Metric Of Nonpositive Curvature Research Articles

Some remarks on the simplicial volume of nonpositively curved manifolds

We show that any closed manifold with a metric of nonpositive curvature that admits either a single point rank condition or a single point curvature condition has positive simplicial volume. We use this to provide a differential geometric proof of a conjecture of Gromov in dimension three.

Systolically extremal nonpositively curved surfaces are flat with finitely many singularities

The regularity of systolically extremal surfaces is a notoriously difficult problem already discussed by Gromov in 1983, who proposed an argument toward the existence of [Formula: see text]-extremizers exploiting the theory of [Formula: see text]-regularity developed by White and others by the 1950s. We propose to study the problem of systolically extremal metrics in the context of generalized metrics of nonpositive curvature. A natural approach would be to work in the class of Alexandrov surfaces of finite total curvature, where one can exploit the tools of the completion provided in the context of Radon measures as studied by Reshetnyak and others. However the generalized metrics in this sense still don’t have enough regularity. Instead, we develop a more hands-on approach and show that, for each genus, every systolically extremal nonpositively curved surface is piecewise flat with finitely many conical singularities. This result exploits a decomposition of the surface into flat systolic bands and nonsystolic polygonal regions, as well as the combinatorial/topological estimates of Malestein–Rivin–Theran, Przytycki, Aougab–Biringer–Gaster and Greene on the number of curves meeting at most once, combined with a kite excision move. The move merges pairs of conical singularities on a surface of genus [Formula: see text] and leads to an asymptotic upper bound [Formula: see text] on the number of singularities.

Embeddings of non-positively curved compact surfaces in flat Lorentzian manifolds

We prove that any metric of non-positive curvature in the sense of Alexandrov on a compact surface can be isometrically embedded as a convex spacelike Cauchy surface in a flat spacetime of dimension $$(2+1).$$ The proof follows from polyhedral approximation.

An obstruction to the smoothability of singular nonpositively curved metrics on 4-manifolds by patterns of incompressible tori

We give new examples of closed smooth 4-manifolds which support singular metrics of nonpositive curvature, but no smooth ones, thereby answering affirmatively a question of Gromov. The obstruction comes from patterns of incompressible 2-tori sufficiently complicated to force branching of geodesics for nonpositively curved metrics.

Obstructions to nonpositive curvature for open manifolds

We study algebraic conditions on a group G under which every properly discontinuous, isometric G-action on a Hadamard manifold has a G-invariant Busemann function. For such G we prove the following structure theorem: every open complete nonpositively curved Riemannian K(G,1) manifold that is homotopy equivalent to a finite complex of codimension >2 is an open regular neighborhood of a subcomplex of the same codimension. In this setting we show that each tangential homotopy type contains infinitely many open K(G,1) manifolds that admit no complete nonpositively curved metric even though their universal cover is the Euclidean space. A sample application is that an open contractible manifold W is homeomorphic to a Euclidean space if and only if the product of W and a circle admits a complete Riemannian metric of nonpositive 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.

Finite-volume complex-hyperbolic surfaces, their toroidal compactifications, and geometric applications

We study the classification of smooth toroidal compactifications of nonuniform ball quotients in the sense of Kodaira and Enriques. Several results concerning the Riemannian and complex algebraic geometry of these spaces are given. In particular we show that there are compact complex surfaces which admit Riemannian metrics of nonpositive curvature, but which do not admit Kahler metrics of nonpositive curvature. An infinite class of such examples arise as smooth toroidal compactifications of ball quotients.

Singular Kähler-Einstein metrics

We study degenerate complex Monge-Ampere equations of the form $(\omega+dd^c\f)^n = e^{t \f}\mu$ where $\omega$ is a big semi-Kahler form on a compact Kahler manifold $X$ of dimension $n$, $t \in \R^+$, and $\mu=f\omega^n$ is a positive measure with density $f\in L^p(X,\omega^n)$, $p>1$. We prove the existence and unicity of continuous $\o$-plurisubharmonic solutions. In case $X$ is projective and $\omega=\psi^*\omega'$, where $\psi:X\to V$ is a proper birational morphism to a normal projective variety, $[\omega']\in NS_{\R} (V)$ is an ample class and $\mu$ has only algebraic singularities, we prove that the solution is smooth in the regular locus of the equation. We use these results to construct singular Kahler-Einstein metrics of non-positive curvature on projective klt pairs, in particular on canonical models of algebraic varieties of general type.

Cohomogeneity one Riemannian manifolds of non-positive curvature

We study a G-manifold M which admits a G-invariant Riemannian metric g of non-positive curvature. We describe all such Riemannian G-manifolds ( M , g ) of non-positive curvature with a semisimple Lie group G which acts on M regularly and classify cohomogeneity one G-manifolds M of a semisimple Lie group G which admit an invariant metric of non-positive curvature. Some results on non-existence of invariant metric of negative curvature on cohomogeneity one G-manifolds of a semisimple Lie group G are given.

