scholarly journals Lower bound estimates of the first eigenvalue for compact manifolds with positive Ricci curvature

1999 ◽  
Vol 190 (2) ◽  
pp. 383-398 ◽  
Author(s):  
DaGang Yang
Keyword(s):  
Lower Bound ◽  
Ricci Curvature ◽  
Compact Manifolds ◽  
First Eigenvalue ◽  
Positive Ricci Curvature ◽  
The First Eigenvalue

scholarly journals Minimal hypersurfaces with small first eigenvalue in manifolds of positive Ricci curvature

2016 ◽  
Vol 09 (03) ◽  
pp. 505-532
Author(s):  
Jonathan J. Zhu
Keyword(s):  
Lower Bound ◽  
Ricci Curvature ◽  
First Eigenvalue ◽  
Positive Ricci Curvature ◽  
Minimal Hypersurfaces ◽  
The First Eigenvalue ◽  
Geometric Data ◽  
Laplace Eigenvalue ◽  
Hypersurfaces In Spheres

In this paper we exhibit deformations of the hemisphere [Formula: see text], [Formula: see text], for which the ambient Ricci curvature lower bound [Formula: see text] and the minimality of the boundary are preserved, but the first Laplace eigenvalue of the boundary decreases. The existence of these metrics suggests that any resolution of Yau’s conjecture on the first eigenvalue of minimal hypersurfaces in spheres would likely need to consider more geometric data than a Ricci curvature 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 Gromov’s convergence theorem and its application

1985 ◽  
Vol 100 ◽  
pp. 11-48 ◽  
Author(s):  
Atsushi Katsuda
Keyword(s):  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Convergence Theorem ◽  
Riemannian Geometry ◽  
Main Tool ◽  
First Eigenvalue ◽  
Sphere Theorem ◽  
Positive Ricci Curvature ◽  
The First Eigenvalue ◽  
Differentiable Sphere Theorem

One of the basic questions of Riemannian geometry is that “If two Riemannian manifolds are similar with respect to the Riemannian invariants, for example, the curvature, the volume, the first eigenvalue of the Laplacian, then are they topologically similar?”. Initiated by H. Rauch, many works are developed to the above question. Recently M. Gromov showed a remarkable theorem ([7] 8.25, 8.28), which may be useful not only for the above question but also beyond the above. But it seems to the author that his proof is heuristic and it contains some gaps (for these, see § 1), so we give a detailed proof of 8.25 in [7]. This is the first purpose of this paper. Second purpose is to prove a differentiable sphere theorem for manifolds of positive Ricci curvature, using the above theorem as a main tool.

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 .

Sign in / Sign up

Export Citation Format

Share Document