Simultaneous metric uniformization of foliations by Riemann surfaces

We consider a two-dimensional linear foliation on torus of arbitrary dimension. For any smooth family of complex structures on the leaves we prove existence of smooth family of uniformizing (conformal complete flat) metrics on the leaves. We extend this result to linear foliations on and families of complex structures with bounded derivatives C ³-close to the standard complex structure. We prove that the analogous statement for arbitrary C two-dimensional foliation on compact manifold is wrong in general, even for suspensions over in dimension 3 the uniformizing metric can be nondifferentiable at some points; in dimension 4 the uniformizing metric of each noncompact leaf can be unbounded.     

Continuity of Dirac spectra

It is well known that on a bounded spectral interval the Dirac spectrum can be described locally by a non-decreasing sequence of continuous functions of the Riemannian metric. In the present article, we extend this result to a global version. We view the spectrum of a Dirac operator as a function $\mathbb Z \,\rightarrow \mathbb R \,$ and endow the space of all spectra with an $\mathrm{arsinh }$ -uniform metric. We prove that the spectrum of the Dirac operator depends continuously on the Riemannian metric. As a corollary, we obtain the existence of a non-decreasing family of functions on the space of all Riemannian metrics, which represents the entire Dirac spectrum at any metric. We also show that, due to spectral flow, these functions do not descend to the space of Riemannian metrics modulo spin diffeomorphisms in general.     

On Einstein Matsumoto metrics

We study a special class of Finsler metrics,namely,Matsumoto metrics F=α2α-β,whereαis a Riemannian metric andβis a 1-form on a manifold M.We prove that F is a(weak)Einstein metric if and only ifαis Ricci flat andβis a parallel 1-form with respect toα.In this case,F is Ricci flat and Berwaldian.As an application,we determine the local structure and prove the 3-dimensional rigidity theorem for a(weak)Einstein Matsumoto metric.     

Scalar Curvature Functions of Almost-Kähler Metrics

For a closed smooth manifold M admitting a symplectic structure, we define a smooth topological invariant Z(M) using almost-Kähler metrics, i.e., Riemannian metrics compatible with symplectic structures. We also introduce \(Z(M, [[\omega ]])\) depending on symplectic deformation equivalence class \([[\omega ]]\). We first prove that there exists a 6-dimensional smooth manifold M with more than one deformation equivalence class with different signs of \(Z(M, [[\omega ]] )\). Using Z invariants, we set up a Kazdan–Warner type problem of classifying symplectic manifolds into three categories. We finally prove that on every closed symplectic manifold \((M, \omega )\) of dimension \(\ge \!\!4\), any smooth function which is somewhere negative and somewhere zero can be the scalar curvature of an almost-Kähler metric compatible with a symplectic form which is deformation equivalent to \(\omega \).     

On Square metrics of scalar flag curvature

In this paper, we study the problem whether a Finsler metric of scalar flag curvature is locally projectively flat. We consider a special class of Finsler metrics — square metrics which are defined by a Riemannian metric and a 1-form on a manifold. We show that in dimension n ≥ 3, any square metric of scalar flag curvature is locally projectively flat.     

On totally geodesic foliations with bundle-like metric

Let be a totally geodesic foliation of dimension n and codimension p on a Riemannian manifold (M, g). Suppose that g is a bundle-like metric for and M has at least one point at which none of its mixed sectional curvatures vanishes. Under these conditions we prove that n ≤ p − 1. We show that this inequality is optimal, and none of the above conditions can be removed.     

