相关论文: A note on first eigenvalue estimates by coupling m…

We prove lower bound for the first closed or Neumann nonzero eigenvalue of the Laplacian on a compact quaternion-K\"ahler manifold in terms of dimension, diameter, and scalar curvature lower bound. It is derived as large time implication of…

We study the eigenvalue problem for the $p$-Laplacian on K\"ahler manifolds. Our first result is a lower bound for the first nonzero eigenvalue of the $p$-Laplacian on compact K\"ahler manifolds in terms of dimension, diameter, and lower…

Let $ (M,\omega_g) $ be a complete K\"ahler manifold of complex dimension $n$. We prove that if the holomorphic sectional curvature satisfies $\mathrm{HSC} \geq 2 $, then the first eigenvalue $\lambda_1$ of the Laplacian on $(M,\omega_g)$…

In this article we study the stability problem for positive quaternion-K\"ahler manifolds. We give a description of infinitesimal Einstein deformations and destabilising directions in terms of Laplace eigenfunctions and a special class of…

We study Laplace eigenvalues $\lambda_k$ on K\"ahler manifolds as functionals on the space of K\"ahler metrics with cohomologous K\"ahler forms. We introduce a natural notion of a $\lambda_k$-extremal K\"ahler metric and obtain necessary…

We prove a Lichnerowicz type lower bound for the first nontrivial eigenvalue of the $p$-Laplacian on K\"ahler manifolds. Parallel to the $p = 2$ case, the first eigenvalue lower bound is improved by using a decomposition of the Hessian on…

We first provide a stochastic formula for the Carath\'eodory distance in terms of general Markovian couplings and prove a comparison result between the Carath\'eodory distance and the complete K\"ahler metric with a negative lower curvature…

We establish lower bound for the first nonzero eigenvalue of the Laplacian on a closed K\"ahler manifold in terms of dimension, diameter, and lower bounds of holomorphic sectional curvature and orthogonal Ricci curvature. On compact…

In K\"ahler-Einstein case of positive scalar curvature and even complex dimension, an improved lower bound for the first eigenvalue of the Dirac operator is given. It is shown by a general construction that there are manifolds for which…

We study the first nonzero eigenvalues for the $p$-Laplacian on quaternionic K\"ahler manifolds. Our first result is a lower bound for the first nonzero closed (Neumann) eigenvalue of the $p$-Laplacian on compact quaternionic K\"ahler…

We will present an estimate for the first eigenvalue of the Dirichlet and Neumann problems in terms of the Bakry-\'Emery Ricci curvature for a compact weighted manifold. As an application we will establish a stability condition for a…

We study the behavior of connections and curvature under the HK/QK correspondence, proving simple formulae expressing the Levi-Civita connection and Riemann curvature tensor on the quaternionic K\"ahler side in terms of the initial…

We establish an explicit lower bound of the first eigenvalue of the Laplacian on K\"ahler manifolds based off the comparison results of Li and Wang. The lower bound will depend on the diameter, dimension, holomorphic sectional curvature and…

We study geometric first order differential operators on quaternionic K\"ahler manifolds. Their principal symbols are related to the enveloping algebra and Casimir elements for $\Sp(1)\Sp(n)$. This observation leads to anti-symmetry of the…

We find upper and lower bounds for the first eigenvalue and the volume entropy of a noncompact real analytic K\"ahler manifold, in terms of Calabi's diastasis function and diastatic entropy, which are sharp in the case of the complex…

In this paper, we derive apriori estimates for constant scalar curvature K\"ahler metrics on a compact K\"ahler manifold. We show that higher order derivatives can be estimated in terms of a $C^0$ bound for the K\"ahler potential. We also…

We give an explicit formula for the quaternionic K\"ahler metrics obtained by the HK/QK correspondence. As an application, we give a new proof of the fact that the Ferrara-Sabharwal metric as well as its one-loop deformation is quaternionic…

We give lower bounds for the first Dirichilet eigenvalues for domains in submanifolds with locally bounded mean curvatures. These bounds depend on the injectivity radius, sectional curvature (upperbound) of the ambient space and on the mean…

In this paper, two interesting eigenvalue comparison theorems for the first non-zero Steklov eigenvalue of the Laplacian have been established for manifolds with radial sectional curvature bounded from above. Besides, sharper bounds for the…

By using Bochner technique and gradient estimate, we give the lower bound estimates of the first eigenvalue of Finsler-Laplacian on Finsler manifolds. These results generalize the corresponding famous theorems in the Riemannian geometry.