Einstein metrics on S^{2}-bundles

New Einstein metrics are constructed on the associated , , and -bundles of principal circle bundles with base a product of K?hler-Einstein manifolds with positive first Chern class and with Euler class a rational linear combination of the first Chern classes. These Einstein metrics represent different generalizations of the well-known Einstein metrics found by Bérard Bergery, D. Page, C. Pope, N. Koiso, and Y. Sakane. Corresponding new Einstein-Weyl structures are also constructed. Received 25 October 1996 / Revised version 1 April 1997     

On the existence of nontrivial extremal metrics on complete noncompact surfaces

In this paper, we consider the 2-dimensional local Calabi flow on a complete noncompact surface . Then, based on the Harnack-type estimate, we show the long-time existence and asymptotic convergence of a subsequence of solutions of such a flow on with and bounded from above by a negative constant on a ball. For its applications, this will lead to the existence of extremal metrics on a complete noncompact surface of finite topological type. In particular, there exists an extremal metric of nonconstant Gaussian curvature on or Received: 21 June 2001 / 18 January 2002 / Published online: 27 June 2002 Research supported in part by NSC and NCTS.     

Invariant metrics of positive Ricci curvature on principal bundles

Let be a compact connected Riemannian manifold with a metric of positive Ricci curvature. Let be a principal bundle over with compact connected structure group . If the fundamental group of is finite, we show that admits a invariant metric with positive Ricci curvature so that is a Riemannian submersion. Received 14 January 1997     

Isospectral deformations of metrics on spheres

We construct non-trivial continuous isospectral deformations of Riemannian metrics on the ball and on the sphere in R ⁿ for every n≥9. The metrics on the sphere can be chosen arbitrarily close to the round metric; in particular, they can be chosen to be positively curved. The metrics on the ball are both Dirichlet and Neumann isospectral and can be chosen arbitrarily close to the flat metric. Oblatum 19-VI-2000 & 21-II-2001?Published online: 4 May 2001     

On the geography of symplectic 6-manifolds

The article investigates the geography of closed, connected and simply connected, six-dimensional manifolds. It is proved that any triple of integers satisfying some necessary arithmetical restrictions occurs as the Chern triple of such a manifold. The main tools used for producing the examples are the symplectic connected sum and the symplectic blow-up. Received: 28 May 1998 / Revised version: 22 January 1999     

On the existence of closed timelike geodesics in compact spacetimes

In [19], Tipler has shown that a compact spacetime having a regular globally hyperbolic covering space with compact Cauchy surfaces necessarily contains a closed timelike geodesic. The restriction to compact spacetimes with just regular globally hyperbolic coverings (i.e., the Cauchy surfaces are not required to be compact) is still an open question. Here, we shall answer this question negatively by providing examples of compact flat Lorentz space forms without closed timelike geodesics, and shall give some criterion for the existence of such geodesics. More generally, we will show that in a compact spacetime having a regular globally hyperbolic covering, each free timelike homotopy class determined by a central deck transformation must contain a closed timelike geodesic. Whether or not a compact flat spacetime contains closed nonspacelike geodesics is, as far as we know, an open question. We shall answer this question affirmatively. We shall also introduce the notion of timelike injectivity radius for a spacetime relative to a free timelike homotopy class and shall show that it is finite whenever the corresponding deck transformation is central. Received: 9 November 1999; in final form: 19 September 2000 / Published online: 25 June 2001     

Kauffman bracket skein module of a connected sum of 3-manifolds

We show that for the Kauffman bracket skein module over the field of rational functions in variable A, the module of a connected sum of 3-manifolds is the tensor product of modules of the individual manifolds. Received: 12 January 1998 / Revised version: 15 September 1999     

