Related papers: A note on a Sung-Wang's paper

We prove existence results for optimization problems for the $k$th Laplace eigenvalue on closed Riemannian manifolds of dimension $m \geq 3$, depending on the choice of normalization. One such normalization leads to eigenvalue optimization…

Let M be a compact, connected, m-dimensional manifold without boundary and p>1. For 1<p\leq m, we prove that the first eigenvalue \lambda_{1,p} of the p-Laplacian is bounded on each conformal class of Riemannian metrics of volume one on M.…

In a Hadamard manifold $M$, it is proved that if $u$ is a $\lambda$-eigenfunction of the Laplacian that belongs to $L^p(M)$ for some $p \ge 2$, then $u$ is bounded and $\|u\|_{\infty} \le C \|u\|_p,$ where $C$ depends only on $p$, $\lambda$…

Given a closed symplectic manifold (M,\omega) of dimension greater than 2, we consider all Riemannian metrics on M, which are compatible with the symplectic structure \omega. For each such metric, we look at the first eigenvalue \lambda_1…

One of the main purposes of this paper is to prove that on a complete K\"ahler manifold of dimension $m$, if the holomorphic bisectional curvature is bounded from below by -1 and the minimum spectrum $\lambda_1(M) \ge m^2$, then it must…

We derive a sharp upper bound for the first eigenvalue $\lambda_{1,p}$ of the $p$-Laplacian on asymptotically hyperbolic manifolds for $1<p<\infty$. We then prove that a particular class of conformally compact submanifolds within…

Suppose $(M,g)$ is a Riemannian manifold having dimension $n$, nonnegative Ricci curvature, maximal volume growth and unique tangent cone at infinity. In this case, the tangent cone at infinity $C(X)$ is an Euclidean cone over the…

In this paper, we consider the optimization problem for the first Dirichlet eigenvalue $\lambda_1(\Omega)$ of the $p$-Laplacian $\Delta_p$, $1< p< \infty$, over a family of doubly connected planar domains $\Omega= B \setminus \overline{P}$,…

Let $M^{4n}$ be a complete quaternionic K\"ahler manifold with scalar curvature bounded below by $-16n(n+2)$. We get a sharp estimate for the first eigenvalue $\lambda_1(M)$ of the Laplacian which is $\lambda_1(M)\le (2n+1)^2$. If the…

Let $H^p(L^2(M))$ be the space of all $L^2$-harmonic $p$-forms $(2\leq p\leq n-2)$ on complete submanifolds $M$ with flat normal bundle in spheres. In this paper, we first show that $H^p(L^2(M))$ is trivial if the total curvature of $M$ is…

Given a simply connected compact generalized flag manifold M together with its invariant K\"ahler Einstein metric g, we investigate the functional given by the first eigenvalue of the Hodge Laplacian on smooth functions restricted to the…

Consider $\left(M,g\right)$ as an $m$-dimensional compact connected Riemannian manifold without boundary. In this paper, we investigate the first eigenvalue $\lambda_{1,p,q}$ of the $\left(p,q\right)$-Laplacian system on $M$. Also, in the…

In this paper, we investigate the Dirichlet problem of Laplacian on complete Riemannian manifolds. By constructing new trial functions, we obtain a sharp upper bound of the gap of the consecutive eigenvalues in the sense of the order, which…

We construct a surface with a cylindrical end which has a finite number of Laplace eigenvalues embedded in its continuous spectrum. The surface is obtained by attaching a cylindrical end to a hyperbolic torus with a hole. To our knowledge,…

For a K\"{a}hler manifold endowed with a weighted measure $e^{-f}\,dv,$ the associated weighted Hodge Laplacian $\Delta _{f}$ maps the space of $(p,q)$-forms to itself if and only if the $(1,0)$-part of the gradient vector field $\nabla f$…

Let $M$ be a connected complete noncompact $n$-dimensional Riemannian manifold with a base point $p \in M$ whose radial sectional curvature at $p$ is bounded from below by that of a noncompact surface of revolution which admits a finite…

The purpose of this paper is to give a self-contained proof that a complete manifold with more than one end never supports an $L^{q,p}$-Sobolev inequality ($2 \leq p$, $q\leq p^{*}$), provided the negative part of its Ricci tensor is small…

In this note we prove the following result: There is a positive constant $\epsilon(n,\Lambda)$ such that if $M^n$ is a simply connected compact K$\ddot{a}$hler manifold with sectional curvature bounded from above by $\Lambda$, diameter…

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…

We show that the Dirichlet problem at infinity is unsolvable for the p-Laplace equation for any nonconstant continuous boundary data, for certain range of p>n, on an n-dimensional Cartan-Hadamard manifold constructed from a complete…