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     

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     

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 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.     

Constant curved minimal 2-spheres in G(2,4)

This paper gives a complete classification for minimal 2-spheres with constant Gaussian curvature immersed in the complex Grassmann manifold G(2,4). Received: 14 May 1998 / Revised version: 12 October 1998     

Asymptotics for the zeros and poles of normalized Pad\'{e} approximants to{\rm e}^{z}

Summary. With denoting the -th partial sum of ${\rm e}^{z}$, the exact rate of convergence of the zeros of the normalized partial sums, , to the Szeg\"o curve was recently studied by Carpenter et al. (1991), where is defined by Here, the above results are generalized to the convergence of the zeros and poles of certain sequences of normalized Pad\'{e} approximants to , where is the associated Pad\'{e} rational approximation to . Received February 2, 1994     

Eigenvalue estimates for the Dirac operator on quaternionic K?hler manifolds

We consider the Dirac operator on compact quaternionic K?hler manifolds and prove a lower bound for the spectrum. This estimate is sharp since it is the first eigenvalue of the Dirac operator on the quaternionic projective space. Received April 21, 1998; in final form June 16, 1998     

Instanton moduli on the quaternion Kähler manifold of type ${\mathbf G_2}$ and singular set

We prove the completeness of an instanton moduli space on quaternion-K?hler manifold . A point in the boundary of the moduli represents an ASD bundle with a particular singular set. It is shown that the singular set is a quaternion submanifold of and the Poincaré dual of the homology class represented by is the second Chern class o f the instanton bundle. Received: 4 July 2000 / Published online: 2 December 2002 Current Address: Faculty of Mathematics, Kyushu University, Ropponmatsu, Fukuoka, 810-8560, Japan (e-mail: nagatomo@math.kyushu-u.ac.jp)     

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 structure of singular spaces of dimension 2

In this paper, we study the structure of locally compact metric spaces of Hausdorff dimension 2. If such a space has non-positive curvautre and a local cone structure, then every simple closed curve bounds a conformal disk. On a surface (a topological manifold of dimension 2), a distance function with non-positive curvature and whose metric topology is equivalent to the surface topology gives a structure of a Riemann surface. The construction of conformal disks in these spaces uses minimal surface theory; in particular, the solution of the Plateau Problem in metric spaces of non-positive curvature. Received: 18 November 1997/ Revised versions: 15 January and 7 June 1999     

Eigenvalues and suspension structure of compact Riemannian orbifolds with positive Ricci curvature

Let M be a compact n-dimensional Riemannian orbifold of Ricci curvature ≥n−1. We prove that for 1 ≤k≤n, the k ^th nonzero eigenvalue of the Laplacian on M is equal to the dimension n if and only if M is isometric to the k-times spherical suspension over the quotient S ^{n − k} }Γ of the unit (n−k)-sphere by a finite group Γ⊂O(n−k+1) acting isometrically on S ^{n − k} ⊂ℝ ^{n − k +}. Received: 21 September 1998 / Revised version: 23 February 1999     

Resolvent and lattice points on symmetric spaces of strictly negative curvature

We study the asymptotics of the lattice point counting function for a Riemannian symmetric space X obtained from a semisimple Lie group of real rank one and a discontinuous group of motions in X, such that has finite volume. We show that as , for each . The constant corresponds to the sum of the positive roots of the Lie group associated to X, and n = dimX. The sum in the main term runs over a system of orthonormal eigenfunctions of the Laplacian, such that the eigenvalues are less than . Received: 4 January 1999