Symmetrization of starlike domains in Riemannian manifolds and a qualitative generalization of Bishop's volume comparison theorem

2004 ◽  
Vol 4 (3) ◽  
Author(s):  
Ryuichi Fukuoka
Keyword(s):  
Comparison Theorem ◽  
Riemannian Manifolds ◽  
Volume Comparison

Comparison Geometry With L1-Norms of Ricci Curvature

2006 ◽  
Vol 49 (1) ◽  
pp. 152-160
Author(s):  
Jong-Gug Yun
Keyword(s):  
Ricci Curvature ◽  
Comparison Theorem ◽  
Sphere Theorem ◽  
Volume Comparison ◽  
Comparison Geometry

AbstractWe investigate the geometry of manifolds with bounded Ricci curvature in L1-sense. In particular, we generalize the classical volume comparison theorem to our situation and obtain a generalized sphere theorem.

On Cheng's eigenvalue comparison theorem

2008 ◽  
Vol 144 (3) ◽  
pp. 673-682 ◽  
Author(s):  
G. P. BESSA ◽  
J. F. MONTENEGRO
Keyword(s):  
Comparison Theorem ◽  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Weak Form ◽  
First Eigenvalue ◽  
Arbitrary Topology ◽  
Eigenvalue Comparison ◽  
The Mean ◽  
Normal Geodesic ◽  
Sectional And Ricci Curvatures

AbstractWe observe that Cheng's Eigenvalue Comparison Theorem for normal geodesic balls [4] is still valid if we impose bounds on the mean curvature of the distance spheres instead of bounds on the sectional and Ricci curvatures. In this version, there is a weak form of rigidity in case of equality of the eigenvalues. Namely, equality of the eigenvalues implies that the distance spheres of the same radius on each ball has the same mean curvature. On the other hand, we construct smooth metrics $g_{\kappa}$ on $[0,r]\times \mathbb{S}^{3}$, non-isometric to the standard metric canκ of constant sectional curvature κ, such that the geodesic balls $B_{g_{\kappa}}(r)=([0,r]\times \mathbb{S}^{3},g_{\kappa})$, $B_{{\rm can}_{\kappa}}(r)=([0,r]\times \mathbb{S}^{3},{\rm can}_{\kappa})$ have the same first eigenvalue, the same volume and the distance spheres $\partial B_{g_{\kappa}}(s)$ and$\partial B_{{\rm can}_{\kappa}}(s)$, $0<s\leq r$, have the same mean curvatures. In the end, we apply this version of Cheng's Eigenvalue Comparison Theorem to construct examples of Riemannian manifolds M with arbitrary topology with positive fundamental tone λ*(M)>0 extending Veeravalli's examples,[7]

A Universal Volume Comparison Theorem for Finsler Manifolds and Related Results

2013 ◽  
Vol 65 (6) ◽  
pp. 1401-1435 ◽  
Author(s):  
Wei Zhao ◽  
Yibing Shen
Keyword(s):  
Isoperimetric Inequality ◽  
Comparison Theorem ◽  
Finsler Geometry ◽  
Finsler Manifolds ◽  
Volume Comparison

AbstractIn this paper, we establish a universal volume comparison theorem for Finsler manifolds and give the Berger–Kazdan inequality and Santalá's formula in Finsler geometry. Based on these, we derive a Berger–Kazdan type comparison theorem and a Croke type isoperimetric inequality for Finsler manifolds.

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