scholarly journals Entropy, minimal surfaces and negatively curved manifolds

2016 ◽  
Vol 38 (1) ◽  
pp. 336-370
Author(s):  
ANDREW SANDERS
Keyword(s):  
Hausdorff Dimension ◽  
Riemannian Manifolds ◽  
Minimal Surfaces ◽  
Second Fundamental Form ◽  
Fuchsian Groups ◽  
Riemannian Surface ◽  
Limit Set ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Convex Cocompact

Taubes [Minimal surfaces in germs of hyperbolic 3-manifolds. Proceedings of the Casson Fest, Geom. Topol. Monogr.7 (2004), 69–100 (electronic)] introduced the space of minimal hyperbolic germs with elements consisting of the first and second fundamental form of an equivariant immersed minimal disk in hyperbolic 3-space. Herein, we initiate a further study of this space by studying the behavior of a dynamically defined function which records the entropy of the geodesic flow on the associated Riemannian surface. We provide a useful estimate on this function which, in particular, yields a new proof of Bowen’s theorem on the rigidity of the Hausdorff dimension of the limit set of quasi-Fuchsian groups. These follow from new lower bounds on the Hausdorff dimension of the limit set which allow us to give a quantitative version of Bowen’s rigidity theorem. To demonstrate the strength of the techniques, these results are generalized to convex-cocompact surface groups acting on $n$-dimensional $\text{CAT}\,(-1)$ Riemannian manifolds.

Some New Results on L2 Cohomology of Negatively Curved Riemannian Manifolds

2005 ◽  
Vol 57 (2) ◽  
pp. 251-266
Author(s):  
M. Cocos
Keyword(s):  
Heat Equation ◽  
Riemannian Manifolds ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Regularity Properties ◽  
L2 Cohomology ◽  
Cohomology Spaces

AbstractThe present paper is concerned with the study of the L2 cohomology spaces of negatively curved manifolds. The first half presents a finiteness and vanishing result obtained under some curvature assumptions, while the second half identifies a class of metrics having non-trivial L2 cohomology for degree equal to the half dimension of the space. For the second part we rely on the existence and regularity properties of the solution for the heat equation for forms.

scholarly journals Surface groups acting on spaces

2017 ◽  
Vol 39 (7) ◽  
pp. 1843-1856
Author(s):  
GEORGIOS DASKALOPOULOS ◽  
CHIKAKO MESE ◽  
ANDREW SANDERS ◽  
ALINA VDOVINA
Keyword(s):  
Hausdorff Dimension ◽  
Group Action ◽  
Metric Space ◽  
Hyperbolic Plane ◽  
Harmonic Map ◽  
Surface Group ◽  
Surface Groups ◽  
Limit Set ◽  
Convex Cocompact

Harmonic map theory is used to show that a convex cocompact surface group action on a $\text{CAT}(-1)$ metric space fixes a convex copy of the hyperbolic plane (i.e. the action is Fuchsian) if and only if the Hausdorff dimension of the limit set of the action is equal to 1. This provides another proof of a result of Bonk and Kleiner. More generally, we show that the limit set of every convex cocompact surface group action on a $\text{CAT}(-1)$ space has Hausdorff dimension $\geq 1$, where the inequality is strict unless the action is Fuchsian.

A variation formula for the topological entropy of convex-cocompact manifolds

2010 ◽  
Vol 31 (6) ◽  
pp. 1849-1864 ◽  
Author(s):  
SAMUEL TAPIE
Keyword(s):  
Hausdorff Dimension ◽  
Explicit Formula ◽  
Topological Entropy ◽  
Geodesic Flow ◽  
Kleinian Group ◽  
Limit Set ◽  
Analytic Family ◽  
Variation Formula ◽  
Sectional Curvatures ◽  
Convex Cocompact

AbstractLet (M,gλ) be a 𝒞2-family of complete convex-cocompact metrics with pinched negative sectional curvatures on a fixed manifold. We show that the topological entropy htop(gλ) of the geodesic flow is a 𝒞1 function of λ and we give an explicit formula for its derivative. We apply this to show that if ρλ(Γ)⊂PSL2(ℂ) is an analytic family of convex-cocompact faithful representations of a Kleinian group Γ, then the Hausdorff dimension of the limit set Λρλ(Γ) is a 𝒞1 function of λ. Finally, we give a variation formula for Λρλ (Γ).

scholarly journals Geodesic flows of negatively curved manifolds with smooth stable and unstable foliations

