Related papers: Sharp inequalities for Neumann eigenvalues on the …

We further research on the accelerated optimization phenomenon on Riemannian manifolds by introducing accelerated global first-order methods for the optimization of $L$-smooth and geodesically convex (g-convex) or $\mu$-strongly g-convex…

We provide an isoperimetric comparison theorem for small volumes in an $n$-dimensional Riemannian manifold $(M^n,g)$ with strong bounded geometry, as in Definition $2.3$, involving the scalar curvature function. Namely in strong bounded…

In this paper, we study the shape optimization problem for the first eigenvalue of the $p$-Laplace operator with the mixed Neumann-Dirichlet boundary conditions on multiply-connected domains in hyperbolic space. Precisely, we establish that…

We investigate in this paper the existence of a metric which maximizes the first eigenvalue of the Laplacian on Riemannian surfaces. We first prove that, in a given conformal class, there always exists such a maximizing metric which is…

In this paper, we prove an upper bound on the second non-zero Laplacian eigenvalue on $n$-dimensional real projective space. The sharp result for 2-dimensions was shown by Nadirashvili and Penskoi and later by Karpukhin when the metric…

Conserved charges in theories with gauge symmetries are supported on codimension-2 surfaces in the bulk spacetime. It has recently been suggested that various classical formulations of gravity dynamics display different symmetries, and…

We prove Cheng's eigenvalue comparison theorems for geodesic balls within the cut locus under weaker geometric hypothesis, and we also show that there are certain geometric rigidity in case of equality of the eigenvalues. This rigidity…

We study the uniqueness of horospheres and equidistant spheres in hyperbolic space under different conditions. First we generalize the Bernstein theorem by Do Carmo and Lawson to the embedded hypersurfaces with constant higher order mean…

We obtain results for the spectral optimisation of Neumann eigenvalues on rectangles in $\mathbb{R}^2$ with a measure or perimeter constraint. We show that the rectangle with measure $1$ which maximises the $k$'th Neumann eigenvalue…

We use optimal mass transport to provide a new proof and a dual formula to the Moser-Onofri inequality on $\s^2$ in the same spirit as the approach of Cordero-Erausquin, Nazaret and Villani to the Sobolev inequality and of…

The Bishop-Gromov theorem upperbounds the rate of growth of volume of geodesic balls in a space, in terms of the most negative component of the Ricci curvature. In this paper we prove a strengthening of the Bishop-Gromov bound for…

Among all $C^{\infty}$ bounded domains with equal volume, we show that the second eigenvalue of the Robin plate is uniquely maximized by an open ball, so long as the Robin parameter lies within a particular range of negative values. Our…

We prove certain optimal systolic inequalities for a closed Riemannian manifold (X,g), depending on a pair of parameters, n and b. Here n is the dimension of X, while b is its first Betti number. The proof of the inequalities involves…

We investigate the nodal volume of random hyperspherical harmonics $\lbrace T_{\ell;d}\rbrace_{\ell\in \mathbb N}$ on the $d$-dimensional unit sphere ($d\ge 2$). We exploit an orthogonal expansion in terms of Laguerre polynomials; this…

Motivated by problems of mathematical physics (quantum chaos) questions of equidistribution of eigenfunctions of the Laplace operator on a Riemannian manifold have been studied by several authors. We consider here, in analogy with…

We prove invariance theorems for general inequalities of different metrics and apply them to limit relations between the sharp constants in the multivariate Markov-Bernstein-Nikolskii type inequalities with the polyharmonic operator for…

Motivated by establishing Neumann Talenti type comparison results, we concern the minimization of the following shape functional under volume constraint: \begin{align*} T(\Omega):=\inf\left\{\frac12 \int_{\Omega} |\nabla u|^2\,dx…

This paper is concerned with the structure of the set of Riemannian metrics on a connected manifold such that the corresponding Laplace--Beltrami operator has an eigenvalue of a given multiplicity. The starting point of our investigation is…

We establish lower bounds for the sum of the reciprocals of eigenvalues of the Laplacian. For bounded domains, our result extends the upper bound provided by Bucur and Henrot on the second Neumann eigenvalue and is related to a result by…

We study the spectrum of the Laplace-Beltrami operator on ellipsoids. For ellipsoids that are close to the sphere, we use analytic perturbation theory to estimate the eigenvalues up to two orders. We show that for biaxial ellipsoids…