Generalized Kahler BHE threefolds

Beatrice Brienza

Home/Archive/arXiv

arXiv · 2610.30130

Generalized Kahler BHE threefolds

Beatrice Brienza

View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the notion of generalized Kahler Bismut--Hermitian--Einstein (GKBHE) manifolds, namely generalized Kähler manifolds for which both Bismut Ricci forms vanish, and we develop a theory for such structures on compact threefolds. We organize the analysis according to the horizontal rank, an invariant taking values in $\{0,1,2\}$ that measures the relative position of the canonical foliations associated with the two Hermitian structures. We characterize the case of horizontal rank $0$ (which corresponds to Kähler Ricci flat geometry) and describe the case of horizontal rank $1$, revealing its close interplay with the steady $ν$-soliton equation in Kähler geometry. This description also provides a tool for constructing new compact GKBHE threefolds.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Beatrice Brienza. 2026-10-21. Generalized Kahler BHE threefolds. https://arxiv.org/abs/2610.30130

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Capillary minimal slicing and scalar curvature rigidity

We develop minimal slicing via capillary hypersurfaces to understand scalar curvature metric on manifolds bounded below with boundary. In particular, in dimension $4$, we prove following comparison and rigidity statement: given a compact Riemannian $4$-manifold $(M^4,g)$ with a mean convex boundary whose boundary is diffeomorphic to boundary of a connected convex domain in $\mathbb R^4$, if the scalar curvature is non-negative and the scaled mean curvature comparison holds along the boundary, then $M$ is isometric to the Euclidean domain. Moreover, we prove a comparison and rigidity result of manifolds with Listing type scaling invariant scalar and mean curvature comparison : $R g \ge -12 g_{\mathbb{H}^{4}}$ and $\hat{H}^{2} g \ge \hat{H}_{0} g_{\mathbb{H}^{4}}$, where $\hat{H}$ is a shifted mean curvature depending on the scale of manifold. This proves the traced version of Gromov's conjecture on comparison and rigidity of hyperbolic convex domains

math.DG↗

arXiv2026

Classification of Smooth Alignable Voss Surfaces

Alignable nets are grid structures that can collapse to a planar strip. This property simplifies on-site assembly and enables compact transport and storage. These structures are then deployed by a scissor motion at vertices in a desired location. In this article, we classify all non-developable surfaces supporting an alignable net that additionally have the geodesic-conjugate net property, namely, the alignable Voss surfaces. Using the surface elements and adapted frames, we show that these surfaces arise as rotation fields of either helicoidal pseudospherical surfaces or Amsler surfaces, resulting in two distinct classes. The first class consists of helicoidal Voss surfaces. Exploiting this helicoidal structure, we combine a kinematic interpretation with the Bour family to derive explicit immersion formulas depending on the isometric deformation and alignability parameters. Furthermore, we examine several subclasses and an extension of this result in detail. In particular, we study the notable case of infinitely many geodesic-conjugate nets on a helicoid and provide a kinematic explanation for why this phenomenon appears in computations. For the second class, we obtain the classification at the level of the fundamental forms and show that they correspond to certain $3$-parameter families of non-helicoidal intrinsic surfaces of revolution. These families are governed by general Painleve III solutions including both regular and nonregular solutions. Finally, a surprising feature of this class is that its members support geodesic isothermal-conjugate nets absent from Eisenhart's classification of the aforementioned nets, thereby identifying a missing case in that classification.

math.DG↗

arXiv2026

On weighted extremal Kähler metrics

The notion of weighted extremal Kähler metrics extends the classical notion of Calabi's extremal Kähler metrics, but includes many well-studied objects in Kähler geometry such as Kähler-Ricci solitons and Sasaki-Einstein metrics. In this paper, after explaining how this notion grew out, we will try to survey recent works concerning the YTD conjecture on weighted extremal Kähler metrics.

math.DG↗