scholarly journals Diastatic entropy and rigidity of complex hyperbolic manifolds

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Roberto Mossa
Keyword(s):  
Lower Bound ◽  
Hyperbolic Manifold ◽  
Hyperbolic Manifolds ◽  
Hyperbolic Metric ◽  
Continuous Map ◽  
The Hyperbolic Metric ◽  
Geometric Rigidity ◽  
Volume Entropy ◽  
Complex Hyperbolic Manifold ◽  
Real Analytic

AbstractLet f : Y → X be a continuous map between a compact real analytic Kähler manifold (Y, g) and a compact complex hyperbolic manifold (X, g0). In this paper we give a lower bound of the diastatic entropy of (Y, g) in terms of the diastatic entropy of (X, g0) and the degree of f . When the lower bound is attained we get geometric rigidity theorems for the diastatic entropy analogous to the ones obtained by G. Besson, G. Courtois and S. Gallot [2] for the volume entropy. As a corollary,when X = Y,we get that the minimal diastatic entropy is achieved if and only if g is isometric to the hyperbolic metric g0.

VOLUMES OF CHAIN LINKS

2012 ◽  
Vol 21 (11) ◽  
pp. 1250115
Author(s):  
JAMES KAISER ◽  
JESSICA S. PURCELL ◽  
CLINT ROLLINS
Keyword(s):  
Lower Bound ◽  
Hyperbolic Manifold ◽  
Hyperbolic Manifolds ◽  
Minimal Volume ◽  
Chain Link ◽  
Computational Data ◽  
Chain Links

Agol has conjectured that minimally twisted n-chain links are the smallest volume hyperbolic manifolds with n cusps, for n ≤ 10. In his thesis, Venzke mentions that these cannot be smallest volume for n ≥ 11, but does not provide a proof. In this paper, we give a proof of Venzke's statement for a number of cases. For n ≥ 60 we use a formula from work of Futer, Kalfagianni and Purcell to obtain a lower bound for volume. The proof for n between 12 and 25 inclusive uses a rigorous computer computation that follows methods of Moser and Milley. Finally, we prove that the n-chain link with 2m or 2m + 1 half-twists cannot be the minimal volume hyperbolic manifold with n cusps, provided n ≥ 60 or |m| ≥ 8, and we give computational data indicating this remains true for smaller n and |m|.

scholarly journals Lagrangian systems on hyperbolic manifolds

1999 ◽  
Vol 19 (5) ◽  
pp. 1157-1173 ◽  
Author(s):  
PHILIP BOYLAND ◽  
CHRISTOPHE GOLÉ
Keyword(s):  
Geodesic Flow ◽  
Hyperbolic Manifold ◽  
Hyperbolic Manifolds ◽  
Hyperbolic Metric ◽  
Invariant Sets ◽  
Lagrangian System ◽  
Mather Theory ◽  
Closed Surfaces ◽  
Bounded Distance ◽  
Time Periodic

This paper gives two results that show that the dynamics of a time-periodic Lagrangian system on a hyperbolic manifold are at least as complicated as the geodesic flow of a hyperbolic metric. Given a hyperbolic geodesic in the Poincaré ball, Theorem A asserts that there are minimizers of the lift of the Lagrangian system that are a bounded distance away and have a variety of approximate speeds. Theorem B gives the existence of a collection of compact invariant sets of the Euler–Lagrange flow that are semiconjugate to the geodesic flow of a hyperbolic metric. These results can be viewed as a generalization of the Aubry–Mather theory of twist maps and the Hedlund–Morse–Gromov theory of minimal geodesics on closed surfaces and hyperbolic manifolds.

scholarly journals ON THE VOLUMES OF COMPLEX HYPERBOLIC MANIFOLDS WITH CUSPS

2004 ◽  
Vol 15 (06) ◽  
pp. 567-572 ◽  
Author(s):  
JUN-MUK HWANG
Keyword(s):  
Hyperbolic Manifold ◽  
Hyperbolic Manifolds ◽  
Geometric Topology ◽  
Toroidal Compactifications ◽  
Geometric Methods ◽  
Complex Hyperbolic Manifold ◽  
Better Than

We study the problem of bounding the number of cusps of a complex hyperbolic manifold in terms of its volume. Applying algebra-geometric methods using Mumford's work on toroidal compactifications and its generalization due to N. Mok and W.-K. To, we get a bound which is considerably better than those obtained previously by methods of geometric topology.

scholarly journals A Question of Norton–Sullivan in the Analytic Case

2020 ◽  
Author(s):  
Jian Wang ◽  
Hui Yang
Keyword(s):  
Negative Answer ◽  
Smoothness Condition ◽  
Continuous Map ◽  
Geometric Conditions ◽  
Analytic Case ◽  
Real Analytic

Abstract In 1996, A. Norton and D. Sullivan asked the following question: If $f:\mathbb{T}^2\rightarrow \mathbb{T}^2$ is a diffeomorphism, $h:\mathbb{T}^2\rightarrow \mathbb{T}^2$ is a continuous map homotopic to the identity, and $h f=T_{\rho } h$, where $\rho \in \mathbb{R}^2$ is a totally irrational vector and $T_{\rho }:\mathbb{T}^2\rightarrow \mathbb{T}^2,\, z\mapsto z+\rho $ is a translation, are there natural geometric conditions (e.g., smoothness) on $f$ that force $h$ to be a homeomorphism? In [ 22], the 1st author and Z. Zhang gave a negative answer to the above question in the $C^{\infty }$ category: in general, not even the infinite smoothness condition can force $h$ to be a homeomorphism. In this article, we give a negative answer in the $C^{\omega }$ category (see also [ 22, Question 3]): we construct a real analytic conservative and minimal totally irrational pseudo-rotation of $\mathbb{T}^2$ that is semi-conjugate to a translation but not conjugate to a translation.

scholarly journals The Harmonic Mapping Whose Hopf Differential Is a Constant

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1310
Author(s):  
Liang Shen
Keyword(s):  
Unit Disk ◽  
Harmonic Mapping ◽  
Hyperbolic Metric ◽  
Beltrami Coefficient ◽  
Hopf Differential ◽  
The Hyperbolic Metric

Suppose that h(z) is a harmonic mapping from the unit disk D to itself with respect to the hyperbolic metric. If the Hopf differential of h(z) is a constant c>0, the Beltrami coefficient μ(z) of h(z) is radially symmetric and takes the maximum at z=0. Furthermore, the mapping γ:c→μ(0) is increasing and gives a homeomorphism from (0,+∞) to (0,1).

Normal Functions: Lp Estimates

1997 ◽  
Vol 49 (1) ◽  
pp. 55-73 ◽  
Author(s):  
Huaihui Chen ◽  
Paul M. Gauthier
Keyword(s):  
Harmonic Function ◽  
Harmonic Functions ◽  
Holomorphic Functions ◽  
Hyperbolic Metric ◽  
Lp Estimates ◽  
Normal Functions ◽  
The Hyperbolic Metric

AbstractFor ameromorphic (or harmonic) function ƒ, let us call the dilation of ƒ at z the ratio of the (spherical)metric at ƒ(z) and the (hyperbolic)metric at z. Inequalities are knownwhich estimate the sup norm of the dilation in terms of its Lp norm, for p > 2, while capitalizing on the symmetries of ƒ. In the present paper we weaken the hypothesis by showing that such estimates persist even if the Lp norms are taken only over the set of z on which ƒ takes values in a fixed spherical disk. Naturally, the bigger the disk, the better the estimate. Also, We give estimates for holomorphic functions without zeros and for harmonic functions in the case that p = 2.

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