Study curvature of complex Finsler metrics on Lie groups.
problem Explicitly calculate and characterize curvature of complex Finsler metrics.
method Analyzes left-invariant complex Finsler metrics on Lie groups using their Lie algebra.
result Characterizes conditions for a Finsler metric to be Kähler or weakly Kähler.
The paper characterizes complex Finsler metrics invariant under U(n) and their properties.
problem Characterizing U(n)-invariant strongly convex complex Finsler metrics. method Analyzing conditions for strong convexity and proving theorems about these metrics.
result A U(n)-invariant strongly convex complex Finsler metric is a real Berwald metric if and only if it comes from a Hermitian metric. Locally finite complexes with polyhedral CAT(0) metrics are arborescent.
problem Characterizing locally finite complexes with CAT(0) metrics. method Proving arborescence for complexes with polyhedral CAT(0) metrics. result Locally finite complexes with polyhedral CAT(0) metrics are arborescent. 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.
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.
In this paper we introduce in study the projectively related complex Finsler metrics. We prove the complex versions of the Rapcsák's theorem and characterize the weakly Kähler and generalized Berwald projectively related complex Finsler metrics. The complex version of Hilbert's Fourth Problem is also pointed out. As an…
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.
Stability of SKT metrics under deformations on complex manifolds.
problem Stability of strong Kähler with torsion metrics under small deformations.
method Finding necessary conditions for stability of SKT metrics along a family of complex manifolds.
result Necessary conditions for the stability of SKT metrics on a smooth curve of Hermitian metrics.
The paper studies deformations of astheno-Kähler metrics on complex manifolds.
problem Stability of astheno-Kähler metrics under complex structure deformations.
method Proves necessary cohomological conditions for astheno-Kähler metrics along deformations.
result Provides obstructions to the existence of astheno-Kähler metrics on specific nilmanifolds.
The paper introduces new metrics on complex domains with specific geometric properties.
problem Developing metrics on complex domains with strong pseudoconvexity and holomorphic invariance.
method Explicit construction of holomorphic invariant strongly pseudoconvex complex Finsler metrics via deformation of Bergman metrics.
result These metrics have bounded curvature properties similar to Bergman metrics.
Study connects two types of metrics on complex surfaces.
problem Interrelations between bi-Hermitian and locally conformally Kähler metrics.
method Analysis of metrics on complex surfaces.
result Found interconnections between bi-Hermitian and locally conformally Kähler metrics.
Study on metric bubbles in complex dimensions 1 and 2.
problem Understanding degenerations of Kähler-Einstein metrics.
method Investigation of metric bubble trees for non-collapsing cases.
result Description of a conjectural higher-dimensional picture.
Study of harmonic maps on 2D simplicial complexes, proving existence and regularity.
problem Existence and regularity of harmonic maps between 2D simplicial complexes.
method Extending previous work, study metrics conformal to flat or ideal hyperbolic, proving existence, uniqueness, and regularity of harmonic maps.
result Existence, uniqueness, and regularity results for harmonic maps between 2D simplicial complexes.
Study examines Kähler immersions of ALE Kähler metrics into complex space forms.
problem Kähler immersions of ALE Kähler metrics into complex space forms.
method Investigation of the relationship between Kähler immersions and the mass of ALE Kähler metrics.
result ALE Kähler metrics with positive mass do not admit a Kähler immersion into complex Euclidean space.
Study on special metrics on complex nilmanifolds, proving existence and properties.
problem Existence and properties of special metrics on complex nilmanifolds.
method Analysis of astheno-Kähler, strongly Gauduchon, and balanced metrics.
result Existence of astheno-Kähler metrics implies specific properties of nilmanifolds.
Study on geometrically formal metrics on complex manifolds.
problem Existence and properties of geometrically formal metrics on complex manifolds.
method Topological and cohomological obstructions, detailed analysis for specific manifolds, and metric constructions.
result Existence and non-existence conditions for geometrically formal metrics on various complex manifolds.