Applications of geometric means on symmetric cones

Let V be a Euclidean Jordan algebra, and let be the corresponding symmetric cone. The geometric mean of two elements a and b in is defined as a unique solution, which belongs to of the quadratic equation where P is the quadratic representation of V. In this paper, we show that for any a in the sequence of iterate of the function defined by converges to a. As applications, we obtain that the geometric mean of can be represented as a limit of successive iteration of arithmetic means and harmonic means, and we derive the L?wner-Heinz inequality on the symmetric cone Furthermore, we obtain a formula which leads a Golden-Thompson type inequality for the spectral norm on V. Received October 5, 1999 / Revised March 6, 2000 / Published online October 30, 2000     

Finding Einstein solvmanifolds by a variational method

We use a variational approach to prove that any nilpotent Lie algebra having a codimension-one abelian ideal, and anyone of dimension , admits a rank-one solvable extension which can be endowed with an Einstein left-invariant riemannian metric. A curve of -dimensional Einstein solvmanifolds is also given. Received: 29 May 2001; in final form: 4 October 2001 / Published online: 4 April 2002     

The projective rank of a Hermitian symmetric space: a geometric approach and consequences

The “Projective Rank” of a compact connected irreducible Hermitian symmetric space M has been defined as the maximal complex dimension of the compact totally geodesic complex submanifolds having positive holomorphic bisectional curvature with the induced K?hler metric. We present a geometric way to compute this invariant for the space M based on ideas developed in [1], [13] and [14]. As a consequence we obtain the following inequality relating the Projective Rank, Pr(M), the usual rank,rk(M), and the 2-number # (which is known to be equal to the Euler-Poincare characteristic in these spaces). Received: 6 June 2000 / Published online: 1 February 2002     

A note on smooth toral reductions of spheres

In this paper we show that there exist mod 2 obstructions to the smoothness of 3-Sasakian reductions of spheres. Specifically, if is a smooth 3-Sasakian manifold obtained by reduction of the 3-Sasakian sphere S _{4 n -1} by a torus, and if the second Betti number then 7, 11, 15, whereas, if then . We also show that the above bounds are sharp, in that we construct explicit examples of 3-Sasakian manifolds in the cases not excluded by these bounds. Received: 6 January 1997 / Revised version: 11 June 1997     

Flat surfaces in the hyperbolic 3-space

In this paper we give a conformal representation of flat surfaces in the hyperbolic 3-space using the complex structure induced by its second fundamental form. We also study some examples and the behaviour at infinity of complete flat ends. Received: 18 September 1997     

Analyticity and smoothing effect for the Korteweg de Vries equation with a single point singularity

We show that a solution of the Cauchy problem for the KdV equation, has a drastic smoothing effect up to real analyticity if the initial data only have a single point singularity at x = 0. It is shown that for () data satisfying the condition the solution is analytic in both space and time variable. The above condition allows us to take as initial data the Dirac measure or the Cauchy principal value of 1/x. The argument is based on the recent progress on the well-posedness result by Bourgain [2] and Kenig-Ponce-Vega [20] and a systematic use of the dilation generator . Received 22 March 1999     

Concavity estimates for a class of nonlinear elliptic equations in two dimensions

We study solutions of the nonlinear elliptic equation on a bounded domain in . It is shown that the set of points where the graph of the solution has negative Gauss curvature always extends to the boundary, unless it is empty. The meethod uses an elliptic equation satisfied by an auxiliary function given by the product of the Hessian determinant and a suitable power of the solutions. As a consequence of the result, we give a new proof for power concavity of solutions to certain semilinear boundary value problems in convex domains. Received: 12 January 2000; in final form: 15 March 2001 / Published online: 4 April 2002     

Critical points of the total scalar curvature functional on the space of metrics of constant scalar curvature

It is well known that critical points of the total scalar curvature functional ? on the space of all smooth Riemannian structures of volume 1 on a compact manifold M are exactly the Einstein metrics. When the domain of ? is restricted to the space of constant scalar curvature metrics, there has been a conjecture that a critical point is also Einstein or isometric to a standard sphere. In this paper we prove that n-dimensional critical points have vanishing n− 1 homology under a lower Ricci curvature bound for dimension less than 8. Received: 12 July 1999