Annals of Global Analysis and Geometry最新文献

We investigate Gauss maps associated to great circle fibrations of the euclidean unit 3-sphere (mathbb {S}^3). We show that the associated Gauss map to such a fibration is harmonic, respectively minimal, if and only if the unit vector field generating the great circle foliation is harmonic, respectively minimal. These results can be viewed as analogues of the classical theorem of Ruh and Vilms about the harmonicity of the Gauss map of a minimal submanifold in the euclidean space. Moreover, we prove that a harmonic or minimal unit vector field on (mathbb {S}^3), whose integral curves are great circles, is a Hopf vector field.

We consider a smooth, complete and non-compact Riemannian manifold ((mathcal {M},g)) of dimension (d ge 3), and we look for solutions to the semilinear elliptic equation

The potential (V :mathcal {M} rightarrow mathbb {R}) is a continuous function which is coercive in a suitable sense, while the nonlinearity f has a subcritical growth in the sense of Sobolev embeddings. By means of (nabla )-theorems introduced by Marino and Saccon, we prove that at least three non-trivial solutions exist as soon as the parameter (lambda ) is sufficiently close to an eigenvalue of the operator (-varDelta _g).

We show the existence of a natural Dirichlet-to-Neumann map on Riemannian manifolds with boundary and bounded geometry, such that the bottom of the Dirichlet spectrum is positive. This map regarded as a densely defined operator in the (L^2)-space of the boundary admits Friedrichs extension. We focus on the spectrum of this operator on covering spaces and total spaces of Riemannian principal bundles over compact manifolds.

The description of the Paley–Wiener space for compactly supported smooth functions (C^infty _c(G)) on a semi-simple Lie group G involves certain intertwining conditions that are difficult to handle. In the present paper, we make them completely explicit for (G=textbf{SL}(2,mathbb {R})^d) ((din mathbb {N})) and (G=textbf{SL}(2,mathbb {C})). Our results are based on a defining criterion for the Paley–Wiener space, valid for general groups of real rank one, that we derive from Delorme’s proof of the Paley–Wiener theorem. In a forthcoming paper, we will show how these results can be used to study solvability of invariant differential operators between sections of homogeneous vector bundles over the corresponding symmetric spaces.

Let X be a compact Thom–Mather stratified pseudomanifold, and let M be the regular part of X endowed with an iterated metric. In this paper, we prove that if the curvature operator of M is bounded, then the (L^2) harmonic space of M is finite dimensional. Next we consider the absolute eigenvalue problems of the Hodge Laplacian of a sequence of compact domains (Omega _j) converging to M. We prove that when the curvature operator of M is bounded, the eigenvalues of (Omega _j) converge to eigenvalues of M, and the eigenforms of (Omega _j) converge to eigenforms of M in the Sobolev norm. This generalizes Chavel and Feldman’s theorem in Chavel and Feldman (J Funct Anal 30:198-222, 1978) from compact manifolds to compact pseudomanifolds and from functions to differential forms. Then, we apply our results to (L^2)-chomology. We will give a correspondence between boundary cohomology and (L^2)-cohomology.

In this paper, we study the problem of prescribing a fourth-order conformal invariant on standard spheres. This problem is variational but it is noncompact due to the presence of nonconverging orbits of the gradient flow, the so called critical points at infinity. Following the method advised by Bahri we determine all such critical points at infinity and compute their contribution to the difference of topology between the level sets of the associated Euler–Lagrange functional. We then derive some existence results under pinching conditions.

Let (M=G/H) be a compact, simply connected, Riemannian homogeneous space, where G is (almost) effective and H is a simple Lie group. In this paper, we first classify all G-naturally reductive metrics on M, and then all G-geodesic orbit metrics on M.

We consider compact Kählerian manifolds X of even dimension 4 or more, endowed with a log-symplectic structure (Phi ), a generically nondegenerate closed 2-form with simple poles on a divisor D with local normal crossings. A simple linear inequality involving the iterated Poincaré residues of (Phi ) at components of the double locus of D ensures that the pair ((X, Phi )) has unobstructed deformations and that D deforms locally trivially.

The variational theory of higher-power energy is developed for mappings between Riemannian manifolds, and more generally sections of submersions of Riemannian manifolds, and applied to sections of Riemannian vector bundles and their sphere subbundles. A complete classification is then given for left-invariant vector fields on three-dimensional unimodular Lie groups equipped with an arbitrary left-invariant Riemannian metric.

The Umehara algebra is studied with motivation on the problem of the non-existence of common complex submanifolds. In this paper, we prove some new results in Umehara algebra and obtain some applications. In particular, if a complex manifolds admits a holomorphic polynomial isometric immersion to one indefinite complex space form, then it cannot admits a holomorphic isometric immersion to another indefinite complex space form of different type. Other consequences include the non-existence of the common complex submanifolds for indefinite complex projective space or hyperbolic space and a complex manifold with a distinguished metric, such as homogeneous domains, the Hartogs triangle, the minimal ball, and the symmetrized polydisc, equipped with their intrinsic Bergman metrics, which generalizes more or less all existing results.