Riemannian hyperbolization

Abstract

Classical flat geometry has formed part of basic human knowledge since ancient times. It is characterized by the almost universally known condition that the sum of the internal angles of a triangle △ is equal to π. We write Σ(△) = π. Other fundamental geometries are defined by replacing the equality Σ(△) = π by inequalities; thus positively curved geometries and negatively curved geometries are determined by the inequalities Σ(△) > π and Σ(△) < π, respectively, where △ runs over all small non-degenerate triangles in a space. It is natural then to try to find spaces that admit such geometries, and this task has been a driving force in Riemannian Geometry for many decades. But surprisingly there are not too many examples of smooth closed manifolds that support either a positively curved or a negatively curved metric. For instance, besides spheres, in dimensions ≥ 17 (and = 24) the only positively curved simply connected known examples are complex and quaternionic projective spaces. In negative curvature the situation is arguably more striking because negative curvature has been studied extensively in many different areas in mathematics. Indeed, from the ergodicity of their geodesic flow in Dynamical Systems to their topological rigidity in Geometric Topology; from the existence of harmonic maps in Geometric Analysis to the well-studied and greatly generalized algebraic properties of their fundamental groups, negatively curved Riemannian manifolds are the main object in many important and well-known results in mathematics. Yet the fact remains that very few examples of closed negatively curved Riemannian manifolds are known. Besides the hyperbolic ones (R, C, H, O), the other known examples are the Mostow-Siu examples (complex dimension 2) which are local branched covers of complex hyperbolic space (1980, [23]), the Gromov-Thurston examples (1987, [18]) which are branched covers of real hyperbolic ones, the exotic Farrell-Jones examples (1989, [12]) which are homeomorphic but not diffeomorphic to real hyperbolic manifolds (and there are other examples of exotic type), and the three examples of Deraux (2005, [10]) which are of the Mostow-Siu type in complex dimension 3. Hence, excluding the Mostow-Siu and Deraux examples (in dimensions 4 and 6, respectively), all known examples of closed negatively curved Riemannian manifolds are homeomorphic to either a hyperbolic one or a branched cover of a hyperbolic one. * The author was partially supported by a NSF grant.objects are matched by the richness and complexity of the singularities obtained, and hyperbolized smooth manifolds are very far from being Riemannian. Interestingly one can relax and lose even more regularity and consider negative curvature from the algebraic point of view, that is consider Gromov's hyperbolic groups, and it can be argued [26] that "almost every group" is hyperbolic. So, negative curvature is in some weak sense generic, but Riemannian negative curvature seems very scarce. It is natural then to inquire about the difference between the class of ...

Deforming an є-close-to-hyperbolic metric to a hyperbolic metric

“…Metrics that are (B a , )-close to hyperbolic are very useful, and are key objects in [5] (see also [2,4]). Our next result answers the following question.…”

“…Next, we give a one-parameter version of hyperbolic forcing with cut at r 0 , with the variable r 0 → ∞ (we change notation and use λ instead of r 0 to express the fact that this number now varies). This family version of hyperbolic forcing is an important ingredient in [5], where it is used to smooth the singularities of a Charney-Davis strict hyperbolization [1].…”

“…The results in this paper are used to construct negatively curved Riemannian smoothings of Charney-Davis strict hyperbolizations of manifolds [1,5]. In the next paragraph we give an idea of how the objects and results in this paper are used in [5].…”

“…We study to what extent the hyperbolically forced metric Hg is close to being hyperbolic, if we assume g to be close to hyperbolic. We also deal with a one-parameter version of this problem.The results in this paper are used in the problem of smoothing Charney-Davis strict hyperbolizations [1], [2].…”

“…The results in this paper are used in the problem of smoothing Charney-Davis strict hyperbolizations [1], [2].…”