scholarly journals Volume growth and closed geodesics on Riemannian manifolds of hyperbolic type

2005 ◽  
Vol 2005 (6) ◽  
pp. 875-893
Author(s):  
Jean-Pierre Ezin ◽  
Carlos Ogouyandjou
Keyword(s):  
Growth Rate ◽  
Riemannian Manifolds ◽  
Compact Manifold ◽  
Volume Growth ◽  
Growth Function ◽  
Hyperbolic Type ◽  
Compact Manifolds ◽  
Closed Geodesics ◽  
Geodesic Spheres

We study the volume growth function of geodesic spheres in the universal Riemannian covering of a compact manifold of hyperbolic type. Furthermore, we investigate the growth rate of closed geodesics in compact manifolds of hyperbolic type.

Entropy and volume growth

2000 ◽  
Vol 20 (1) ◽  
pp. 77-84 ◽  
Author(s):  
KURT COGSWELL
Keyword(s):  
Growth Rate ◽  
Probability Measure ◽  
Topological Entropy ◽  
Compact Manifold ◽  
Volume Growth ◽  
Maximal Entropy ◽  
Measure Of Maximal Entropy

We consider a $C^{1+1}$ diffeomorphism $f$ of a compact manifold $M$ which preserves an ergodic probability measure $\mu$. We conclude that $\mu$-a.e. $x \in M$ is contained in a disk $D_x \subset W^u(x)$, with $D_x$ open in the $W^u(x)$ topology, which exhibits an exponential volume growth rate greater than or equal to the measure-theoretic entropy of $f$ with respect to $\mu$. Drawing on results of Newhouse and Yomdin, we then find that when $f$ is $C^\infty$ and $\mu$ is a measure of maximal entropy, this exponential volume growth rate equals the topological entropy of $f$ for $\mu$-a.e. $x$.

Numerical Analysis of the Effect of Diffusion and Creep Flow on Cavity Growth

2007 ◽  
Author(s):  
Frederick W. Brust ◽  
Joonyoung Oh
Keyword(s):  
Growth Rate ◽  
Numerical Method ◽  
Cavity Growth ◽  
Volume Growth ◽  
Numerical Results ◽  
Shape Evolution ◽  
Cavity Volume ◽  
Cavity Shape ◽  
Cavity Growth Rate ◽  
Fully Coupled

In this paper, intergranular cavity growth in regimes, where both surface diffusion and deformation enhanced grain boundary diffusion are important, is studied. In order to continuously simulate the cavity shape evolution and cavity growth rate, a fully-coupled numerical method is proposed. Based on the fully-coupled numerical method, a gradual cavity shape change is predicted and this leads to an adverse effect on the cavity growth rates. As the portion of the cavity volume growth due to jacking and viscoplastic deformation in the total cavity volume growth increases, the initially spherical cavity evolves to V-shaped cavity. The numerical results are physically more realistic compared to results in the previous studies. The present numerical results suggest that the cavity shape evolution and cavity growth rate based on an assumed cavity shape, whether spherical or crack-like, cannot be used in this regime due to transitional coupled growth mechanisms.

Jacobi fields and geodesic spheres

1979 ◽  
Vol 82 (3-4) ◽  
pp. 233-240 ◽  
Author(s):  
L. Vanhecke ◽  
T. J. Willmore
Keyword(s):  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Holomorphic Sectional Curvature ◽  
Principal Curvatures ◽  
Jacobi Fields ◽  
Geodesic Spheres ◽  
Jacobi Vector ◽  
Nearly Kähler ◽  
Nearly Kähler Manifolds

SynopsisThis is a contribution to the general problem of determining the extent to which the geometry of a riemannian manifold is determined by properties of its geodesic spheres. In particular we show that total umbilicity of geodesic spheres determines riemannian manifolds of constant sectional curvature; quasi-umbilicity of geodesic spheres determines Kähler and nearly-Kähler manifolds of constant holomorphic sectional curvature; and the condition that geodesic spheres have only two different principal curvatures, one having multiplicity 3, determines manifolds locally isometric to the quaternionic projective spaces. The use of Jacobi vector fields leads to a unified treatment of these different cases.

scholarly journals A volume estimate for strong subharmonicity and maximum principle on complete Riemannian manifolds

1998 ◽  
Vol 151 ◽  
pp. 25-36 ◽  
Author(s):  
Kensho Takegoshi
Keyword(s):  
Riemannian Manifold ◽  
Maximum Principle ◽  
Growth Condition ◽  
Riemannian Manifolds ◽  
Volume Growth ◽  
Complete Riemannian Manifold ◽  
Volume Estimate ◽  
Complete Riemannian Manifolds ◽  
Generalized Maximum Principle

Abstract.A generalized maximum principle on a complete Riemannian manifold (M, g) is shown under a certain volume growth condition of (M, g) and its geometric applications are given.

Closed geodesics in Riemannian manifolds with convex boundary

1994 ◽  
Vol 124 (6) ◽  
pp. 1247-1258 ◽  
Author(s):  
Anna Maria Candela ◽  
Addolorata Salvatore
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Closed Geodesics ◽  
Convex Boundary

In this paper we look for closed geodesies on a noncomplete Riemannian manifold ℳ. We prove that if ℳ has convex boundary, then there exists at least one closed nonconstant geodesic on it.

scholarly journals ON THE PROPERTIES OF THE CAYLEY GRAPH OF RICHARD THOMPSON'S GROUP F

2004 ◽  
Vol 14 (05n06) ◽  
pp. 677-702 ◽  
Author(s):  
V. S. GUBA
Keyword(s):  
Growth Rate ◽  
Lower Bound ◽  
Cayley Graph ◽  
Upper Bound ◽  
Growth Function ◽  
Vertex Degree ◽  
General Growth ◽  
Thompson’S Group ◽  
Average Vertex ◽  
Thompson's Group F

We study some properties of the Cayley graph of R. Thompson's group F in generators x0, x1. We show that the density of this graph, that is, the least upper bound of the average vertex degree of its finite subgraphs is at least 3. It is known that a 2-generated group is not amenable if and only if the density of the corresponding Cayley graph is strictly less than 4. It is well known this is also equivalent to the existence of a doubling function on the Cayley graph. This means there exists a mapping from the set of vertices into itself such that for some constant K>0, each vertex moves by a distance at most K and each vertex has at least two preimages. We show that the density of the Cayley graph of a 2-generated group does not exceed 3 if and only if the group satisfies the above condition with K=1. Besides, we give a very easy formula to find the length (norm) of a given element of F in generators x0, x1. This simplifies the algorithm by Fordham. The length formula may be useful for finding the general growth function of F in generators x0, x1 and the growth rate of this function. In this paper, we show that the growth rate of F has a lower bound of [Formula: see text].

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