Werner Ballmann

Manifolds of nonpositive curvature

Preface.-Introduction.-Lectures on Manifolds of Nonpositive Curvature.-Simply Connected Manifolds of Nonpositive Curvature.-Groups of Isometries.-Finiteness theorems.-Strong Rigidity of Locally Symmetric Spaces.-Appendix 1. Manifolds of Higher Rank.-Appendix 2: Finiteness Results for Nonanalytic Manifolds.-Appendix 3: Tits Metric and the Action of Isometries at Infinity.-Appendix 4: Tits Metric and Asymptotic Rigidity.-Appendix 5: Symmetric Spaces of Noncompact types.-References.-Subject Index.

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.

Polygonal Complexes and Combinatorial Group Theory

We study the structure of certain simply connected 2-dimensional complexes with non-positive curvature. We obtain a precise description of how these complexes behave at infinity and prove an existence theorem which gives an abundance of such complexes. We also investigate the structure of groups which act transitively on the set of vertices of such a complex.

On the Dirichlet Problem at Infinity for Manifolds of Nonpositive Curvature

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Axial Isometries of Manifolds of Non-Positive Curvature.

Let M be a complete C ~~ Riemannian manifold of non-positive sectional curvature. We say that a geodesic 9: IR~ M bounds a fiat strip of width c > 0 (a fiat half plane) if there is a totally geodesic, isometric immersion i: [0, c) x IR~M(i: [0, oo) x IR~M) such that i(0, t) = 9(0. A 9eodesic without fiat strip (without fiat half plane) is a geodesic, which does not bound a flat strip (a flat half plane). We will prove that the existence of a closed geodesic without flat half plane has rather strong consequences for the geometry and topology of M. In fact, many of the properties of a manifold of strictly negative curvature (resp. of a visibility manifold) still remain true if one assumes only the existence of a closed geodesic without flat half plane. We will discuss the existence of free (non-Abelian) subgroups of gl(M), the existence of infinitely many closed geodesics, the density of closed geodesics, and a transitivity property of the geodesic flow. It is, therefore, interesting to give conditions which ensure the existence of a closed geodesic without flat half plane. We will prove that M has a closed geodesic without flat half plane if vol(M)< oo and if M contains a geodesic without flat half plane. Note that a geodesic is not boundary of a flat strip (and a fortiori not boundary of a flat half plane) if it passes through a point p e M such that the sectional curvature of all tangent planes at p is negative. In the proofs of our results we investigate the action of rtl(M ) as group of isometries on the universal covering space H of M. In the proofs of many of our results we do not use the fact that this action is properly discontinuous and free. We, therefore, formulate these results for arbitrary groups D of isometries of H. The paper is organized as follows: In Sect. 1 we fix some definitions and notations and quote some standard results of non-positive curvature. Section 2 is the central section of this paper. We investigate the properties of those isometries of H which correspond to closed geodesics in M. We also prove

Boundary value problems for elliptic differential operators of first order

We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instance. We provide a selfcontained introduction to (nonlocal) elliptic boundary conditions, boundary regularity of solutions, and index theory. In particular, we simplify and generalize the traditional theory of elliptic boundary value problems for Dirac type operators. We also prove a related decomposition theorem, a general version of Gromov and Lawsons relative index theorem and a generalization of the cobordism theorem.

Diameter rigidity of spherical polyhedra

1. Introduction. The diameter rigidity question considered in this paper is motivated by the rank rigidity problem for spaces of nonpositive curvature. The rank of a complete, simply connected space Y of nonpositive curvature is greater than or equal to 2 if every geodesic segment in Y is contained in an isometrically embedded, convex Euclidean plane. The rank rigidity problem asks for a classification of such spaces, at least when the isometry group of Y is large. Recall that a topological space is called a polyhedron if it admits a triangulation. A polyhedron with a length metric is called Euclidean (respectively, spherical) if it admits a triangulation into Euclidean (respectively, spherical) simplices. Here a Euclidean (respectively, spherical) k-simplex is a k-simplex A such that A with the induced length metric is isometric to the intersection of k + 1 closed half-spaces in R k (respectively, closed hemispheres in S k ) in general position. We are interested in the rank rigidity of simply connected Euclidean polyhedra of nonpositive curvature and of rank greater than or equal to 2. We expect them to be Euclidean buildings or products. The rank rigidity for dim Y = 2 is not difficult and is contained in [BB, Section 6]. For a Euclidean polyhedron Y and p ∈ Y , we denote by SpY the link of Y at p, that is, the set of directions at p. Clearly, SpY is a spherical polyhedron, and, if Y has nonpositive curvature, then the injectivity radius of SpY is π . Furthermore, if the rank of Y is greater than or equal to 2, then SpY is geodesically complete and has diameter π . Hence, for dim Y = 3 the link X = SpY is a geodesically complete, compact, 2-dimensional spherical polyhedron of diameter and injectivity radius π . The aim of this paper is to classify such spaces X. Here are examples of geodesically complete compact spherical polyhedra of diameter and injectivity radius π . 1.1. Spherical building. If X is a spherical building, then X carries a natural metric, for which the apartments are unit spheres. For this metric, the diameter and injectivity radius of X are π .I fY is a Euclidean building of dimension n ≥ 2 with the natural metric, then every geodesic in Y is contained in an isometrically embedded, convex Euclidean n-space. The link of a vertex in Y is a spherical building of dimension n−1, which has injectivity radius and diameter π .A nn-dimensional building X

Eigenvalues and holonomy

We estimate the eigenvalues of connection Laplacians in terms of the non-triviality of the holonomy.

ON THE SPECTRAL THEORY OF MANIFOLDS WITH CUSPS

We are interested in the spectral properties of Dirac operators on Riemannian manifolds with cuspidal ends. We derive estimates for the essential spectrum and get formulas for the index in the Fredholm case.

On the Construction of Isospectral Manifolds

I discuss two general methods, namely Sunada’s method and a more recent method, due to Gordon and Schueth, for the construction of isospectral closed Riemannian manifolds. I prove a theorem unifying and extending the two methods and apply it to obtain isospectral metrics on S 2 × S 3 .