Annals of Global Analysis and Geometry最新文献

We investigate the space of Hermitian metrics on a fixed complex vector bundle. This infinite-dimensional space has appeared in the study of Hermitian-Einstein structures, where a special (L^2)-type Riemannian metric is introduced. We compute the metric spray, geodesics and curvature associated to this metric, and show that the exponential map is a diffeomorphism. Though being geodesically complete, the space of Hermitian metrics is metrically incomplete, and its metric completion is proved to be the space of “(L^2) integrable” singular Hermitian metrics. In addition, both the original space and its completion are CAT(0). In the holomorphic case, it turns out that Griffiths seminegative/semipositive singular Hermitian metric is always (L^2) integrable in our sense. Also, in the Appendix, the Nash-Moser inverse function theorem is utilized to prove that, for any (L^2) metric on the space of smooth sections of a given fiber bundle, the exponential map is always a local diffeomorphism, provided that each fiber is nonpositively curved.

Let G be a Lie group equipped with a left-invariant Riemannian metric. Let K be a semisimple and normal subgroup of G generating a left-invariant conformal foliation (mathcal {F}) on G. We then show that the foliation (mathcal {F}) is Riemannian and minimal. This means that locally the leaves of (mathcal {F}) are fibres of a harmonic morphism. We also prove that if the metric restricted to K is biinvariant then (mathcal {F}) is totally geodesic.

We provide a rigorous analysis of the generalized Jang equation in the asymptotically anti-de Sitter setting modelled on constant time slices of anti-de Sitter spacetimes in dimensions (3le n le 7) for a very general class of asymptotics. Potential applications to spacetime positive mass theorems for asymptotically anti-de Sitter initial data sets are discussed.

For a small polarised deformation of a constant scalar curvature Kähler manifold, under some cohomological vanishing conditions, we prove that K-polystability along nearby polarisations implies the existence of a constant scalar curvature Kähler metric. In this setting, we reduce K-polystability to the computation of the classical Futaki invariant on the cscK degeneration. Our result holds on specific families and provides local wall-crossing phenomena for the moduli of cscK manifolds when the polarisation varies.

Jang has proven a remarkable classification of 6-dimensional manifolds having an almost complex circle action with 4 fixed points. Jang classifies the weights and associated multigraph into six cases, leaving the existence of connected manifolds fitting into two of the cases unknown. We show that one of the unknown cases may be constructed by a surgery construction of Kustarev, and the underlying manifold is diffeomorphic to (S^4 times S^2). We show that the action is not equivariantly diffeomorphic to a linear one, thus giving an exotic (S^1)-action of on a product of spheres that preserves an almost complex structure. We also prove a uniqueness statement for the almost complex structures produced by Kustarev’s construction and prove some topological applications of Jang’s classification.

We prove a transversality theorem for the moduli space of perturbed special Lagrangian submanifolds in a 6-manifold equipped with a generalization of a Calabi–Yau structure. These perturbed special Lagrangian submanifolds arise as solutions to an infinite-dimensional Lagrange multipliers problem which is part of a proposal for counting special Lagrangians outlined by Donaldson and Segal in [9]. More specifically, we prove that this moduli space is generically a set of isolated points.

In this note, we give a characterization of immersed submanifolds of simply-connected space forms for which the quotient of the extrinsic diameter by the focal radius achieves the minimum possible value of 2. They are essentially round spheres, or the “Veronese” embeddings of projective spaces. The proof combines the classification of submanifolds with planar geodesics due to K. Sakamoto with a version of A. Schur’s Bow Lemma for space curves. Open problems and the relation to recent work by M. Gromov and A. Petrunin are discussed.

Functionals involving surface curvatures are objects with applications in physics, mathematics, and related areas. It is then natural to study the minimizers of these functionals, as well as the stability of its critical points. In this paper, we begin by examining a general functional on n-dimensional hypersurfaces, which depends on the 1-mean curvature and the 2-mean curvature. We compute its first variation formula, obtaining the Euler-Lagrange equation that characterizes critical points. As a consequence, we also obtain the Euler-Lagrange equation for hypersurfaces immersed in Einstein manifolds as well as in manifolds with constant sectional curvature. In the case where the ambient space is a manifold with constant sectional curvature, we also compute the second variation, obtaining a stability criterion for these points in terms of geometric invariants that depend solely on the first and second fundamental forms. These results generalize those obtained in [7]. To demonstrate the applicability of the results, we studied the functional given by the (L^2)-norm of the traceless second fundamental form. From a geometric perspective, (Phi ) is a functional that measures how much M deviates from being totally umbilical, that is, from having equal principal curvatures at every point. We investigated the Euler-Lagrange equation and checked the stability of some known critical points.

We investigate the topology of the compact submanifolds in round spheres that satisfy a lower bound on the Ricci curvature depending only on the length of the mean curvature vector of the immersion. Just in special cases, the limited strength of the assumption allows some strong additional information on the extrinsic geometry of the submanifold.