Annals of Global Analysis and Geometry最新文献

We prove that the stability condition for Fano manifolds defined by Saito–Takahashi, given in terms of the sum of the Ding invariant and the Chow weight, is equivalent to the existence of anticanonically balanced metrics. Combined with the result by Rubinstein–Tian–Zhang, we obtain the following algebro-geometric corollary: the (delta _m)-invariant of Fujita–Odaka satisfies (delta _m >1) if and only if the Fano manifold is stable in the sense of Saito–Takahashi, establishing a Hilbert–Mumford-type criterion for (delta _m >1). We also extend this result to the Kähler–Ricci g-solitons and the coupled Kähler–Einstein metrics, and as a by-product we obtain a formula for the asymptotic slope of the coupled Ding functional in terms of multiple test configurations.

We introduce the coupled Ricci–Calabi functional and the coupled H-functional which measure how far a Kähler metric is from a coupled Kähler–Einstein metric in the sense of Hultgren–Witt Nyström. We first give corresponding moment weight type inequalities which estimate each functional in terms of algebraic invariants. Secondly, we give corresponding Hessian formulas for these functionals at each critical point, which have an application to a Matsushima type obstruction theorem for the existence of a coupled Kähler–Einstein metric.

Given a (C^1) function (mathcal {H}) defined in the unit sphere (mathbb {S}^2), an (mathcal {H})-surface M is a surface in the Euclidean space (mathbb {R}^3) whose mean curvature (H_M) satisfies (H_M(p)=mathcal {H}(N_p)), (pin M), where N is the Gauss map of M. Given a closed simple curve (Gamma subset mathbb {R}^3) and a function (mathcal {H}), in this paper we investigate the geometry of compact (mathcal {H})-surfaces spanning (Gamma ) in terms of (Gamma ). Under mild assumptions on (mathcal {H}), we prove non-existence of closed (mathcal {H})-surfaces, in contrast with the classical case of constant mean curvature. We give conditions on (mathcal {H}) that ensure that if (Gamma ) is a circle, then M is a rotational surface. We also establish the existence of estimates of the area of (mathcal {H})-surfaces in terms of the height of the surface.

We introduce and study a special class of almost contact metric manifolds, which we call anti-quasi-Sasakian (aqS). Among the class of transversely Kähler almost contact metric manifolds ((M,varphi , xi ,eta ,g)), quasi-Sasakian and anti-quasi-Sasakian manifolds are characterized, respectively, by the (varphi )-invariance and the (varphi )-anti-invariance of the 2-form (textrm{d}eta ). A Boothby–Wang type theorem allows to obtain aqS structures on principal circle bundles over Kähler manifolds endowed with a closed (2, 0)-form. We characterize aqS manifolds with constant (xi )-sectional curvature equal to 1: they admit an (Sp(n)times 1)-reduction of the frame bundle such that the manifold is transversely hyperkähler, carrying a second aqS structure and a null Sasakian (eta )-Einstein structure. We show that aqS manifolds with constant sectional curvature are necessarily flat and cokähler. Finally, by using a metric connection with torsion, we provide a sufficient condition for an aqS manifold to be locally decomposable as the Riemannian product of a Kähler manifold and an aqS manifold with structure of maximal rank. Under the same hypothesis, (M, g) cannot be locally symmetric.

Pittie (Proc Indian Acad Sci Math Sci 98:117-152, 1988) proved that the Dolbeault cohomology of all left-invariant complex structures on compact Lie groups can be computed by looking at the Dolbeault cohomology induced on a conveniently chosen maximal torus. We generalized Pittie’s result to left-invariant Levi-flat CR structures of maximal rank on compact Lie groups. The main tools we used was a version of the Leray–Hirsch theorem for CR principal bundles and the algebraic classification of left-invariant CR structures of maximal rank on compact Lie groups (Charbonnel and Khalgui in J Lie Theory 14:165-198, 2004) .

This paper introduces a notion of decompositions of integral varifolds into countably many integral varifolds, and the existence of such decomposition of integral varifolds whose first variation is representable by integration is established. However, the decompositions may fail to be unique. Furthermore, this result can be generalized by replacing the class of integral varifolds with some classes of rectifiable varifolds whose density is uniformly bounded from below; for these classes, we also prove a general version of the compactness theorem for integral varifolds.

We present a construction of closed 7-manifolds of holonomy (G_2), which generalises Kovalev’s twisted connected sums by taking quotients of the pieces in the construction before gluing. This makes it possible to realise a wider range of topological types, and Crowley, Goette and the author use this to exhibit examples of closed 7-manifolds with disconnected moduli space of holonomy (G_2) metrics.

We solve a super Toda system on a closed Riemann surface of genus (gamma >1) and with some particular spin structures. This generalizes the min–max methods and results for super Liouville equations and gives new existence results for super Toda systems.

In 2012, M. M. Graev associated to a compact homogeneous space G/H a nerve ({text {X}}_{G/H}), whose non-contractibility implies the existence of a G-invariant Einstein metric on G/H. The nerve ({text {X}}_{G/H}) is a compact, semi-algebraic set, defined purely Lie theoretically by intermediate subgroups. In this paper we present a detailed description of the work of Graev and the curvature estimates given by Böhm in 2004.