p-harmonic 1-forms on complete manifolds

Let (M ^m , g) be a complete non-compact manifold with asymptotically non-negative Ricci curvature and finite first Betti number. We prove that any bounded set of p-harmonic 1-forms in L ^q (M), 0 < q < ∞, is relatively compact with respect to the uniform convergence topology.     

Closed holomorphic 1-forms without zeros on Stein manifolds

We prove that a stein mainfold admits a closed holomorphic 1-form without zeros in every class of the first cohomology group. We also prove an approximation result for closed holomorphic 1-forms without zeros defined in a neighborhood of a compact subset.     

Harmonic and holomorphic 1-forms on compact balanced Hermitian manifolds

On compact balanced Hermitian manifolds we obtain obstructions to the existence of harmonic 1-forms, ∂-harmonic (1,0)-forms and holomorphic (1,0)-forms in terms of the Ricci tensors with respect to the Riemannian curvature and the Hermitian curvature. Necessary and sufficient conditions the (1,0)-part of a harmonic 1-form to be holomorphic and vice versa, a real 1-form with a holomorphic (1,0)-part to be harmonic are found. The vanishing of the first Dolbeault cohomology groups of the twistor space of a compact irreducible hyper-Kähler manifold is shown.     

Conformal vector fields on Kaehler manifolds

It is known that a conformal vector field on a compact Kaehler manifold is a Killing vector field. In this paper, we are interested in finding conditions under which a conformal vector field on a non-compact Kaehler manifold is Killing. First we prove that a harmonic analytic conformal vector field on a 2n-dimensional Kaehler manifold (n ≠ 2) of constant nonzero scalar curvature is Killing. It is also shown that on a 2n-dimensional Kaehler Einstein manifold (n > 1) an analytic conformal vector field is either Killing or else the Kaehler manifold is Ricci flat. In particular, it follows that on non-flat Kaehler Einstein manifolds of dimension greater than two, analytic conformal vector fields are Killing.     

Conformal vector fields on some Finsler manifolds

We study conformal vector fields on a Finsler manifold whose metric is defined by a Riemannian metric, a 1-form and its norm. We find PDEs characterizing conformal vector fields. Then we obtain the explicit expressions of conformal vector fields for certain spherically symmetric metrics on R~n.     

Conformal Ricci flow on asymptotically hyperbolic manifolds

In this article, we study the short-time existence of conformal Ricci flow on asymptotically hyperbolic manifolds. We also prove a local Shi's type curvature derivative estimate for conformal Ricci flow.     

Conformal scalar curvature equations on complete manifolds

This note contains considerations on the existence and non-existence problem of conformal scalar curvature equations on some complete manifolds. We impose two general types of conditions on complete manifolds. The first type is in terms of bounds on curvature and injectivity radius. The second type is in terms of some particular structures on ends of manifolds, for examples, manifolds with cones or cusps and conformally compact manifolds. We obtain non-existence results on both types of conditions. Then we study in more details the existence problem on manifolds with cones, manifolds with cusps and conformally flat manifolds of bounded positive scalar curvature.     

The natural operators lifting 1-forms on manifolds to the bundles ofA-velocities

LetA be a Weil algebra withp variables. We prove that forn-manifolds (np+2) the set of all natural operatorsT ^*T ^* T ^A is a free finitely generated module over a ring canonically dependent onA. We construct explicitly the basis of this module.     

Conformal actions of on Lorentz manifolds

We prove that, for , a locally faithful action of or of by conformal transformations of a connected Lorentz manifold must be a proper action.

     

A class of 3-dimensional manifolds with bounded first eigenvalue on 1-forms

Let be the framebundle over an oriented, Riemannian surface . Denote by the first nonzero eigenvalue of the Laplace operator acting on differential forms of degree 1. We prove that for all with canonical metrics of volume 1.