The Degree of Symmetry of Certain Compact Smooth Manifolds II*

In this paper, we give the sharp estimates for the degree of symmetry and the semi-simple degree of symmetry of certain four dimensional fiber bundles by virtue of the rigidity theorem of harmonic maps due to Schoen and Yau. As a corollary of this estimate, we compute the degree of symmetry and the semi-simple degree of symmetry of ${\Bbb C}P^2\times V$, where $V$ is closed smooth manifold admitting a real analytic Riemannian metric of non-positive curvature. In addition, by the Albanese map, we obtain the sharp estimate of the degree of symmetry of a compact smooth manifold with some restrictions on its one dimensional cohomology.

Extremal bounded holomorphic functions and an embedding theorem for arithmetic varieties of rank ? 2

Let Ω be a bounded symmetric domain of rank ≥ 2, and Γ ⊂ Aut(Ω) be a torsion-free irreducible cocompact lattice, X := Ω/Γ. On the projective manifold X there is the canonical Kahler-Einstein metric, which is of nonpositive holomorphic bisectional curvature. In Mok [M1,2] we established a Hermitian metric rigidity theorem for such projective manifolds X, which in the case when Ω is irreducible says that any Hermitian metric of nonpositive curvature in the sense of Griffiths is necessarily a constant multiple of the Kahler-Einstein metric. As a consequence, we proved in the latter case that any nontrivial holomorphic mapping f : X → Z into a Hermitian manifold Z of nonpositive curvature in the sense of Griffiths is necessarily an isometric immersion totally geodesic with respect to the Hermitian connection on Z. The Hermitian metric rigidity theorem can be taken as a tool in establishing the statement that any f : X → Z is necessarily a holomorphic immersion, a statement which only concerns the complex structure of X and Z. The Hermitian metric rigidity theorem and its consequences were generalized by To [To] to be applicable to any torsion-free lattice Γ ⊂ Aut(Ω), where X := Ω/Γ may be noncompact.

Metrics of Nonpositive Curvature on Graph-Manifolds and Electromagnetic Fields on Graphs

A 3-dimensional graph-manifold consists of simple blocks that are products of compact surfaces with boundary by the circle. Its global structure may be as complicated as desired and is described by a graph, which can be an arbitrary graph. A metric of nonpositive curvature on such a manifold, if it exists, can be described essentially by a finite number of parameters satisfying a geometrization equation. In the paper, it is shown that this equation is a discrete version of the Maxwell equations of classical electrodynamics, and its solutions, i.e., metrics of nonpositive curvature, are critical configurations of the same sort of action that describes the interaction of an electromagnetic field with a scalar charged field. This analogy is established in the framework of the spectral calculus (noncommutative geometry) of A. Connes. Bibliography: 17 titles.

Combinatorial conditions that imply word-hyperbolicity for 3-manifolds

Thurston conjectured that a closed triangulated 3-manifold in which every edge has degree 5 or 6, and no two edges of degree 5 lie in a common 2-cell, has word-hyperbolic fundamental group. We establish Thurston's conjecture by proving that such a manifold admits a piecewise Euclidean metric of non-positive curvature and the universal cover contains no isometrically embedded flat planes. The proof involves a mixture of computer computation and techniques from small cancellation theory.

The double of a hyperbolic manifold and non-positively curved exotic $PL$ structures

We give examples of non-compact finite volume real hyperbolic manifolds of dimension greater than five, such that their doubles admit at least three non-equivalent smoothable PL structures, two of which admit a Riemannian metric of non-positive curvature while the third does not. We also prove that the doubles of non-compact finite volume real hyperbolic manifolds of dimension greater than four are differentiably rigid.

On the CAT(0) dimension of 2–dimensional Bestvina–Brady groups

Let K be a 2-dimensional finite flag complex. We study the CAT(0) dimension of the `Bestvina-Brady group', or `Artin kernel', Gamma_K. We show that Gamma_K has CAT(0) dimension 3 unless K admits a piecewise Euclidean metric of non-positive curvature. We give an example to show that this implication cannot be reversed. Different choices of K lead to examples where the CAT(0) dimension is 3, and either (i) the geometric dimension is 2, or (ii) the cohomological dimension is 2 and the geometric dimension is not known.

Exotic smooth structures on nonpositively curved symmetric spaces

We construct series of examples of exotic smooth structures on compact locally symmetric spaces of noncompact type. In particular, we obtain higher rank examples, which do not support Riemannian metric of nonpositive curvature. The examples are obtained by taking the connected sum with an exotic sphere. To detect the change of the smooth structure we use a tangential map from the locally symmetric space its dual compact type twin.

Geometry of the Space of Phylogenetic Trees

We consider a continuous space which models the set of all phylogenetic trees having a fixed set of leaves. This space has a natural metric of nonpositive curvature, giving a way of measuring distance between phylogenetic trees and providing some procedures for averaging or combining several trees whose leaves are identical. This geometry also shows which trees appear within a fixed distance of a given tree and enables construction of convex hulls of a set of trees. This geometric model of tree space provides a setting in which questions that have been posed by biologists and statisticians over the last decade can be approached in a systematic fashion. For example, it provides a justification for disregarding portions of a collection of trees that agree, thus simplifying the space in which comparisons are to be made.

