scholarly journals A sharp lower bound for the first eigenvalue on Finsler manifolds

2013 ◽  
Vol 30 (6) ◽  
pp. 983-996 ◽  
Author(s):  
Guofang Wang ◽  
Chao Xia
Keyword(s):  
Lower Bound ◽  
First Eigenvalue ◽  
Finsler Manifolds ◽  
The First Eigenvalue ◽  
Sharp Lower Bound

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 A variation on the Donsker–Varadhan inequality for the principal eigenvalue

2017 ◽  
Vol 473 (2204) ◽  
pp. 20160877
Author(s):  
Jianfeng Lu ◽  
Stefan Steinerberger
Keyword(s):  
Lower Bound ◽  
Exit Time ◽  
Principal Eigenvalue ◽  
Short Paper ◽  
First Eigenvalue ◽  
Drift Diffusion ◽  
First Exit Time ◽  
The First Eigenvalue ◽  
Order Elliptic Operator ◽  
The Mean

The purpose of this short paper is to give a variation on the classical Donsker–Varadhan inequality, which bounds the first eigenvalue of a second-order elliptic operator on a bounded domain Ω by the largest mean first exit time of the associated drift–diffusion process via λ 1 ≥ 1 sup x ∈ Ω E x τ Ω c . Instead of looking at the mean of the first exit time, we study quantiles: let d p , ∂ Ω : Ω → R ≥ 0 be the smallest time t such that the likelihood of exiting within that time is p , then λ 1 ≥ log ( 1 / p ) sup x ∈ Ω d p , ∂ Ω ( x ) . Moreover, as p → 0 , this lower bound converges to λ 1 .

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