The sharp lower bound of the first Dirichlet eigenvalue for geodesic balls

@article{Wang2021TheSL,
  title={The sharp lower bound of the first Dirichlet eigenvalue for geodesic balls},
  author={Haibin Wang and Guoyi Xu and Jie Zhou},
  journal={Mathematische Zeitschrift},
  year={2021},
  volume={300},
  pages={2063-2068}
}
On complete noncompact Riemannian manifolds with non-negative Ricci curvature, Li–Schoen proved the uniform Poincaré inequality for any geodesic ball. In this note, we obtain the sharp lower bound of the first Dirichlet eigenvalue of such geodesic balls, which implies the sharp Poincaré inequality for geodesic balls. 
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References

SHOWING 1-10 OF 12 REFERENCES

Sharp modulus of continuity for parabolic equations on manifolds and lower bounds for the first eigenvalue

We derive sharp estimates on modulus of continuity for solutions of the heat equation on a compact Riemannian manifold with a Ricci curvature bound, in terms of initial oscillation and elapsed time.…

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

We present some new lower bound estimates of the first eigenvalue for compact manifolds with positive Ricci curvature in terms of the diameter and the lower Ricci curvature bound of the manifolds.…

The Estimate of the First Eigenvalue of a Compact Riemannian Manifold

The main theorem proved in this chapter is: Let M be a compact Riemannian manifold with nonnegative Ricci curvature. Then the first eigenvalue −λ1 of the Laplace operator of M satisfies λ1≥π2/ d2 ,…

Proof of the fundamental gap conjecture

We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spec- tral gap) of a Schrodinger operator with convex potential and…

Estimates on the modulus of expansion for vector fields solving nonlinear equations

Estimates on the modulus of expansion for vector fields solving nonlinear equations

In this article, by extending the method of [AC] we prove a sharp estimate on the expansion modulus of the gradient of the logarithm of the parabolic kernel to the Schördinger operator with convex…

A remark on Zhong-Yang's eigenvalue estimate

Moreover we know exactly when the equality holds: if and only if D is a disc. For any optimal geometric inequality it is important to have a complete understanding of the equality case. Sometimes…

Spectra of domains in compact manifolds

Lp and mean value properties of subharmonic functions on Riemannian manifolds

defined on M. To ensure a uniqueness property on equation (0.I) it is necessary to res t r ic t f to lie in a suitable function space. Some of the most natural spaces are those consisting of L p…

Estimates of eigenvalues of a compact Riemannian manifold