Study bi-Hermitian metrics on complex surfaces and solve geometric PDEs.
problem Construct canonical metrics on complex surfaces with split tangent bundle.
method Introduced new fully non-linear geometric PDEs and established smooth solutions.
result Solved the prescribed Bismut Ricci problem on complex surfaces.
Study cohomological and metric properties of non-Kähler complex manifolds.
problem Understanding cohomological invariants and metrics of non-Kähler complex manifolds.
method Partial account of problems through cohomological and metric properties.
result Partial insights into cohomological and metric properties of non-Kähler manifolds.
Characterizes Kähler-Berwald metrics on complex manifolds.
problem Identifying Kähler-Berwald metrics among strongly convex complex Finsler metrics.
method Geometric characterization using Cartan and Chern-Finsler connections.
result Characterizes Kähler-Berwald metrics in terms of parallelism of the canonical complex structure.
The paper explores Ricci forms on noncompact complex manifolds, including Kähler-Einstein and canonical metrics.
problem Existence of specific metrics on noncompact complex manifolds with prescribed Ricci curvature.
method Analyzes geometric problems on noncompact complex manifolds using Ricci curvature.
result Improves the main theorem in Cheng-Yau [4] and constructs Hesse-Einstein metrics.
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 studies complex Finsler metrics on complex Lie groups.
problem Characterizing properties of left invariant complex Finsler metrics on complex Lie groups.
method Using invariant frames, the paper proves properties of the metric and its spray.
result The strongly Kähler, Kähler, and weakly Kähler properties are equivalent for the metric.
The paper studies extremal Kaehler metrics induced by complex space forms, proving properties in both finite and infinite dimensions.
problem Analyzing extremal Kaehler metrics induced by complex space forms in finite and infinite dimensions.
method Proving properties of extremal Kaehler metrics under specific conditions in both finite and infinite dimensional settings.
result Extremal Kaehler metrics induced by infinite dimensional elliptic complex space forms have constant non-positive holomorphic sectional curvature under stability conditions.
The study defines and analyzes semi-invariant submanifolds in complex contact metric manifolds.
problem Characterizing semi-invariant submanifolds in complex contact metric manifolds.
method Definition and derivation of relations, integrability conditions of distributions.
result Obtained useful relations and integrability conditions for semi-invariant submanifolds.
We introduce complex generalizations of the classical Legendre transform, operating on Kähler metrics on a compact complex manifold. These Legendre transforms give explicit local isometric symmetries for the Mabuchi metric on the space of Kähler metrics around any real analytic Kähler metric, answering a question origi…
Study of Calabi-Yau manifold degenerations near complex structure limits.
problem Understanding polarized degenerations of Calabi-Yau manifolds.
method Improvement of metric convergence results on generic regions.
result Metric convergence for collapsing Ricci-flat Kähler metrics on generic regions.
Inspired by a construction due to Hitchin, we produce strongly bihermitian metrics on certain Hopf complex surfaces, which integrate the locally conformally Kaehler metrics found by Gauduchon and Ornea. We also show that the Inoue complex surfaces with zero second Betti number do not admit bihermitian metrics. This com…
New approach finds Kähler metrics on compact complex manifolds.
problem Finding Kähler metrics on compact complex manifolds.
method Defining a new functional whose critical points are Kähler metrics.
result Critical points of the new functional are precisely the Kähler metrics.
Paper solves open problem in complex Finsler geometry.
problem Existence of non-Kähler weakly Kähler Finsler metrics.
method Constructs a family of weakly Kähler Finsler metrics.
result Proves uniformization theorem for unitary invariant complex Randers metrics.
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.
The paper proves properties of complex Finsler metrics on specific domains.
problem Investigating invariant complex Finsler metrics on complex domains.
method Analyzing holomorphic automorphism groups and constructing metrics.
result Explicitly constructed metrics on polydisks with properties similar to Bergman metric.
