Compactness theorems and the Calabi flow on Kähler surfaces with stable tangent bundle

Abstract. In this paper, we prove some compactness theorems and collapse phenomenon on compact K?hler surfaces with stable tangent bundle. We then apply the results to the Calabi flow. More precisely, we prove, under suitable curvature conditions, the longtime existence and asymptotic convergence for solutions of the Calabi flow on compact K?hler surfaces admitting no nonzero holomorphic tangent vector fields and with stable tangent bundle. We also give some examples where the Calabi flow blows up. Received January 7, 1999 / Revised February 2, 2000 / Published online July 20, 2000     

Maximum principles, uniqueness and existence for harmonic maps with potential and Landau-Lifshitz equations

In this paper, we consider the harmonic maps with potential from compact Riemannian manifold with boundary into a convex ball in any Riemannian manifold. We will establish some general properties such as the maximum principles, uniqueness and existence for these maps, and as an application of them, we derive existence and uniqueness result for the Dirichlet problem of the Landau-Lifshitz equations. Received: December 10, 1997 / Accepted: June 29, 1998     

Projections in free product C$^*$–algebras, II

Let M be a connected sum of complete Riemannian manifolds satisfying the volume doubling condition and the Poincaré inequality. We prove that the space of polynomial growth harmonic functions on M is finite dimensional whenever M has finitely many ends and satisfies the finite covering condition on each end. This result directly generalizes that of Tam, and it also partially generalizes that of Colding and Minicozzi II. Received July 17, 1998; in final form December 2, 1998     

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     

Continua of H-graphs: convexity and isoperimetric stability

In this paper we consider nonparametric solutions of the volume constrained Plateau problem with respect to a convex planar curve. Existence and regularity is obtained from standard elliptic theory, and convexity results for small volumes are obtained as an immediate consequence. Finally, the regularity is applied to show a strong stability condition (Theorem 8) for all volumes considered. This condition, in turn, allows us to adapt an argument of Cabré and Chanillo [CC97] which yields that any solution enclosing a non-zero volume has a unique nondegenerate critical point. Received March 25, 1998 / Accepted December 1, 1998     

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     

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     

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     

Rough isometry and Dirichlet finite harmonic functions on Riemannian manifolds

We prove that the dimension of harmonic functions with finite Dirichlet integral is invariant under rough isometries between Riemannian manifolds satisfying the local conditions, expounded below. This result directly generalizes those of Kanai, of Grigor'yan, and of Holopainen. We also prove that the dimension of harmonic functions with finite Dirichlet integral is preserved under rough isometries between a Riemannian manifold satisfying the same local conditions and a graph of bounded degree; and between graphs of bounded degree. These results generalize those of Holopainen and Soardi, and of Soardi, respectively. Received: 23 July 1998 / Revised version: 10 February 1999     

On height functions on Jacobian surfaces

The canonical height on an abelian variety is useful and important for the study of the Mordell-Weil group. But it is difficult to calculate the canonical height in general. We give an effective method to calculate the canonical height on a Jacobian surface. As an application, we verify the Birch-Swinnerton-Dyer conjecture for the Jacobian surface of a twisted modular curve. Received: 15 July 1996 / Revised version: 19 January 1997     

A Bernstein theorem for complete spacelike constant mean curvature hypersurfaces in Minkowski space

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As an application, we prove a Bernstein theorem which says that if the image of the Gauss map is bounded from one side, then the spacelike constant mean curvature hypersurface must be linear. This result extends the previous theorems obtained by B. Palmer [Pa] and Y.L. Xin [Xin1] where they assume that the image of the Gauss map is bounded. We also prove a Bernstein theorem for spacelike complete surfaces with parallel mean curvature vector in four-dimensional spaces. Received July 4, 1997 / Accepted October 9, 1997     

Liouville theorem for harmonic maps with potential

Let M, N be complete manifolds, u:M→N be a harmonic map with potential H, namely, a critical point of the functional , where e(u) is the energy density of u. We will give a Liouville theorem for u with a class of potentials H's. Received: Received: 10 July 1997     

Total excess on length surfaces

We study the space of directions on a length space and examine examples having particular spaces of directions. Then we generalize the notion of total excess on length spaces satisfying some suitable conditions, which we call good surfaces. For good surfaces we generalize the Euler characteristic, and prove the generalized Gauss-Bonnet Theorem and other relations between the total excess and the Euler characteristic. Furthermore, we see that the Gaussian curvature can be defined almost everywhere on a good surface with non-positive total excess. Received April 3, 1997 / Published online February 5, 2001