyzes compact almost complex manifolds with Hermitian metrics and integral conditions involving ∂-harmonic (0,1)-forms. result The integral condition is automatically satisfied for strongly Gauduchon metrics, and equivalent to being strongly Gauduchon for integrable almost complex structures.
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.
Paper solves Gauduchon scalar curvature problem on almost Hermitian manifolds.
problem Prescribed Gauduchon scalar curvature problem on almost Hermitian manifolds.
method Reduced to solving a semi-linear partial differential equation with exponential nonlinearity using super and sub-solution method.
result Existence of solution depends on the sign of a constant associated to Gauduchon degree.
Study Gauduchon connections on Kähler-like metrics in 6D solvmanifolds.
problem Investigate Hermitian metrics with Gauduchon connections having symmetries similar to Levi-Civita and Chern connections.
method Analyze 6-dimensional solvmanifolds with invariant complex structures and trivial canonical bundles. result Evidence supports two conjectures about Kähler-like conditions for different Gauduchon connections.
We show existence of unique smooth solutions to the Monge-Ampere equation for (n-1)-plurisubharmonic functions on Hermitian manifolds, generalizing previous work of the authors. As a consequence we obtain Calabi-Yau theorems for Gauduchon and strongly Gauduchon metrics on a class of non-Kahler manifolds: those satisfyi…
Study examines flow behavior on non-Kähler manifolds.
problem Asymptotic behavior of Hermitian-Yang-Mills flow on non-Kähler manifolds.
method Analysis of Hermitian-Yang-Mills flow over compact non-Kähler manifolds with specific conditions.
result Characterized the limiting behavior of the flow.
Study of t-Gauduchon Ricci-flat condition under Chern-Ricci flow on non-Kähler manifolds.
problem Investigating the t-Gauduchon Ricci-flat condition on non-Kähler manifolds. method Chern-Ricci flow approach, examples of non-Kähler Calabi-Yau manifolds, and geometric flow analysis.
result Examples of Chern-Ricci flow on non-Kähler Calabi-Yau manifolds that do not preserve the t-Gauduchon Ricci-flat condition. The paper explores applications of Gauduchon metrics in complex geometry.
problem Implications of Gauduchon metrics in complex geometry.
method Existence and properties of Gauduchon metrics.
result Non-existence of holomorphic sections and restrictions on ∂∂ˉ-closedness. 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.
This text is dedicated to the real Killing equation on 3-dimensional Weyl manifolds. Any manifold admitting a real Killing spinor of weight 0 satisfies the conditions of a Gauduchon-Tod geometry. Conversely, any simply connected Gauduchon-Tod geometry has a 2-dimensional space of solutions of the real Killing equation …
Investigates special metrics in hypercomplex geometry.
problem Characterizing and understanding special hyperhermitian metrics.
method Characterization of hypercomplex structures with Obata holonomy, investigation of quaternionic Gauduchon and balanced metrics, incompatibility results, and introduction of Einstein-type conditions.
result Joyce's manifolds always admit special metrics.
In this paper, we characterize Riemannian 4-manifold in terms of its almost Hermitian twistor spaces (Z,gt,J±). Some special metric conditions (including Balanced metric condition, first Gauduchon metric condition) on (Z,gt,J±) are studied. For the first Chern form of a natural unitary…
Study extends continuity equation for Gauduchon metrics.
problem Continuity equation for Gauduchon metrics.
method Solution to Gauduchon conjecture by Székelyhidi, Tosatti, and Weinkove.
result Extended interval of maximal existence for continuity equation.
This paper is intended as the first step of a programme aiming to prove in the long run the long-conjectured closedness under holomorphic deformations of compact complex manifolds that are bimeromorphically equivalent to compact Kähler manifolds, known as Fujiki {\it class} C manifolds. Our main idea is to exp…
The Lee-Gauduchon cone is a convex cone of cohomology classes for complex manifolds.
problem Understanding the Lee-Gauduchon cone for complex manifolds.
method Analyzing the Lee-Gauduchon cone as a convex cone of cohomology classes.
result The Lee-Gauduchon cone is a bimeromorphic invariant.
We define strongly Gauduchon spaces and the class SG which are generalization of strongly Gauduchon manifolds in complex spaces. Comparing with the case of Kahlerian, the strongly Gauduchon space and the class SG are similar to the Kahler space and the Fujiki class C respectively. Some properties about these complex sp…
Study proves properties of compact Hermitian surfaces with specific curvature conditions.
problem Characterizing compact Hermitian surfaces with pointwise constant Gauduchon holomorphic sectional curvature.
method Analyzes surfaces with Gauduchon connections and Lichnerowicz holomorphic sectional curvature.
result Compact Hermitian surfaces with pointwise constant Gauduchon holomorphic sectional curvature are either Kähler or isosceles Hopf surfaces.
