Related papers: Sharp inequalities for Neumann eigenvalues on the …

We study the effect of two types of degeneration of the Riemannian metric on the first eigenvalue of the Laplace operator on surfaces. In both cases we prove that the first eigenvalue of the round sphere is an optimal asymptotic upper…

For any $p \in ( 1, +\infty)$, we give a new inequality for the first nontrivial Neumann eigenvalue $\mu _ p (\Omega, \varphi)$ of the $p$-Laplacian on a convex domain $\Omega \subset \mathbb{R}^N$ with a power-concave weight $\varphi$. Our…

We compute three-term semiclassical asymptotic expansions of counting functions and Riesz-means of the eigenvalues of the Laplacian on spheres and hemispheres, for both Dirichlet and Neumann boundary conditions. Specifically for Riesz-means…

We derive the equation of a free vibrating thin plate whose mass is concentrated at the boundary, namely a Steklov problem for the biharmonic operator. We provide Hadamard-type formulas for the shape derivatives of the corresponding…

The $n$-dimensional hypercube has $n+1$ distinct eigenvalues $n-2i$, $0\leq i\leq n$, with corresponding eigenspaces $U_i(n)$. In 2021 it was proved by the author that if a function with non-empty support belongs to the direct sum…

We prove two weighted geometric inequalities that hold for strictly mean convex and star-shaped hypersurfaces in Euclidean space. The first one involves the weighted area and the area of the hypersurface and also the volume of the region…

For $ p\in (1, \infty)$, we consider the following weighted Neumann eigenvalue problem on $B_1^c$, the exterior of the closed unit ball in $R^N$: \begin{equation}\label{Neumann eqn} \begin{aligned} -\Delta_p \phi & = \lambda g |\phi|^{p-2}…

We prove sharp inequalities for determinants of Toeplitz operators and twisted Laplace operators on the two-sphere, generalizing the Moser-Trudinger-Onofri inequality. In particular a sharp version of conjectures of Gillet-Soule and Fang…

We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex…

Let $(\Sigma,g)$ be a closed Riemannian surface, $W^{1,2}(\Sigma,g)$ be the usual Sobolev space, $\textbf{G}$ be a finite isometric group acting on $(\Sigma,g)$, and $\mathscr{H}_\textbf{G}$ be a function space including all functions $u\in…

Given a Riemannian submersion $(M,g) \to (B,j)$ each of whose fibers is connected and totally geodesic, we consider a certain 1-parameter family of Riemannian metrics $(g_{t})_{t > 0}$ on $M$, which is called the canonical variation. Let…

Let $M$ be a product of rank-one symmetric spaces of compact type, each of dimension at least $3$. We establish sharp $L^p$ bounds for the restriction of Laplace--Beltrami eigenfunctions on $M$ to arbitrary submanifolds contained in a…

Given a compact manifold equipped with a volume element and a Riemannian metric, we formulate and study a dual pair of optimization problems: one concerning smooth maps from the manifold into the Hilbert space $l^2$ and the other concerning…

Sharp comparison theorems are derived for all eigenvalues of the (weighted) Laplacian, for various classes of weighted-manifolds (i.e. Riemannian manifolds endowed with a smooth positive density). Examples include Euclidean space endowed…

We establish several Poincar\'e--Sobolev type inequalities for the Lapalce--Beltrami operator $\Delta_g$ in the hyperbolic space $\mathbb H^n$ with $n\geq 5$. These inequalities could be seen as the improved second order Poincar\'e…

The main goal in this paper is to point out that quantity $||\nabla R||^2(p)$ on a harmonic space can not be determined by the spectra of local geodesic spheres or balls, therefore the main results of [AM-S] (quoted in the title) are wrong.…

Let $M$ be a compact, connected Riemannian manifold whose Riemannian volume measure is denoted by $\sigma$. Let $f: M \rightarrow \mathbb{R}$ be a non-constant eigenfunction of the Laplacian. The random wave conjecture suggests that in…

We consider second order elliptic systems of partial differential equations subject to Dirichlet and Neumann boundary conditions. We prove analyticity of the elementary symmetric functions of the eigenvalues, and compute Hadamard-type…

Let $M$ be an oriented geometrically finite hyperbolic manifold of infinite volume with dimension at least $3$. For all $k \geq 0$, we provide a lower bound on the $k$th eigenvalue of the Laplace-Beltrami operator of $M$ by the $k$th…

Let $X$ be an arithmetic hyperbolic surface, $\psi$ a Hecke-Maass form, $\ell$ a geodesic segment on $X$, and $\mu$ a Borel measure supported on $\ell$ with dimension greater than 1/2. We obtain a power saving over the local bound of…