arXiv research

Study locally conformally balanced metrics on specific Lie algebras.

Study non-Kähler metrics on complex nilmanifolds, proving torus structure under certain conditions.

The abstract discusses conjectures about metrics on complex manifolds.

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…

Flow stabilizes on non-Kähler metrics near Calabi-Yau.

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 $\partial \overline \partial F^{n - 2} =0$. If $n =3$, then the metric is {\em strong KT}, i.e. $F$ is $\partial \overline \partial$-closed. By using blow-ups and the …

Study balanced Hermitian structures on almost abelian Lie algebras, classifying six-dimensional cases.

Study shows compact Vaisman manifolds cannot have certain special Hermitian metrics.

Classifies Kähler structures with special foliations using symplectic techniques.

Study on balanced Hermitian structures on Lie algebras twisted by representations.

The paper analyzes systoles of complex projective spaces under various metrics.

The paper solves a problem related to curvature in complex geometry.

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).

Characterizes complex Finsler metrics and their properties.

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…

Locally conformal SKT structures are introduced and studied on Lie groups and their compact quotients.

Study of complex structures on specific solvmanifolds, proving existence and non-existence results.

We study complex non-Kähler manifolds with Hermitian metrics being locally conformal to metrics with special cohomological properties. In particular, we provide examples where the existence of locally conformal holomorphic-tamed structures implies the existence of locally conformal Kähler metrics, too.

Constructs metrics with negative curvature on specific manifold types.

The paper explores Hermitian structures on tangent bundles of affine manifolds with Riemannian metrics.

We obtain an example of a compact locally conformal symplectic nilmanifold which admits no locally conformal Kähler metrics. This gives a new positive answer to a question raised by L. Ornea and M. Verbitsky.

In this note we study the conformal metrics of constant $Q$ curvature on closed locally conformally flat manifolds. We prove that for a closed locally conformally flat manifold of dimension $n\geq 5$ and with Poincarë exponent less than $\frac {n-4}2$, the set of conformal metrics of positive constant $Q$ and positive …

New compact examples of pseudo-Kähler manifolds with essential conformal transformations found.

Characterizes pluriclosed metrics on Oeljeklaus-Toma manifolds.

Local Lorentzian theorem preserves metrics or makes them flat.

Study of special Kato manifolds derived from toric geometry.

Adapted metrics found on complex manifolds.

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…

Study on special Hermitian metrics on cohomogeneity one manifolds.

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…

We review the properties of the Morse-Novikov cohomology and compute it for all known compact complex surfaces with locally conformally Kähler metrics. We present explicit computations for the Inoue surfaces $\mathcal{S}^0$, $\mathcal{S}^+$, $\mathcal{S}^-$ and classify the locally conformally Kähler (and the tamed loc…

Study connects two types of metrics on complex surfaces.

We show that every Kato surface (or surface with a global spherical shell) admits a locally conformally Kaehler metric.

We prove that the normal metric contact pairs with orthogonal characteristic foliations, which are either Bochner flat or locally conformally flat, are locally isometric to the Hopf manifolds. As a corollary we obtain the classification of locally conformally flat and Bochner-flat non-Kähler Vaisman manifolds.

A locally conformally Kähler (lcK) manifold is a complex manifold $(M,J)$ together with a Hermitian metric $g$ which is conformal to a Kähler metric in the neighbourhood of each point. In this paper we obtain three classification results in locally conformally Kähler geometry. The first one is the classification of con…

We investigate isometric immersions of locally conformally Kaehler metrics into Hopf manifolds. In particular, we study Hopf-induced metrics on compact complex surfaces.

There is a one-to-one correspondence between associated families of generic conformally flat (local-)hypersurfaces in 4-dimensional space forms and conformally flat 3-metrics with the Guichard condition. In this paper, we study the space of conformally flat 3-metrics with the Guichard condition: for a conformally flat …

Study local curvature estimates and existence of conformal metrics on noncompact manifolds.

In this paper, we establish some compactness results of conformally compact Einstein metrics on $4$-dimensional manifolds. Our results were proved under assumptions on the behavior of some local and non-local conformal invariants, on the compactness of the boundary metrics at the conformal infinity, and on the topology…

Paper surveys balanced metrics and proves a geodesic convexity result.

We study a fully nonlinear flow for conformal metrics. The long-time existence and the sequential convergence of flow are established for locally conformally flat manifolds. As an application, we solve the $\sk$-Yamabe problem for locally conformal flat manifolds when $k \neq n/2$.

We characterize the existence of a locally conformally Kähler metric on a compact complex manifold in terms of currents, adapting the celebrated result of Harvey and Lawson for Kähler metrics.

Unified flow approach to curvature problem on specific manifolds.

Our principal goal is to study the Prescribed Curvature Tensor problem in locally conformally flat manifolds. The solution to this problem is given explicitly for the special cases of the tensor R, including a case where the metric g is complete on Rn. Similar problems are considered for locally conformally flat manifo…

No locally conformally Kähler metrics found on Oeljeklaus-Toma manifolds.

We discuss a correspondence between certain contact pairs on the one hand, and certain locally conformally symplectic forms on the other. In particular, we characterize these structures through suspensions of contactomorphisms. If the contact pair is endowed with a normal metric, then the corresponding lcs form is loca…

Compact locally conformal Kähler manifolds with constant Chern holomorphic sectional curvature are necessarily Kähler.

The local classification of conformally flat Lorentzian manifolds with special holonomy groups is obtained. The corresponding local metrics are certain extensions of Riemannian spaces of constant sectional curvature to Walker metrics.