Sectional curvature,compact cores,and local quasiconvexity

We define a new notion of sectional curvature for 2-complexes, and describe a variety of examples with nonpositive or negative sectional curvature. The 2-complexes with nonpositive sectional curvature have coherent and locally indicable fundamental groups. The 2-complexes with negative sectional curvature have the compact core property for covers with finitely generated fundamental group. The fundamental groups of compact 2-complexes with metric negative sectional curvature have locally-quasiconvex fundamental groups.     

Curvature of invariant metrics on Pyatetskii-Shapiro's generalized homogeneous domains

We study the curvature of invariant metrics on the generalization of the classical homogeneous domain of Pyatetskii-Shapiro, as given by D'Atri in [3]. We obtain all invariant Kähler metrics of either, nonpositive sectional curvature or nonpositive holomorphic sectional curvature, and determine the corresponding connected groups of isometries in each case. This yields a continuous family of nonsymmetric homogeneous Kähler metrics with nonpositive curvature.Supported in part by CONICOR and SECyT (UNC).     

Warped product structure of submanifolds with nonpositive extrinsic curvature in space forms

By means of a simple warped product construction we obtain examples of submanifolds with nonpositive extrinsic curvature and minimal index of relative nullity in any space form. We then use this to extend to arbitrary space forms four known splitting results for Euclidean submanifolds with nonpositive sectional curvature.     

Examples of 4-manifolds with almost nonpositive curvature

We construct an infinite family of non-homeomorphic 4-manifolds with almost nonpositive sectional curvature whose universal covering space is not contractible. As a consequence, these manifolds do not support metrics with nonpositive sectional curvature. To achieve this, we use a generalization of Bavard's surgery construction, combined with an open book decomposition and knot theory.     

Properly discontinuous actions on Hilbert manifolds

In this article we study properly discontinuous actions on Hilbert manifolds giving new examples of complete Hilbert manifolds with nonnegative, respectively nonpositive, sectional curvature with infinite fundamental group. We also get examples of complete infinite dimensional Kähler manifolds with positive holomorphic sectional curvature and infinite fundamental group in contrastwith the finite dimensional case and we classify abelian groups acting linearly, isometrically and properly discontinuously on Stiefel manifolds. Finally, we classify homogeneous Hilbert manifolds with constant sectional curvature.     

Partially hyperbolic geodesic flows

We construct a category of examples of partially hyperbolic geodesic flows which are not Anosov, deforming the metric of a compact locally symmetric space of nonconstant negative curvature. Candidates for such an example as the product metric and locally symmetric spaces of nonpositive curvature with rank bigger than one are not partially hyperbolic. We prove that if a metric of nonpositive curvature has a partially hyperbolic geodesic flow, then its rank is one. Other obstructions to partial hyperbolicity of a geodesic flow are also analyzed.     

The uniqueness of the maximal measure for geodesic flows on symmetric spaces of higher rank

In this paper we show that the geodesic flow on a compact locally symmetric space of nonpositive curvature has a unique invariant measure of maximal entropy. As an application to dynamics we show that closed geodesics are uniformly distributed with respect to this measure. Furthermore, we prove that the volume entropy is minimized at a compact locally symmetric space of nonpositive curvature among all conformally equivalent metrics with the same total volume.     

Cusp closing in rank one symmetric spaces

We investigate conditions under which cusps of pinched negative curvature can be closed as manifolds or orbifolds with nonpositive sectional curvature. We show that all cusps of complex hyperbolic type can be closed in this way whereas cusps of quaternionic or Cayley hyperbolic type cannot be closed. For cusps of real hyperbolic type we derive necessary and sufficient closing conditions. In this context we prove that a noncompact finite volume quotient of a rank one symmetric space can be approximated in the Gromov Hausdorff topology by closed orbifolds with nonpositive curvature if and only if it is real or complex hyperbolic. Using cusp closing methods we obtain new examples of real analytic manifolds of nonpositive sectional curvature and rank one containing flats. By the same methods we get an explicit resolution of the singularities in the Baily-Borel resp. Siu-Yau compactification of finite volume quotients of the complex hyperbolic space.Oblatum 2-IX-1994 & 7-VIII-1995     

Cusp closing in rank one symmetric spaces

We investigate conditions under which cusps of pinched negative curvature can be closed as manifolds or orbifolds with nonpositive sectional curvature. We show that all cusps of complex hyperbolic type can be closed in this way whereas cusps of quaternionic or Cayley hyperbolic type cannot be closed. For cusps of real hyperbolic type we derive necessary and sufficient closing conditions. In this context we prove that a noncompact finite volume quotient of a rank one symmetric space can be approximated in the Gromov Hausdorff topology by closed orbifolds with nonpositive curvature if and only if it is real or complex hyperbolic. Using cusp closing methods we obtain new examples of real analytic manifolds of nonpositive sectional curvature and rank one containing flats. By the same methods we get an explicit resolution of the singularities in the Baily–Borel resp.Siu–Yau compactification of finite volume quotients of the complex hyperbolic space. Oblatum 2-IX-1994 & 7-VIII-1995     