1988 ◽  
Vol 8 (2) ◽  
pp. 215-239 ◽  
Author(s):  
Masahiko Kanai
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Geodesic Flows ◽  
Constant Negative Curvature ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Closed Riemannian Manifold

AbstractWe are concerned with closed C∞ riemannian manifolds of negative curvature whose geodesic flows have C∞ stable and unstable foliations. In particular, we show that the geodesic flow of such a manifold is isomorphic to that of a certain closed riemannian manifold of constant negative curvature if the dimension of the manifold is greater than two and if the sectional curvature lies between − and −1 strictly.

scholarly journals Topological uniqueness of negatively curved surfaces

2010 ◽  
Vol 199 ◽  
pp. 137-149
Author(s):  
Hsungrow Chan
Keyword(s):  
Black Hole ◽  
Minimal Surfaces ◽  
Total Curvature ◽  
Second Fundamental Form ◽  
Curved Surfaces ◽  
Index Method ◽  
Negatively Curved ◽  
Asymptotic Curves ◽  
Euclidean Three Space ◽  
Three Space

AbstractIn this paper we consider complete, noncompact, negatively curved surfaces that are twice continuously differentiably embedded in Euclidean three-space, showing that if such surfaces have square integrable second fundamental form, then their topology must, by the index method, be an annulus. We then show how this relates to some minimal surface theorems and has a corollary on minimal surfaces with finite total curvature. In addition, we discuss, by the index method, the relation between the topology and asymptotic curves. Finally, we apply the results yielded to the problem of isometrical immersions into Euclidean three-space of black hole models.

scholarly journals Critical Exponent and Hausdorff Dimension in Pseudo-Riemannian Hyperbolic Geometry

2019 ◽  
Author(s):  
Olivier Glorieux ◽  
Daniel Monclair
Keyword(s):  
Hausdorff Dimension ◽  
Critical Exponent ◽  
Hyperbolic Geometry ◽  
Limit Set ◽  
Rigidity Result ◽  
Famous Theorem ◽  
Limit Sets ◽  
Convex Cocompact

AbstractThe aim of this article is to understand the geometry of limit sets in pseudo-Riemannian hyperbolic geometry. We focus on a class of subgroups of $\textrm{PO}(p,q+1)$ introduced by Danciger, Guéritaud, and Kassel, called ${\mathbb{H}}^{p,q}$-convex cocompact. We define a pseudo-Riemannian analogue of critical exponent and Hausdorff dimension of the limit set. We show that they are equal and bounded from above by the usual Hausdorff dimension of the limit set. We also prove a rigidity result in ${\mathbb{H}}^{2,1}={\mathbb{A}}\textrm{d}{\mathbb{S}}^3$, which can be understood as a Lorentzian version of a famous Theorem of R. Bowen in $3$D hyperbolic geometry.

Harmonic Mappings of Negatively Curved Manifolds

1978 ◽  
Vol 30 (03) ◽  
pp. 631-637 ◽  
Author(s):  
Zvi Har'el
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Harmonic Mapping ◽  
Harmonic Mappings ◽  
Space Forms ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Image Position

Volume decreasing properties of harmonic mappings of space forms were investigated by S. S. Chern and S. I. Goldberg [3] and the author. In a previous paper [6], a step toward generalization of the results was made proving the following theorem: Theorem. Let ƒ: M —> N be a harmonic mapping of n-dimensional Riemannian manifolds, with C ≦ 0. Suppose the scalar curvature of M is not less than — S, and the Ricci curvature of N is not greater than —S/n, where S ≧ 0 and S > 0 are constants. Then, if u has a maximum on M, i.e. ƒ is volume decreasing up to a constant.

Holomorphic families of quasi-Fuchsian groups

1994 ◽  
Vol 14 (2) ◽  
pp. 207-212 ◽  
Author(s):  
K. Astala ◽  
M. Zinsmeister
Keyword(s):  
Hausdorff Dimension ◽  
Fuchsian Groups ◽  
Limit Set ◽  
Finitely Generated ◽  
Holomorphic Family ◽  
Finitely Generated Groups

AbstractWe produce a holomorphic family of infinitely generated quasi-Fuchsian groups such that the Hausdorff dimension of the limit set L (Гλ) is identical to 1 for small λ, but strictly greater than 1 for λ ˜ 1. In particular, this shows that Hausdorff dimension does not depend real analytically on the parameter λ, contrary to the case of finitely generated groups.

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