Annals of Global Analysis and Geometry最新文献

In this paper we consider the prescribed mean curvature flow of a non-compact space-like Cauchy hypersurface of bounded geometry in a generalized Robertson–Walker space-time. We prove that the flow preserves the space-likeness condition and exists for infinite time. We also prove convergence in the setting of manifolds with boundary. Our discussion generalizes previous work by Ecker, Huisken, Gerhardt and others with respect to a crucial aspects: we consider any non-compact Cauchy hypersurface under the assumption of bounded geometry. Moreover, we specialize the aforementioned works by considering globally hyperbolic Lorentzian space-times equipped with a specific class of warped product metrics.

The space of full-ranked one-forms on a smooth, orientable, compact manifold (possibly with boundary) is metrically incomplete with respect to the induced geodesic distance of the generalized Ebin metric. We show a distance equality between the induced geodesic distances of the generalized Ebin metric on the space of full-ranked one-forms and the corresponding Riemannian metric defined on each fiber. Using this result, we immediately have a concrete description of the metric completion of the space of full-ranked one-forms. Additionally, we study the relationship between the space of full-ranked one-forms and the space of all Riemannian metrics, leading to quotient structures for the space of Riemannian metrics and its completion.

Manifolds with fibered corners arise as resolutions of stratified spaces, as ‘many-body’ compactifications of vector spaces, and as compactifications of certain moduli spaces including those of non-abelian Yang–Mills–Higgs monopoles, among other settings. However, Cartesian products of manifolds with fibered corners do not generally have fibered corners themselves and thus fail to reflect the appropriate structure of products of the underlying spaces in the above settings. Here, we determine a resolution of the Cartesian product of fibered corners manifolds by blow-up which we call the ‘ordered product,’ which leads to a well-behaved category of fibered corners manifolds in which the ordered product satisfies the appropriate universal property. In contrast to the usual category of manifolds with corners, this category of fibered corners not only has all finite products, but all finite transverse fiber products as well, and we show in addition that the ordered product is a natural product for wedge (aka incomplete edge) metrics and quasi-fibered boundary metrics, a class which includes QAC and QALE metrics.

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.