scholarly journals A Note on Finsler Version of Calabi-Yau Theorem

2018 ◽  
Vol 2018 ◽  
pp. 1-4
Author(s):  
Songting Yin ◽  
Ruixin Wang ◽  
Pan Zhang
Keyword(s):  
Ricci Curvature ◽  
Volume Growth ◽  
Finsler Manifold ◽  
Infinite Volume ◽  
Growth Theorem ◽  
Negative Function ◽  
Weighted Ricci Curvature ◽  
Bounded Below

We generalize Calabi-Yau’s linear volume growth theorem to Finsler manifold with the weighted Ricci curvature bounded below by a negative function and show that such a manifold must have infinite volume.

Gradient Estimates for Harmonic Functions on Manifolds With Lipschitz Metrics

1998 ◽  
Vol 50 (6) ◽  
pp. 1163-1175 ◽  
Author(s):  
Jingyi Chen ◽  
Elton P. Hsu
Keyword(s):  
Ricci Curvature ◽  
Harmonic Functions ◽  
Volume Growth ◽  
Gradient Estimate ◽  
Gradient Estimates ◽  
Smooth Manifolds ◽  
Lipschitz Continuous ◽  
Bounded Below

AbstractWe introduce a distributional Ricci curvature on complete smooth manifolds with Lipschitz continuous metrics. Under an assumption on the volume growth of geodesics balls, we obtain a gradient estimate for weakly harmonic functions if the distributional Ricci curvature is bounded below.

scholarly journals A Lichnerowicz–Obata–Cheng Type Theorem on Finsler Manifolds

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 311
Author(s):  
Songting Yin ◽  
Pan Zhang
Keyword(s):  
Lower Bound ◽  
Ricci Curvature ◽  
Distance Function ◽  
Type Theorem ◽  
Finsler Manifold ◽  
First Eigenvalue ◽  
Comparison Result ◽  
Finsler Manifolds ◽  
The First Eigenvalue ◽  
Bounded Below

Let ( M , F , d μ ) be a Finsler manifold with the Ricci curvature bounded below by a positive number and constant S-curvature. We prove that, if the first eigenvalue of the Finsler–Laplacian attains its lower bound, then M is isometric to a Finsler sphere. Moreover, we establish a comparison result on the Hessian trace of the distance function.

scholarly journals Ricci curvature and volume growth

1991 ◽  
Vol 148 (1) ◽  
pp. 161-167
Author(s):  
Martin Strake ◽  
Gerard Walschap
Keyword(s):  
Ricci Curvature ◽  
Volume Growth

scholarly journals Complete logarithmic Sobolev inequalities via Ricci curvature bounded below

2022 ◽  
Vol 394 ◽  
pp. 108129
Author(s):  
Michael Brannan ◽  
Li Gao ◽  
Marius Junge
Keyword(s):  
Ricci Curvature ◽  
Sobolev Inequalities ◽  
Logarithmic Sobolev Inequalities ◽  
Bounded Below

Weighted Ricci Curvature

2021 ◽  
pp. 115-128
Author(s):  
Shin-ichi Ohta
Keyword(s):  
Ricci Curvature ◽  
Weighted Ricci Curvature

SOME RESULTS ON FUNDAMENTAL GROUPS AND BETTI NUMBERS OF FINSLER MANIFOLDS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250063 ◽  
Author(s):  
YIBING SHEN ◽  
WEI ZHAO
Keyword(s):  
Ricci Curvature ◽  
Betti Number ◽  
Betti Numbers ◽  
Finsler Manifold ◽  
Fundamental Groups ◽  
Finsler Manifolds ◽  
Nonnegative Ricci Curvature ◽  
Finiteness Theorems ◽  
The Relationship

In this paper the relationship between the Ricci curvature and the fundamental groups of Finsler manifolds are studied. We give an estimate of the first Betti number of a compact Finsler manifold. Two finiteness theorems for fundamental groups of compact Finsler manifolds are proved. Moreover, the growth of fundamental groups of Finsler manifolds with almost-nonnegative Ricci curvature are considered.

scholarly journals Splitting theorems for Finsler manifolds of nonnegative Ricci curvature

2015 ◽  
Vol 2015 (700) ◽  
Author(s):  
Shin-ichi Ohta
Keyword(s):  
Vector Field ◽  
Ricci Curvature ◽  
Splitting Theorem ◽  
Gradient Vector ◽  
Busemann Function ◽  
Finsler Manifolds ◽  
Nonnegative Ricci Curvature ◽  
Straight Line ◽  
Weighted Ricci Curvature ◽  
Gradient Vector Field

AbstractWe investigate the structure of a Finsler manifold of nonnegative weighted Ricci curvature including a straight line, and extend the classical Cheeger–Gromoll–Lichnerowicz Splitting Theorem. Such a space admits a diffeomorphic, measure-preserving splitting in general. As for a special class of Berwald spaces, we can perform the isometric splitting in the sense that there is a one-parameter family of isometries generated from the gradient vector field of the Busemann function. A Betti number estimate is also given for Berwald spaces.

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