Autodual Einstein Versus Kähler–Einstein

Any strictly pseudoconvex domain in carries a complete Kähler-Einstein metric, the Cheng–Yau metric, with “conformal infinity” the CR structure of the boundary.It is well known that not all CR structures on S³ arise in this way. In this paper, we study CR structures on the 3-sphere satisfying a different filling condition: boundaries at infinity of (complete) selfdual Einstein metrics. We prove that (modulo contactomorphisms) they form an infinite dimensional manifold, transverse to the space of CR structures which are boundaries of complex domains (and therefore of Kähler–Einstein metrics).Received: March 2004 Revision: July 2004 Accepted: August 2004     

Compactness for conformal metrics with constant Q curvature on locally conformally flat manifolds

In this note we study the conformal metrics of constant Q curvature on closed locally conformally flat manifolds. We prove that for a closed locally conformally flat manifold of dimension n ≥ 5 and with Poincaré exponent less than , the set of conformal metrics of positive constant Q and positive scalar curvature is compact in the C∞ topology.     

Hermitian structures on six-dimensional nilmanifolds

Let (J,g) be a Hermitian structure on a six-dimensional compact nilmanifold M with invariant complex structure J and compatible metric g, which is not required to be invariant. We show that, up to equivalence of the complex structure, the strong Kahler with torsion structures (J,g) on M are parametrized by the points in a subset of the Euclidean space, in particular, the region inside a certain ovaloid corresponds to such structures on the Iwasawa manifold and the region outside to strong Kahler with torsion structures with nonabelian J on the nilmanifold where H³ is the Heisenberg group. A classification of six-dimensional nilmanifolds admitting balanced Hermitian structures (J,g) is given, and as an application we classify the nilmanifolds having invariant complex structures which do not admit Hermitian structure with restricted holonomy of the Bismut connection contained in SU(3). It is also shown that on the nilmanifold the balanced condition is not stable under small deformations. Finally, we prove that a compact quotient of where H(2,1) is the five-dimensional generalized Heisenberg group, is the only six-dimensional nilmanifold having locally conformal Kahler metrics, and the complex structures underlying such metrics are all equivalent. Moreover, this nilmanifold is a Vaisman manifold for any invariant locally conformal Kahler metric.     

Generic metrics and the mass endomorphism on spin three-manifolds

Let (M, g) be a closed Riemannian spin manifold. The constant term in the expansion of the Green function for the Dirac operator at a fixed point \({p\in M}\) is called the mass endomorphism in p associated to the metric g due to an analogy to the mass in the Yamabe problem. We show that the mass endomorphism of a generic metric on a three-dimensional spin manifold is nonzero. This implies a strict inequality which can be used to avoid bubbling-off phenomena in conformal spin geometry.     

First eigenvalues of geometric operators under the Yamabe flow

Suppose \((M,g_0)\) is a compact Riemannian manifold without boundary of dimension \(n\ge 3\). Using the Yamabe flow, we obtain estimate for the first nonzero eigenvalue of the Laplacian of \(g_0\) with negative scalar curvature in terms of the Yamabe metric in its conformal class. On the other hand, we prove that the first eigenvalue of some geometric operators on a compact Riemannian manifold is nondecreasing along the unnormalized Yamabe flow under suitable curvature assumption. Similar results are obtained for manifolds with boundary and for CR manifold.     

关于切丛和单位切球丛的度量的一个注记

本文在黎曼流形$(M,g)$的切丛$TM$ 上研究与参考文献[10]中平行的一类度量$G$以及相容的近复结构$J$.证明了切丛$TM$关于这些度量和相应的近复结构是局部共形近K\"{a}hler流形,并且把这些结构限制在单位切球丛上得到了切触度量结构的新例子.     

An ampleness criterion with the extendability of singular positive metrics

Coman, Guedj and Zeriahi proved that, for an ample line bundle L on a projective manifold X, any singular positive metric on the line bundle L| _V along a subvariety ${V \subset X}$ can be extended to a global singular positive metric on L. In this paper, we prove that the extendability of singular positive metrics on a line bundle along a subvariety implies the ampleness of the line bundle.     

Deforming metrics with negative curvature by a fully nonlinear flow

By studying a fully nonlinear flow deforming conformal metrics on compact and connected manifold, we prove that for , any metric g with its modified Schouten tensor always can be deformed in a natural way to a conformal metric with constant -scalar curvature at exponential rate.Received: 2 December 2003, Accepted: 10 May 2004, Published online: 16 July 2004Jiayu Li: Supported in part by a grant from the National Science Foundation of China. Weimin Sheng: Supported by the Zhejiang Provincial Natural Science Foundation of China (No.102033).     