In this paper, we study normal complex contact metric manifolds and we get some general results on them. Moreover, we obtained the general expression of the curvature tensor field for arbitrary vector fields. Furthermore, we show that the necessary and succient conditions to be normal a complex contact metric manifold.…
Researchers create a new metric on complex projective space bundles.
problem Constructing a hyperkähler metric on complex projective space bundles.
method Explicit construction in local coordinates, using holomorphic isomorphism to coadjoint orbits.
result A hyperkähler metric on twisted cotangent bundles of CPn. In this article we introduce a generalization of locally conformally Kaehler metrics from complex manifolds to complex analytic spaces with singularities and study which properties of locally conformally Kaehler manifolds still hold in this new setting. We prove that if a complex analytic space has only quotient singul…
The aim of this note is the study of Einstein condition for para-holomorphic Riemannian metrics in the para-complex geometry framework. Firstly, we make some general considerations about para-complex Riemannian manifolds (not necessarily para-holomorphic). Next, using an one-to-one correspondence between para-holomorph…
Paper proves polynomial equivalence of quantum complexity metrics.
problem Quantum complexity metrics equivalence.
method Study of right-invariant metrics on unitary group.
result All metrics in the equivalence class have polynomial slowdown in approximation.
In this paper, we construct complex metric structures on complex hypersurfaces in hyperkahler manifolds. This construction is that in contact geometry.
Study of metrics on a specific Lie group with complex structure.
problem Determine moduli space of left-invariant metrics on a six-dimensional Lie group.
method Investigate left-invariant metrics on a Lie group with complex structure and first Betti number 4.
result Classify metrics and complex structures on the Lie group.
Homotopy equivalence shown between complex and thickened versions of manifolds.
problem Homotopy equivalence between manifold complexes and thickened versions.
method Natural bijections and homotopy equivalences of Vietoris-Rips and Čech complexes and thickened versions.
result Natural bijections between complexes and thickened versions are homotopy equivalences.
Study continuity of complex Sobolev functions, with applications to Kaehler metrics.
problem Continuity of functions in complex Sobolev spaces.
method Analysis of function regularity in Sobolev spaces, with applications to Kaehler metrics.
result Hermitian generalizations of recent results on Kaehler metrics.
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.
In this paper, we study the geometry of compact complex manifolds with Levi-Civita Ricci-flat metrics and prove that compact complex surfaces admitting Levi-Civita Ricci-flat metrics are Kahler Calabi-Yau surfaces or Hopf surfaces.
Study proposes a functional for LCK metrics on complex manifolds.
problem Existence problem for locally conformally Kähler metrics on compact complex manifolds.
method Introducing and studying a functional that varies by complex dimension.
result Functional approach to existence problem for LCK metrics.
The paper studies complex Finsler metrics and their equivalence to the Kobayashi metric.
problem Investigating properties and equivalence of complex Finsler metrics.
method Using curvature properties of Bergman metrics and Schwarz lemma, the paper analyzes complex Finsler metrics and their equivalence to the Kobayashi metric.
result Uniform equivalences of the Kobayashi metric and Carathéodory metric on bounded strongly convex domains with smooth boundaries are proven.
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. We classify all Kahler metrics in an open subset of C2 whose real geodesics are circles. All such metrics are equivalent (via complex projective transformations) to Fubini metrics (i.e. to Fubini-Study metric on CP2 restricted to an affine chart, to the complex hyperbolic metric in the unit ball model or to the E…
Solves classical problem with Kähler-Einstein metrics in complex projective spaces.
problem Classical problem of non-isometric bidimensional Kähler-Einstein submanifolds.
method Listed complete non-isometric bidimensional rotation invariant Kähler-Einstein submanifolds.
result Solves the classical problem in the specified case.
A four-parametric family of linear connections preserving the almost complex structure is defined on an almost complex manifold with Norden metric. Necessary and sufficient conditions for these connections to be natural are obtained. A two-parametric family of complex connections is studied on a conformal Kähler manifo…