Bach-flat noncompact steady quasi-Einstein manifolds

The goal of this article is to study the geometry of Bach-flat noncompact steady quasi-Einstein manifolds. We show that a Bach-flat noncompact steady quasi-Einstein manifold \((M^{n},\,g)\) with positive Ricci curvature such that its potential function has at least one critical point must be a warped product with Einstein fiber. In addition, the fiber has constant curvature if \(n = 4\).     

Static flow on complete noncompact manifolds I: short-time existence and asymptotic expansions at conformal infinity

We study short-time existence of static flow on complete noncompact asymptotically static manifolds from the point of view that the stationary points of the evolution equations can be interpreted as static solutions of the Einstein vacuum equations with negative cosmological constant.For a static vacuum(Mn,g,V),we also compute the asymptotic expansions of g and V at conformal infinity.     

On Einstein Hermitian Surfaces

We show that every compact Einstein Hermitian surface with constant conformal scalar curvature is a Kahler surface and that, in contrast to the compact case, there exits a noncompact Einstein Hermitian and non-Kahler surface with constant conformal scalar curvature.     

An index formula on manifolds with fibered cusp ends

We consider a compact manifold X whose boundary is a locally trivial fiber bundle, and an associated pseudodifferential algebra that models fibered cusps at infinity. Using tracelike functionals that generate the 0-dimensional Hochschild cohomology groups we first express the index of a fully elliptic fibered cusp operator as the sum of a local contribution from the interior of X and a term that comes from the boundary. This leads to an abstract answer to the index problem formulated in [11]. We give a more precise answer for firstorder differential operators when the base of the boundary fiber bundle is S¹. In particular, for Dirac operators associated to a metric of the form near ∂X = {x = 0} with twisting bundle T we obtain

Locally minimizing harmonic maps from noncompact manifolds

By introducing the “relative energy”, we develop a new method for finding harmonic maps from noncompact complete Riemannian manifolds with prescribed asympototic behaviour at infinity. This method is an extension of the well known direct method of energy-minimization for compact domains. As an application of our method, we show that the Dirichlet problem at infinity with Hölder continuous boundary data for harmonic maps from a Cartan-Hadarmard manifold with bounded negative curvature into a compact manifold, has a locally minimizing solution which is smooth near infinity.     

Solvability of some classes of nonlinear integro-differential equations with noncompact operator

For a class of nonlinear integrodifferential equations with a noncompact Urysohn-type operator we prove the existence of nonnegative bounded solutions. We study the asymptotic behavior of solutions at infinity. We give some examples that are of practical interest.     

A note on gradient Einstein-type manifolds

In this note, we show that a nontrivial, compact, degenerate or nondegenerate, gradient Einstein-type manifold of constant scalar curvature is isometric to the standard sphere with a well defined potential function. Moreover, under some geometric assumptions the noncompact case is also treated. In this case, the main result is that a homogeneous, proper, noncompact, nondegenerate, gradient Einstein-type manifold is an Einstein manifold.     

Extremal of Log Sobolev inequality and W entropy on noncompact manifolds

Let M be a complete, connected noncompact manifold with bounded geometry. Under a condition near infinity, we prove that the Log Sobolev functional (1.1) has an extremal function decaying exponentially near infinity. We also prove that an extremal function may not exist if the condition is violated. This result has the following consequences. 1. It seems to give the first example of connected, complete manifolds with bounded geometry where a standard Log Sobolev inequality does not have an extremal. 2. It gives a negative answer to the open question on the existence of extremal of Perelman?s W entropy in the noncompact case, which was stipulated by Perelman (2002) [22, p. 9, 3.2 Remark]. 3. It helps to prove, in some cases, that noncompact shrinking breathers of Ricci flow are gradient shrinking solitons.     

Topology at infinity of polynominal mappings and Thom regularity condition

We consider polynomial mappings which have atypical fibres due to the asymptotic behavior at infinity. Fixing some proper extension of the polynomial mapping, we study the localizability at infinity of the variation of topology of fibres and the possibility of interpreting local results at infinity into global results. We prove local and global Bertini–Sard–Lefschetz type statements for noncompact spaces and nonproper mappings and we deduce results on the homotopy type or the connectivity of the fibres of polynomial mappings.     

On the Underlying Lower Order Bundle Functors

For every bundle functor we introduce the concept of subordinated functor. Then we describe subordinated functors for fiber product preserving functors defined on the category of fibered manifolds with m-dimensional bases and fibered manifold morphisms with local diffeomorphisms as base maps. In this case we also introduce the concept of the underlying functor. We show that there is an affine structure on fiber product preserving functors. Tis work was supported by a grant of the GA CR No. 201/02/0225.     

Geometry and quantum transport

This is abrief and informal introduction to a differential geometric interpetation of adiabatic charge transport in quantum mechanics. It involves the study of afamily of Schrödinger operators. For compact multiply connected surfaces the charge transported around the “holes” is related to the first Chern character of spectral bundles. For noncompact surfaces the charge transported to infinity is related to the index of a certain Fredholm operator which involves the comparison of appropriate spectral projections. There are also relations to Connes noncommutative differential geometry. Simple examples are given.     

