Remarks on the injectivity radius estimate for almost 1/4-pinched manifolds

1986 ◽  
pp. 156-164
Author(s):  
W. Klingenberg ◽  
T. Sakai
Keyword(s):  
Injectivity Radius

A note on the injectivity radius of quasiconformal immersions

2016 ◽  
Vol 71 (1) ◽  
pp. 161-163 ◽  
Author(s):  
V A Zorich
Keyword(s):  
Injectivity Radius

Systoles of hyperbolic 3-manifolds

2000 ◽  
Vol 128 (1) ◽  
pp. 103-110 ◽  
Author(s):  
COLIN C. ADAMS ◽  
ALAN W. REID
Keyword(s):  
Finite Volume ◽  
Closed Geodesic ◽  
Injectivity Radius ◽  
Hyperbolic Surfaces

Let M be a complete hyperbolic n-manifold of finite volume. By a systole of M we mean a shortest closed geodesic in M. By the systole length of M we mean the length of a systole. We denote this by sl (M). In the case when M is closed, the systole length is simply twice the injectivity radius of M. In the presence of cusps, injectivity radius becomes arbitrarily small and it is for this reason we use the language of ‘systole length’.In the context of hyperbolic surfaces of finite volume, much work has been done on systoles; we refer the reader to [2, 10–12] for some results. In dimension 3, little seems known about systoles. The main result in this paper is the following (see below for definitions):

A relationship between volume, injectivity radius, and eigenvalues

1990 ◽  
Vol 65 (1) ◽  
pp. 448-453
Author(s):  
Burton Randol
Keyword(s):  
Injectivity Radius

Injectivity radius estimate for 1/4-pinched manifolds

1980 ◽  
Vol 34 (1) ◽  
pp. 371-376 ◽  
Author(s):  
W. Klingenberg ◽  
T. Sakai
Keyword(s):  
Injectivity Radius

scholarly journals Density Problems for second order Sobolev Spaces and Cut-off Functions on Manifolds With Unbounded Geometry

2019 ◽  
Author(s):  
Debora Impera ◽  
Michele Rimoldi ◽  
Giona Veronelli
Keyword(s):  
Maximum Principle ◽  
Ricci Curvature ◽  
Sobolev Spaces ◽  
Upper Bound ◽  
Ricci Tensor ◽  
Injectivity Radius ◽  
Quadratic Growth ◽  
Riemann Curvature ◽  
Main Application ◽  
Compactly Supported Functions

Abstract We consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show the existence on such a manifold of a distance-like function with bounded gradient and mild growth of the Hessian. As a main application, we prove that smooth compactly supported functions are dense in $W^{2,p}$. The result is improved for $p=2$ avoiding both the upper bound on the Ricci tensor, and the injectivity radius assumption. As further applications we prove new disturbed Sobolev and Calderón–Zygmund inequalities on manifolds with possibly unbounded curvature and highlight consequences about the validity of the full Omori–Yau maximum principle for the Hessian.

Conjugate Radius and Sphere Theorem

1999 ◽  
Vol 42 (2) ◽  
pp. 214-220 ◽  
Author(s):  
Seong-Hun Paeng ◽  
Jong-Gug Yun
Keyword(s):  
Lower Bound ◽  
Injectivity Radius ◽  
Sphere Theorem ◽  
Standard Sphere

AbstractBessa [Be] proved that for given n and i0, there exists an depending on n, i0 such that if M admits a metric g satisfying , then M is diffeomorphic to the standard sphere. In this note, we improve this result by replacing a lower bound on the injectivity radius with a lower bound of the conjugate radius.

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