Vortex type equations and canonical metrics

We introduce a notion of Gieseker stability for a filtered holomorphic vector bundle over a projective manifold. We relate it to an analytic condition in terms of hermitian metrics on coming from a construction of the Geometric Invariant Theory (G.I.T). We prove that if there is a τ-Hermite-Einstein metric h _HE on , then there exists a sequence of such balanced metrics that converges and its limit is h _HE . As a corollary, we obtain an approximation theorem for quiver Vortex equations and other classical equations.     

The Sasaki Join,Hamiltonian 2-Forms,and Constant Scalar Curvature

We describe a general procedure for constructing new explicit Sasaki metrics of constant scalar curvature (CSC), including Sasaki–Einstein metrics, from old ones. We begin by taking the join of a regular Sasaki manifold of dimension \(2n+1\) and constant scalar curvature with a weighted Sasakian 3-sphere. Then by deforming in the Sasaki cone we obtain CSC Sasaki metrics on compact Sasaki manifolds \(M_{l_1,l_2,\mathbf{w}}\) of dimension \(2n+3\) which depend on four integer parameters \(l_1,l_2,w_1,w_2\). Most of the CSC Sasaki metrics are irregular. We give examples which show that the CSC rays are often not unique on the underlying fixed strictly pseudoconvex CR manifold. Moreover, it is shown that when the first Chern class of the contact bundle vanishes, there is a two-dimensional subcone of Sasaki–Ricci solitons in the Sasaki cone, and a unique Sasaki–Einstein metric in each of the two-dimensional subcones.     

Einstein–Weyl structures on contact metric manifolds

In this paper we study Einstein-Weyl structures in the framework of contact metric manifolds. First, we prove that a complete K-contact manifold admitting both the Einstein-Weyl structures W ^± = (g, ±ω) is Sasakian. Next, we show that a compact contact metric manifold admitting an Einstein-Weyl structure is either K-contact or the dual field of ω is orthogonal to the Reeb vector field, provided the Reeb vector field is an eigenvector of the Ricci operator. We also prove that a contact metric manifold admitting both the Einstein-Weyl structures and satisfying is either K-contact or Einstein. Finally, a couple of results on contact metric manifold admitting an Einstein-Weyl structure W = (g, f η) are presented.      

On the Completions of the Spaces of Metrics on an Open Manifold

We construct a complete metric d₀ on the space of continuous complete Riemannian metrics on a smooth manifold X of dimension n. Using that metric, we are able to show, that the space ^b,m?(X) defined in [1] is complete when suppplied with the uniform structure defined in the same paper.     

Stable bundles and the first eigenvalue of the Laplacian

In this article we study the first eigenvalue of the Laplacian on a compact manifold using stable bundles and balanced bases. Our main result is the following: Let M be a compact Kähler manifold of complex dimension n and E a holomorphic vector bundle of rank r over M. If E is globally generated and its Gieseker point T_e is stable, then for any Kähler metric g on M\(\lambda _1 (M,g) \leqslant \frac{{4\pi h^0 (E)}}{{r(h^0 (E) - r)}} \cdot \frac{{\left\langle {C_1 (E) \cup [\omega ]^{n - 1} ,[M]} \right\rangle }}{{(n - 1)!vol(M,[\omega ])}}\) where ω = ω_g is the Kähler form associated to g.By this method we obtain, for example, a sharp upper bound for λ₁ of Kähler metrics on complex Grassmannians.     

Surgery and harmonic spinors

Let M be a compact spin manifold with a chosen spin structure. The Atiyah-Singer index theorem implies that for any Riemannian metric on M the dimension of the kernel of the Dirac operator is bounded from below by a topological quantity depending only on M and the spin structure. We show that for generic metrics on M this bound is attained.