Annals of Global Analysis and Geometry最新文献

Bounds of total curvature and entropy are two common conditions placed on mean curvature flows. We show that these two hypotheses are equivalent for the class of ancient complete embedded smooth planar curve shortening flows, which are one-dimensional mean curvature flows. As an application, we give a short proof of the uniqueness and classification of tangent flow at infinity of an ancient smooth complete non-compact curve shortening flow with finite entropy embedded in (mathbb {R}^2).

Let (Lambda ) be the unit tangent bundle of the unit 3-sphere acted on transitively by the contact group of Lie sphere transformations. We study the Lie sphere geometry of generic curves in (Lambda ) which are everywhere transversal to the contact distribution of (Lambda ). By the method of moving frames, we prove that such curves can be parametrized by a Lie-invariant parameter, the Lie arclength, and that in this parametrization they are uniquely determined, up to Lie sphere transformation, by four local invariants, the Lie curvatures. We then consider the simplest Lie-invariant functional on generic transversal curves defined by integrating the differential of the Lie arclength. The corresponding Euler–Lagrange equations are computed and the critical curves are characterized in terms of their Lie curvatures. In our discussion, we adopt Griffiths’ exterior differential systems approach to the calculus of variations.

In this paper, we study holomorphic vector bundles on the homogeneous varieties (G_2/P_1cong mathbb {Q}^5) and (G_2/P_2). We prove that if a rank 2 vector bundle E on (G_2/P_i~(i=1,2)) is uniform with respect to the special family of lines, then E is either a direct sum of line bundles or an indecomposable 2-bundle, which is unique up to twist. As a consequence, we give a new characterization of the Cayley bundles on (mathbb {Q}^5).

We show that for any Riemannian foliation with a simply connected and negatively curved leaf space the normal exponential map of a leaf is a diffeomorphism. As an application, if the leaves are furthermore minimal submanifolds, we give a sharp estimate for the bottom of the spectrum of such a Riemannian manifold. Our proof of the spectral estimate also yields an estimate for the bottom of the spectrum of the horizontal Laplacian.

We exploit an identity for the gradients of Laplacian eigenfunctions on compact homogeneous Riemannian manifolds with irreducible linear isotropy group to obtain asymptotically sharp universal eigenvalue inequalities and sharp Weyl bounds on Riesz means. The approach is non variational and is based on identities for spectral quantities in the form of sum rules.

We prove that there are relative ({textrm{SO}}_0(2,q))-character varieties of the punctured sphere which are compact, totally non-hyperbolic and contain a dense representation. This work fills a remaining case of the results of N. Tholozan and J. Toulisse. Our approach relies on the non-abelian Hodge correspondence and we study the moduli space of parabolic ({textrm{SO}}_0(2,q))-Higgs bundles with some fixed weight. Additionally, we provide a construction based on Geometric Invariant Theory (GIT) to demonstrate that the considered moduli spaces can be viewed as a projective variety over (mathbb {C}).

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.