The asymptotic blow-up of a surface in Euclidean 3-space

Given a smoothly immersed surface in Euclidean (or affine) 3-space, the asymptotic directions define a subset in the Grassmann bundle of unoriented one-dimensional subspaces over the surface. This links the Euler characteristic of the region where the Gauss curvature is nonpositive with the index of singularities in a natural line field defined on this subset. To apply this we need only identify mechanisms which restrict the index of the singularities. In Section 2.1 we show that specific configurations of nonpositive Gauss curvature cannot be realized by an immersed surface and that specific configurations in 2-sphere cannot be realized as Gauss images of surfaces. In Section 2.2 we prove an existence theorem for surfaces which satisfy regularity conditions and a Symplectic Monge Ampere PDE. In general, a PDE of this type will not restrict the indices of the singularities over a solution. However, we show that over a surface of nonzero constant mean curvature the indices are restricted and, hence, that specific configurations of nonpositive Gauss curvature cannot be realized by a constant mean curvature surface.     

Wu's theorem for Kähler–Finsler spaces

In this paper, we study strongly convex Kähler–Finsler manifolds. We prove two theorems: A strongly convex Kähler–Berwald manifold with a pole is a Stein manifold if it has nonpositive horizontal radial flag curvature; A strongly convex Kähler–Finsler manifold with a complex pole is a Stein manifold if it has nonpositive horizontal radial flag curvature.     

On an invariant on isometric immersions into spaces of constant sectional curvature

In the present paper, we give an invariant on isometric immersions into spaces of constant sectional curvature. This invariant is a direct consequence of the Gauss equation and the Codazzi equation of isometric immersions. We apply this invariant on some examples. Further, we apply it to codimension 1 local isometric immersions of 2-step nilpotent Lie groups with arbitrary leftinvariant Riemannian metric into spaces of constant nonpositive sectional curvature. We also consider the more general class, namely, three-dimensional Lie groups G with nontrivial center and with arbitrary left-invariant metric. We show that if the metric of G is not symmetric, then there are no local isometric immersions of G into Q _c ⁴.     

Orbihedra of nonpositive curvature

A 2-dimensional orbihedron of nonpositive curvature is a pair (X, Γ), where X is a 2-dimensional simplicial complex with a piecewise smooth metric such that X has nonpositive curvature in the sense of Alexandrov and Busemann and Γ is a group of isometries of X which acts properly discontinuously and cocompactly. By analogy with Riemannian manifolds of nonpositive curvature we introduce a natural notion of rank 1 for (X, Γ) which turns out to depend only on Γ and prove that, if X is boundaryless, then either (X, Γ) has rank 1, or X is the product of two trees, or X is a thick Euclidean building. In the first case the geodesic flow on X is topologically transitive and closed geodesics are dense. Partially supported by MSRI, SFB256 and University of Maryland. Partially supported by MSRI, SFB256 and NSF DMS-9104134.     

Pseudo-harmonic maps from closed pseudo-Hermitian manifolds to Riemannian manifolds with nonpositive sectional curvature

In this paper, we use heat flow method to prove the existence of pseudo-harmonic maps from closed pseudo-Hermitian manifolds to Riemannian manifolds with nonpositive sectional curvature, which is a generalization of Eells–Sampson’s existence theorem. Furthermore, when the target manifold has negative sectional curvature, we analyze horizontal energy of geometric homotopy of two pseudo-harmonic maps and obtain that if the image of a pseudo-harmonic map is neither a point nor a closed geodesic, then it is the unique pseudo-harmonic map in the given homotopic class. This is a generalization of Hartman’s theorem.     

Topology of orbits and orbit spaces of some product G-manifolds

We give a topological classification of cohomogeneity two Riemannian G-manifolds of nonpositive curvature and their orbits, under the condition that \({M^{G} \neq \emptyset}\) and the universal Riemannian covering of M has a suitable decomposition.     

Warped products of Hadamard spaces

In the Riemannian case, our approach to warped products illuminates curvature formulas that previously seemed formal and somewhat mysterious. Moreover, the geometric approach allows us to study warped products in a much more general class of spaces. For complete metric spaces, it is known that nonpositive curvature in the Alexandrov sense is preserved by gluing on isometric closed convex subsets and by Gromov–Hausdorff limits with strictly positive convexity radius; we show it is also preserved by warped products with convex warping functions. Received: 9 January 1998/ Revised version: 12 March 1998     

Jacobi fields on a Riemann manifold

Some properties of Jacobi fields on a manifold of nonpositive curvature are considered. As a result, we obtain relations for derivatives of one class of functions on the manifold. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1602–1613, December, 2006.     

On the average curvature of a convex curve in a surface of nonpositive Gaussian curvature

In this paper, the upper bound of the average curvature of a convex curve in a simply connected surface of nonpositive Gaussian curvature is obtained.

     

Estimates and nonexistence of solutions of the scalar curvature equation on noncompact manifolds

This paper is to study the conformal scalar curvature equation on complete noncompact Riemannian manifold of nonpositive curvature. We derive some estimates and properties of supersolutions of the scalar curvature equation, and obtain some nonexistence results for complete solutions of scalar curvature equation.