Study of Hermitian and Gauduchon connections on Lie groups with almost Hermitian structures.
problem Characterizing connections on Lie groups with almost Hermitian structures.
method Analyzing left-invariant Hermitian and Gauduchon connections on Lie groups equipped with almost Hermitian structures.
result Explicit formulas for torsion components and curvature of Gauduchon connections on Lie groups.
In this paper, we study strongly Gauduchon metrics on compact complex manifolds. We study the cohomology cones SG in the de Rham cohomology groups generated by all strongly Gauduchon metrics and its direct images under proper modifications. We also study the moduli of strongly Gauduchon manifolds. We prove an existence…
The paper shows conditions under which certain cohomology groups vanish for Hermitian manifolds.
problem Conditions for vanishing certain cohomology groups on Hermitian manifolds.
method Analyzes the cohomology groups of Hermitian manifolds with specific conditions on metrics and forms.
result If the Aeppli class of ωn−p vanishes, then HBCp,0(M)=0. Compact lcK manifolds with holomorphic Lee field are Vaisman under certain conditions.
problem Characterizing compact locally conformally Kähler manifolds with holomorphic Lee fields.
method Analyzing conditions for a compact lcK manifold to be Vaisman when it has a holomorphic Lee vector field.
result Compact lcK manifolds with holomorphic Lee field are Vaisman if the Lee field has constant norm or the metric is Gauduchon.
The paper explores properties of Gauduchon curvature in Hermitian manifolds.
problem Investigating properties of Gauduchon curvature in Hermitian manifolds.
method Analyzing the Ricci curvature of Gauduchon connections and proving existence of metrics.
result Monotonicity theorem for Gauduchon holomorphic sectional curvature.
Study on stable vector bundles over Gauduchon manifolds.
problem Existence and stability of vector bundles over Gauduchon manifolds.
method Uhlenbeck--Yau's continuity method for approximate Hermitian--Einstein structures.
result Equivalence of semi-stability and existence of Hermitian--Einstein structures.
The paper studies deformations of Calabi-Yau manifolds using Gauduchon metrics.
problem Deformations of Calabi-Yau manifolds under co-polarised conditions.
method Analyzes local deformations of Calabi-Yau ∂∂ˉ-manifolds using Gauduchon metrics and constructs a new hp-HS form. result Proves the p-SKT h-∂∂ˉ-property is deformation open. Gauduchon's theorem extended to singular spaces with smoothing.
problem Extending Gauduchon's theorem to singular spaces.
method Using smoothing techniques for singular spaces.
result Existence of conformally equivalent metrics on singular spaces.
This paper discusses partial answers and a proof for conjectures about Gauduchon connections on Hermitian manifolds.
problem Conjectures about Gauduchon connections and Hermitian metrics on compact manifolds.
method Analyzes partial answers to conjectures and provides a proof for a related conjecture, discovering a duality phenomenon.
result Proof of the second conjecture about two Kähler-like Gauduchon connections implying a Kähler metric.
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.
We prove that on any compact complex manifold one can find Gauduchon metrics with prescribed volume form. This is equivalent to prescribing the Chern-Ricci curvature of the metrics, and thus solves a conjecture of Gauduchon from 1984.
Paper solves a singular version of Gauduchon's conjecture.
problem Finding Gauduchon metrics with prescribed Ricci curvature on compact complex manifolds.
method Study of the Monge-Ampère equation for (n−1)-plurisubharmonic functions with a gradient term, adapted to singular settings. result Obtained a C0-estimate for the singular problem, proving smoothness of solutions on holomorphic Kähler families. 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.
The paper defines two types of hyperbolicity for complex manifolds and proves related results.
problem Defining and studying hyperbolicity for a broader class of complex manifolds.
method Introducing SKT hyperbolicity and Gauduchon hyperbolicity, proving results using SKT and Gauduchon metrics.
result Every SKT hyperbolic manifold is also Kobayashi/Brody hyperbolic and every Gauduchon hyperbolic manifold is divisorially hyperbolic.
Study on prescribing curvature on specific manifolds with negative Gauduchon degree.
problem Prescribing Chern scalar curvatures on compact Hermitian manifolds with negative Gauduchon degree.
method Analysis of geometric flow convergence to obtain existence results.
result Existence results for curvature functions that are nonzero and nonpositive, and sign-changing cases.