On the construction of a nonnegative solution for one class of Urysohn-type nonlinear integral equations on a semiaxis

We investigate one class of Urysohn-type nonlinear integral equations with noncompact operator. It is assumed that a Wiener–Hopf–Hankel-type linear integral operator is a local minorant for the initial Urysohn operator. We prove an alternative theorem on the existence of positive solutions and investigate the asymptotic behavior of the obtained solutions at infinity.     

Graham-Witten's Conformal Invariant for Closed Four Dimensional Submanifolds

It was proved by Graham and Witten in 1999 that conformal invariants of submanifolds can be obtained via volume renormalization of minimal surfaces in conformally compact Einstein manifolds. The conformal invariant of a submanifold $\Sigma$ is contained in the volume expansion of the minimal surface which is asymptotic to $\Sigma$ when the minimal surface approaches the conformaly infinity. In the paper we give the explicit expression of Graham-Witten's conformal invariant for closed four dimensional submanifolds and find critical points of the conformal invariant in the case of Euclidean ambient spaces.     

Universal hyperbolic geometry I: trigonometry

Hyperbolic geometry is developed in a purely algebraic fashion from first principles, without a prior development of differential geometry. The natural connection with the geometry of Lorentz, Einstein and Minkowski comes from a projective point of view, with trigonometric laws that extend to ‘points at infinity’, here called ‘null points’, and beyond to ‘ideal points’ associated to a hyperboloid of one sheet. The theory works over a general field not of characteristic two, and the main laws can be viewed as deformations of those from planar rational trigonometry. There are many new features; this paper gives 92 foundational theorems.     

K?hler–Einstein metrics on strictly pseudoconvex domains

Extending the results of Cheng and Yau it is shown that a strictly pseudoconvex domain ${M\subset X}$ in a complex manifold carries a complete K?hler–Einstein metric if and only if its canonical bundle is positive, i.e. admits an Hermitian connection with positive curvature. We consider the restricted case in which the CR structure on ${\partial M}$ is normal. In this case M must be a domain in a resolution of the Sasaki cone over ${\partial M}$ . We give a condition on a normal CR manifold which it cannot satisfy if it is a CR infinity of a K?hler–Einstein manifold. We are able to mostly determine those normal CR three-manifolds which can be CR infinities. We give many examples of K?hler–Einstein strictly pseudoconvex manifolds on bundles and resolutions. In particular, the tubular neighborhood of the zero section of every negative holomorphic vector bundle on a compact complex manifold whose total space satisfies c ₁?<?0 admits a complete K?hler–Einstein metric.     

Asymptotic behaviour of families of real curves

We consider the family of fibres of a polynomial function f on a smooth noncompact algebraic real surface and we characterise the regular fibres of f which are atypical due to their asymptotic behaviour at infinity. We compare to the similar problem in the complex case. Received: 5 May 1998 / Revised version: 20 March 1999     

Normalized Ricci flows and conformally compact Einstein metrics

In this paper, we investigate the behavior of the normalized Ricci flow on asymptotically hyperbolic manifolds. We show that the normalized Ricci flow exists globally and converges to an Einstein metric when starting from a non-degenerate and sufficiently Ricci pinched metric. More importantly we use maximum principles to establish the regularity of conformal compactness along the normalized Ricci flow including that of the limit metric at time infinity. Therefore we are able to recover the existence results in Graham and Lee (Adv Math 87:186–255, 1991), Lee (Fredholm Operators and Einstein Metrics on Conformally Compact Manifolds, 2006), and Biquard (Surveys in Differential Geometry: Essays on Einstein Manifolds, 1999) of conformally compact Einstein metrics with conformal infinities which are perturbations of that of given non-degenerate conformally compact Einstein metrics.     

Noncompact Homogeneous Einstein 5-Manifolds

This article is devoted to the classification of noncompact homogeneous Einstein 5-manifolds. In particular, we prove that each noncompact homogeneous Einstein 5-manifolds is locally isometric to some standard Einstein solvmanifoldMathematics Subject Classifications (2000). 53C25, 53C30     

Resolvents of cone pseudodifferential operators,asymptotic expansions and applications

We study the structure and asymptotic behavior of the resolvent of elliptic cone pseudodifferential operators acting on weighted Sobolev spaces over a compact manifold with boundary. We obtain an asymptotic expansion of the resolvent as the spectral parameter tends to infinity, and use it to derive corresponding heat trace and zeta function expansions as well as an analytic index formula.      

The log-term of the Bergman kernel of the disc bundle over a homogeneous Hodge manifold

We show the vanishing of the log-term in the Fefferman expansion of the Bergman kernel of the disk bundle over a compact simply-connected homogeneous Kähler–Einstein manifold of classical type. Our results extends that in (Engli? and Zhang, Math Z 264(4):901–912, 2010) for the case of Hermitian symmetric spaces of compact type.