Annals of Global Analysis and Geometry最新文献

We study the nonexistence and rigidity of an important class of particular cases of trapped submanifolds, more precisely, n-dimensional spacelike mean curvature flow solitons related to the closed conformal timelike vector field (mathcal K=f(t)partial _t) ((tin Isubset mathbb R)) which is globally defined on an ((n+p+1))-dimensional generalized Robertson–Walker (GRW) spacetime (-Itimes _fM^{n+p}) with warping function (fin C^infty (I)) and Riemannian fiber (M^{n+p}), via applications of suitable generalized maximum principles and under certain constraints on f and on the curvatures of (M^{n+p}). In codimension 1, we also obtain new Calabi–Bernstein-type results concerning the spacelike mean curvature flow soliton equation in a GRW spacetime.

In this paper, we study the first (frac{2}{n})-stability eigenvalue on singular minimal hypersurfaces in space forms. We provide a characterization of catenoids in space forms in terms of (frac{2}{n})-stable eigenvalue. We emphasize that this result is even new in the regular setting.

Let (X, J) be a nilmanifold with an invariant nilpotent complex structure. We study the existence of p-Kähler structures (which include Kähler and balanced metrics) on X. More precisely, we determine an optimal p such that there are no p-Kähler structures on X. Finally, we show that, contrarily to the Kähler case, on compact complex manifolds there is no relation between the existence of balanced metrics and the degeneracy step of the Frölicher spectral sequence. More precisely, on balanced manifolds the degeneracy step can be arbitrarily large.

In the present paper, we consider star-shaped mean convex hypersurfaces of the real, complex and quaternionic hyperbolic space evolving by a class of nonhomogeneous expanding flows. For any choice of the ambient manifold, the initial conditions are preserved and the long-time existence of the flow is proved. The geometry of the ambient space influences the asymptotic behaviour of the flow: after a suitable rescaling, the induced metric converges to a conformal multiple of the standard Riemannian round metric of the sphere if the ambient manifold is the real hyperbolic space; otherwise, it converges to a conformal multiple of the standard sub-Riemannian metric on the odd-dimensional sphere. Finally, in every case, we are able to construct infinitely many examples such that the limit does not have constant scalar curvature.

For an orbifold, there are two naturally associated differential graded algebras, one is the de Rham algebra of orbifold differential forms and the other one is the differential graded algebra of piecewise polynomial differential forms of a triangulation of the coarse space. In this paper, we prove that these two differential graded algebras are weakly equivalent; hence, the formality of these two differential graded algebras is consistent, when the triangulation is smooth. We show that global quotient orbifolds and global homogeneous isotropy orbifolds admit smooth triangulations; hence, the two kinds of formality coincide with each other for these orbifolds.

This article studies hypoellipticity on general filtered manifolds. We extend the Rockland criterion to a pseudodifferential calculus on filtered manifolds, construct a parametrix and describe its precise analytic structure. We use this result to study Rockland sequences, a notion generalizing elliptic sequences to filtered manifolds. The main application that we present is to the analysis of the Bernstein–Gelfand–Gelfand (BGG) sequences over regular parabolic geometries. We do this by generalizing the BGG machinery to more general filtered manifolds (in a non-canonical way) and show that the generalized BGG sequences are Rockland in a graded sense.

The flat torus ({{mathbb T}}=mathbb S^1left( frac{1}{2} right) times mathbb S^1left( frac{1}{2} right) ) admits a proper biharmonic isometric immersion into the unit 4-dimensional sphere (mathbb S^4) given by (Phi =i circ varphi ), where (varphi :{{mathbb T}}rightarrow mathbb S^3(frac{1}{sqrt{2}})) is the minimal Clifford torus and (i:mathbb S^3(frac{1}{sqrt{2}}) rightarrow mathbb S^4) is the biharmonic small hypersphere. The first goal of this paper is to compute the biharmonic index and nullity of the proper biharmonic immersion (Phi ). After, we shall study in the detail the kernel of the generalised Jacobi operator (I_2^Phi ). We shall prove that it contains a direction which admits a natural variation with vanishing first, second and third derivatives, and such that the fourth derivative is negative. In the second part of the paper, we shall analyse the specific contribution of (varphi ) to the biharmonic index and nullity of (Phi ). In this context, we shall study a more general composition ({tilde{Phi }}=i circ {tilde{varphi }}), where ({tilde{varphi }}: M^m rightarrow mathbb S^{n-1}(frac{1}{sqrt{2}})), ( m ge 1), (n ge {3}), is a minimal immersion and (i:mathbb S^{n-1}(frac{1}{sqrt{2}}) rightarrow mathbb S^n) is the biharmonic small hypersphere. First, we shall determine a general sufficient condition which ensures that the second variation of ({tilde{Phi }}) is nonnegatively defined on (mathcal {C}big ({tilde{varphi }}^{-1}Tmathbb S^{n-1}(frac{1}{sqrt{2}})big )). Then, we complete this type of analysis on our Clifford torus and, as a complementary result, we obtain the p-harmonic index and nullity of (varphi ). In the final section, we compare our general results with those which can be deduced from the study of the equivariant second variation.

We prove that in a Finsler manifold with vanishing (chi )-curvature (in particular with constant flag curvature) some non-Riemannian geometric structures are geodesically invariant and hence they induce a set of non-Riemannian first integrals. Two alternative expressions of these first integrals can be obtained either in terms of the mean Berwald curvature, or as functions of the mean Cartan torsion and the mean Landsberg curvature.

In this article, we construct two one-parameter families of properly embedded minimal surfaces in a three-dimensional Lie group (widetilde{E(2)}), which is the universal covering of the group of rigid motions of Euclidean plane endowed with a left-invariant Riemannian metric. The first one can be seen as a family of helicoids, while the second one is a family of catenoidal minimal surfaces. The main tool that we use for the construction of these surfaces is a Weierstrass-type representation introduced by Meeks, Mira, Pérez and Ros for minimal surfaces in Lie groups of dimension three. In the end, we study the limit of the catenoidal minimal surfaces. As an application of this limit case, we get a new proof of a half-space theorem for minimal surfaces in (widetilde{E(2)}).

Suppose ((M^n, g, f)) is a complete shrinking gradient Ricci soliton. Assume that (|Ric|<frac{n-2}{2sqrt{n}}), where (n ge 3), then it has only one end. Similar results hold for the expanding gradient Ricci soliton.