A relationship between volume, injectivity radius, and eigenvalues

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):

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Injectivity radius estimate for 1/4-pinched manifolds

Archiv der Mathematik ◽

10.1007/bf01224973 ◽

1980 ◽

Vol 34 (1) ◽

pp. 371-376 ◽

Cited By ~ 10

Author(s):

W. Klingenberg ◽

T. Sakai

Keyword(s):

Injectivity Radius

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

International Mathematics Research Notices ◽

10.1093/imrn/rnz131 ◽

2019 ◽

Cited By ~ 2

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.

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Conjugate Radius and Sphere Theorem

Canadian Mathematical Bulletin ◽

10.4153/cmb-1999-026-6 ◽

1999 ◽

Vol 42 (2) ◽

pp. 214-220 ◽

Cited By ~ 1

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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