A geometric characteristic splitting in all dimensions

We prove the existence of a geometric characteristic submanifold for non-positively curved manifolds of any dimension greater than or equal to three. In dimension three, our result is a geometric version of the topological characteristic submanifold theorem due to Jaco, Shalen and Johannson. In the 1970's, Jaco and Shalen (JS) and Johannson (J) showed that a closed ori- entable Haken 3-manifold M has a canonical family of disjoint embedded incom- pressible tori, no two of which are parallel, such that the complementary pieces of M are either Seifert bre spaces or are atoroidal. They dened the characteristic submanifold V (M )o f Mto be essentially the union of the Seifert manifold pieces of M. Further, they showed that any essential map of the torus into M is homo- topic into V (M ). Johannson called this last property the Enclosing Property. For brevity, we will refer to these results as the JSJ results. In this paper, we show that if M is a closed manifold of dimension three or more, and if M has a Riemannian metric of non-positive curvature, then either the metric on M is flat or there is a precisely analogous decomposition of M along codimension one submanifolds. Further these submanifolds are totally geodesic in M and are flat in the metric induced from M. Note that in dimension three, a flat manifold must be a Seifert bre space, so that, in particular, our arguments give a new proof of the JSJ results for the special case when M is assumed to have a metric of non-positive curvature. In dimension four or more, a flat manifold need not be a Seifert manifold, see the example near the end of section 1, so this case really is dierent in higher dimensions. We also prove that essentially the same

Polyhedral Metrics and 3-Manifolds which are Virtual Bundles

Throughout this paper, all 3-manifolds are closed and orientable. Our aim is to give some new examples of 3-manifolds which are virtual bundles, that is, have finite sheeted covers which are surface bundles over the circle. Thurston has raised the question as to whether all irreducible atoroidal 3-manifolds might have this property. The geometrisation conjecture would imply that all such 3-manifolds have hyperbolic metrics, and the virtual bundle question is then equivalent to showing the existence of geometrically infinite incompressible surfaces (compare [14]). Not much progress has been made on this problem. Recently, Reid [9] has given an explicit example using an arithmetic hyperbolic 3-manifold, and Cooper, Long and Reid [3] have given methods of constructing immersed incompressible surface in surfaces bundles over the circle, which are sometimes geometrically infinite. In [10], examples are given of horizontal immersed incompressible surfaces in graph manifolds which do not lift to embeddings in any finite sheeted covering spaces. Similar examples are given in toroidal manifolds with cubings of non-positive curvatures in [9]. So the assumption that the manifold has the atoroidal property is essential in Thurston's question. Our aim is to give several simple constructions of large classes of examples using different polyhedral metrics of non-positive curvature on 3-manifolds (compare [2]). We follow closely the construction of Thurston (compare [13]), where he observes that the right-angled regular hyperbolic dodecahedron can be viewed as a cube with arcs drawn on each face through midpoints of a pair of opposite sides so that no two such arcs share a common vertex. Then the foliation of the cube by planes orthogonal to a diagonal gives an induced foliation of any 3-manifold arising from a torsion-free finite index subgroup of the symmetry group of the tessellation of hyperbolic 3-space by the right-angled dodecahedron. Clearly, any leaf of this foliation is then an immersed geometrically infinite surface. One explicit example is then the 4-fold cyclic branched cover of the Borromean rings, since the Borromean rings has a flat 2-fold branched cover which is obtained by gluing two cubes together. This example is formed by a subgroup of index 4 in the above symmetry group. Our examples come directly from this observation of Thurston, using different ways of dividing a 3-manifold into simple polyhedra so that there is an overall metric of non-positive curvature. We use some well-known tessellations of Euclidean 3-dimensional space, described in Coxeter [4], by truncated octahedra and tetrahedra, plus flying saucers (compare [2]), as well as the easy case of cubes. In the final section we show that our classes of cubed examples and flying saucers admit geometric decompositions in the sense of Thurston, since they are virtual bundles. We would like to thank the referee for a number of helpful comments which improved the exposition. 1991 Mathematics Subject Classification 57M50, 57N10, 57R30.

3-manifolds with(out) metrics of nonpositive curvature

In the context of Thurstons geometrisation program we address the question which compact aspherical 3-manifolds admit Riemannian metrics of nonpositive curvature. We show that non-geometric Haken manifolds generically, but not always, admit such metrics. More precisely, we prove that a Haken manifold with, possibly empty, boundary of zero Euler characteristic admits metrics of nonpositive curvature if the boundary is non-empty or if at least one atoroidal component occurs in its canonical topological decomposition. Our arguments are based on Thurstons Hyperbolisation Theorem. We give examples of closed graph-manifolds with linear gluing graph and arbitrarily many Seifert components which do not admit metrics of nonpositive curvature.