Paper uses flow to prove theorem on Higgs bundles.
problem Proving generalized Donaldson-Uhlenbeck-Yau theorem on Higgs bundles.
method Affine Hermitian-Yang-Mills flow
result Generalized Donaldson-Uhlenbeck-Yau theorem proved.
The study proves leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces.
problem Proving the existence of leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces.
method Using the ∂∂-class, the study proves the existence of leafwise flat forms for Gauduchon metrics on Inoue-Bombieri surfaces. result Uniform convergence of the normalized Chern-Ricci flow starting at any Gauduchon metric on all Inoue-Bombieri surfaces, with smooth convergence and bounded curvature for initial metrics in the ∂∂-class of the Tricerri/Vaisman metric. The paper studies LCAK metrics on complex manifolds and their properties.
problem Characterizing and understanding LCAK metrics on complex manifolds.
method Analyzes the geometric structures induced by LCAK metrics and their properties.
result Pluricanonical LCAK metrics have parallel Lee form on compact manifolds.
Researchers find a way to estimate potential functions for quaternionic metrics.
problem Existence of quaternionic Gauduchon metrics with prescribed volume form.
method Reframed as a fully nonlinear elliptic equation and established a uniform estimate.
result Uniform estimate for the potential function.
In this paper, we generalize the Gauduchon metrics on a compact complex manifold and define the γk functions on the space of its hermitian metrics.
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. Extends Gauduchon's result to higher dimensions, showing balanced metrics.
problem Understanding critical metrics in higher-dimensional Hermitian manifolds.
method Analyzes the functional of L2-norm of torsion 1-form and full Chern torsion. result Critical metrics are balanced in all dimensions.
In this paper, we consider the existence of approximate Hermitian-Einstein structure and the semi-stability on Higgs bundles over compact Gauduchon manifolds. By using the continuity method, we show that they are equivalent.
Existence of metrics on non-Kähler varieties, generalizing previous work.
problem Existence of metrics on non-Kähler varieties.
method Definition of slope stability and existence of singular Hermite-Einstein metrics.
result Existence and uniqueness of singular Hermite-Einstein metrics for slope-stable sheaves.
The study characterizes finite vector bundles on specific complex manifolds.
problem Characterizing finite vector bundles on complex manifolds.
method Proves a theorem for compact complex manifolds with Gauduchon astheno-Kahler metrics.
result Establishes a condition for finite vector bundles similar to Nori's theorem.
Paper proves a theorem for Higgs bundles on certain non-compact manifolds.
problem Proving a generalized Donaldson-Uhlenbeck-Yau theorem for Higgs bundles.
method Generalized Donaldson-Uhlenbeck-Yau theorem on Higgs bundles over non-compact Gauduchon manifolds.
result Proved a generalized Donaldson-Uhlenbeck-Yau theorem for Higgs bundles.
Enhanced Schwarz lemma for Hermitian manifolds with new curvature constraints.
problem Improving Schwarz lemma for holomorphic maps between Hermitian manifolds.
method Introducing new curvature constraints on source and target manifolds, controlling by holomorphic sectional curvature.
result Significant improvements on the Wu--Yau theorem and Schwarz lemma for Gauduchon connections.
We propose the study of a Monge-Ampère-type equation in bidegree (n−1,n−1) rather than (1,1) on a compact complex manifold X of dimension n for which we prove uniqueness of the solution subject to positivity and normalisation restrictions. Existence will hopefully be dealt with in future work. The aim is to…
Conformal vector fields on LCP manifolds are orthogonal and Killing.
problem Understanding conformal vector fields on specific geometric manifolds.
method Analyzing properties of conformal vector fields on compact locally conformally product manifolds.
result Conformal vector fields are orthogonal to the flat distribution and Killing.
The study proves estimates for transverse nonlinear equations on Sasakian manifolds with applications in geometry.
problem Estimating transverse fully nonlinear equations on Sasakian manifolds.
method Proving a priori estimates for transverse fully nonlinear equations.
result The study proves estimates for transverse fully nonlinear equations on Sasakian manifolds and gives geometric applications.
Proves Liouville theorem for non-Kähler manifolds.
problem Extending Liouville theorem to non-Kähler manifolds.
method Proves Liouville theorem for a specific class of complete Gauduchon manifolds.
result Generalizes Yau's result to non-Kähler manifolds.