Related papers: Spectra, rigidity and stability of sine-cones

We investigate the spectrum of the Laplacian on complete, non-compact manifolds $M^n$ whose Ricci curvature satisfies $\mathrm{Ric} \geq -(n-1)\mathrm{H}(r)$, for some continuous, non-increasing $\mathrm{H}$ with $\mathrm{H}-1 \in…

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 prove stability of integrable ALE manifolds with a parallel spinor under Ricci flow, given an initial metric which is close in $L^p \cap L^\infty$, for any $p \in (1, n)$, where $n$ is the dimension of the manifold. In particular, our…

The eigenvalues of the Laplace-Beltrami operator and the integrals of products of eigenfunctions must satisfy certain consistency conditions on compact Riemannian manifolds. These consistency conditions are derived by using spectral…

Based on the Weierstrass representation of second variation we develop a non-spectral theory of stability for isoperimetric problem with minimized and constrained two-dimensional functionals of general type and free endpoints allowed to…

We study the behaviour of eigenvalues, below the bottom of the essential spectrum, of the Laplacian under finite Riemannian coverings of complete and connected Riemannian manifolds. We define spectral stability and instability of such…

The goal of this paper is to investigate some rigidity properties of stable solutions of elliptic equations set on manifolds with boundary. We provide several types of results, according to the dimension of the manifold and the sign of its…

We study stability of a generalized sine-Gordon model with two coupled scalar fields in two dimensions. Topological soliton solutions are found from the first-order equations that solve the equations of motion. The perturbation equations…

In this paper, we prove rigidity results on gradient shrinking Ricci solitons with weakly harmonic Weyl curvature tensors. Let $(M^n, g)$ be a compact gradient shrinking Ricci soliton satisfying ${\rm Ric}_g + Ddf = \rho g$ with $\rho >0$…

We explicitely compute the absolutely continuous spectrum of the Laplace-Beltrami operator for $p$-forms for the class of warped product metrics $d\sigma^2= y^{2a}dy^2 + y^{2b}d\theta_{\Sphere^{N-1}}^2$, where $y$ is a boundary defining…

We consider the Laplace operator in a planar waveguide, i.e., an infinite two-dimensional straight strip of constant width, with particular types of Robin boundary conditions. We study the essential spectrum of the corresponding Laplacian…

We will consider the Navier-Stokes equation on a Riemannian manifold M with Ricci tensor bounded below, the involved Laplacian operator is De Rham-Hodge Laplacian. The novelty of this work is to introduce a family of connections which are…

We prove that spectral projections of Laplace-Beltrami operator on the m-complex unit sphere E_{Delta_{S^{2m-1}}}([0,R)) are uniformly bounded as an operator from H^p(S^{2m-1}) to L^p(S^{2m-1}) for all p --> (1,infinity). We also show that…

We consider the inverse boundary value problem for the first order perturbation of the polyharmonic operator $\mathcal L_{g,X,q}$, with $X$ being a $W^{1,\infty}$ vector field and $q$ being an $L^\infty$ function on compact Riemannian…

We investigate the stability of the wave equation with spatial dependent coefficients on a bounded multidimensional domain. The system is stabilized via a scattering passive feedback law. We formulate the wave equation in a port-Hamiltonian…

Throughout the history of Einstein manifolds, differential geometers have shown great interest in finding the relationships between curvature and the topology of Einstein manifolds. In the paper, first, we prove that a compact Einstein…

Let \Sigma be a compact oriented surface immersed in a four dimensional K\"ahler-Einstein manifold M. We consider the evolution of \Sigma in the direction of its mean curvature vector. It is proved that being symplectic is preserved along…

Let M be an even dimensional compact Riemannian manifold with boundary and let D be a Dirac operator acting on the sections of the Clifford module E over M. We impose certain local elliptic boundary conditions for D obtaining a selfadjoint…

To what extent does the eigenvalue spectrum of the Laplace-Beltrami operator on a compact Riemannian manifold determine the geometry of the manifold? We give examples of isospectral manifolds with different local geometry including…

Let $\Omega$ be an unbounded two dimensional strip on a ruled surface in $\mathbb{R}^d$, $d\geq2$. Consider the Laplacian operator in $\Omega$ with Dirichlet and Neumann boundary conditions on opposite sides of $\Omega$. We prove some…