     

Conformal reference frames for Lorentzian manifolds

We define a conformal reference frame, i.e., a special projection of the six-dimensional sky bundle of a Lorentzian manifold (or the five-dimensional twistor space) to a three-dimensional manifold. We construct an example, a conformal compactification, for Minkowski space. Based on the complex structure on the skies, we define the celestial transformation of Lorentzian vectors, a kind of spinor correspondence. We express a 1-form generating the contact structure in the twistor space (when it is smooth) explicitly as a form taking line-bundle values. We prove a theorem on the projection of this 1-form to the fiberwise normal bundle of a reference frame; its corollary is an equation for the flow of time.     

Continuous families of Riemannian manifolds, Isospectral on functions but not on 1-forms

The purpose of this paper is to present the first continuous families of Riemannian manifolds that are isospectral on functions but not on 1-forms, and, simultaneously, the first continuous families of Riemannian manifolds with the same marked length spectrum but not the same 1-form spectrum. Examples of isospectral manifolds that are not isospectral on forms are sparse, as most examples of isospectral manifolds can be explained by Sunada’s method or its generalizations, hence are strongly isospectral. The examples here are three-step Riemannian nilmanifolds, arising from a general method for constructing isospectral Riemannian nilmanifolds previously presented by the author. Gordon and Wilson constructed the first examples of nontrivial isospectral deformations, continuous families of Riemannian nilmanifolds. Isospectral manifolds constructed using the Gordon-Wilson method, a generalized Sunada method, are strongly isospectral and must have the same marked length spectrum. Conversely, Ouyang and Pesce independently showed that all isospectral deformations of two-step nilmanifolds must arise from the Gordon-Wilson method, and Eberlein showed that all pairs of two-step nilmanifolds with the same marked length spectrum must come from the Gordon-Wilson method. To the memory of Hubert Pesce, a valued friend and colleague.     

$$L^mathrm{2}$$ L 2 -transverse conformal Killing forms on complete foliated manifolds

Mathematische Zeitschrift - In this article, we study the $$L^2$$ -transverse conformal Killing forms on complete foliated Riemannian manifolds and prove some vanishing theorems. Also, we study the...     

A note on four-dimensional (anti-)self-dual quasi-Einstein manifolds

In this short note we prove that any complete four-dimensional anti-self-dual (or self-dual) quasi-Einstein manifold is either Einstein or locally conformally flat. This generalizes a recent result of X. Chen and Y. Wang.     

Conformal capacities and conformally invariant functions on Riemannian manifolds

This paper develops the theory of conformal invariants initiated inJ. Differential Geom 8 (1973), 487–510 for a Riemannian manifoldM with dimensionn2. We construct and study four conformally invariant functions M, M, M, M resp. depending on 4, 3 or 2 points onM, defined as extremal capacities for condensers associated with those points. These functions have similarities with the classical invariants onS ⁿ ,R ⁿ orH ⁿ . Their properties, and especially their continuity, are efficient tools for solving some problems of conformal geometry in the large.     

Killing vector fields of constant length on Riemannian manifolds

We study the nontrivial Killing vector fields of constant length and the corresponding flows on complete smooth Riemannian manifolds. Various examples are constructed of the Killing vector fields of constant length generated by the isometric effective almost free but not free actions of S ¹ on the Riemannian manifolds close in some sense to symmetric spaces. The latter manifolds include “almost round” odd-dimensional spheres and unit vector bundles over Riemannian manifolds. We obtain some curvature constraints on the Riemannian manifolds admitting nontrivial Killing fields of constant length.     

On conformal Killing symmetric tensor fields on Riemannian manifolds

A vector field on Riemannian manifold is called conformal Killing if it generates oneparameter group of conformal transformation. The class of conformal Killing symmetric tensor fields of an arbitrary rank is a natural generalization of the class of conformal Killing vector fields, and appears in different geometric and physical problems. In this paper, we prove that a trace-free conformal Killing tensor field is identically zero if it vanishes on some hypersurface. This statement is a basis of the theorem on decomposition of a symmetric tensor field on a compact manifold with boundary to a sum of three fields of special types. We also establish triviality of the space of trace-free conformal Killing tensor fields on some closed manifolds.