Related papers: Two balls maximize the third Neumann eigenvalue in…

In this article we will explore Dirichlet Laplace eigenvalues on balls on spherically symmetric manifolds. We will compare any Dirichlet Laplace eigenvalue with the corresponding Dirichlet Laplace eigenvalue on balls in Euclidean space with…

Let $\Sigma$ be a $k$-dimensional complete proper minimal submanifold in the Poincar\'{e} ball model $B^n$ of hyperbolic geometry. If we consider $\Sigma$ as a subset of the unit ball $B^n$ in Euclidean space, we can measure the Euclidean…

We study overdetermined problems for fully nonlinear elliptic equations in subdomains $\O$ of the Euclidean sphere $\mathbb{S}^{N}$ and the hyperbolic space $\mathbb{H}^{N}$. We prove, the existence of a classical solution to the underlined…

A soft presentation of hyperbolic spaces, free of differential apparatus, is offered. Fifth Euclid's postulate in such spaces is overthrown and, among other things, it is proved that spheres (equipped with great-circle distances) and…

In the seminal work of Balogh-Buckley [Invent. Math. 2003], the authors asked the following fundamental open problem: for proper subdomains in the Euclidean space $\mathbb{R}^n$, does the ball separation condition alone imply the…

In this paper, we prove an isoperimetric inequality for lower order eigenvalues of the free membrane problem on bounded domains of a Euclidean space or a hyperbolic space which strengthens the well-known Szeg\"o-Weinberger inequality and…

We prove that among all triangles of given diameter, the equilateral triangle minimizes the sum of the first $n$ eigenvalues of the Neumann Laplacian, when $n \geq 3$. The result fails for $n=2$, because the second eigenvalue is known to be…

We prove that the second positive Neumann eigenvalue of a bounded simply-connected planar domain of a given area does not exceed the first positive Neumann eigenvalue on a disk of a twice smaller area. This estimate is sharp and attained by…

We consider a volume maximization program to construct hyperbolic structures on triangulated 3-manifolds, for which previous progress has lead to consider angle assignments which do not correspond to a hyperbolic metric on each simplex. We…

The second named author and David Kalaj introduced a pseudometric on any domain in the real Euclidean space $\mathbb R^n$, $n\ge 3$, defined in terms of conformal harmonic discs, by analogy with Kobayashi's pseudometric on complex…

Previous works provided several counterexamples to monotonicity of the lowest eigenvalue for the magnetic Laplacian in the two-dimensional case. However, the three-dimensional case is less studied. We use the results obtained by Helffer,…

We prove that if $C$ is a symmetric convex body of revolution in $\mathbb R^4$ containing the unit Euclidean ball $\mathbb B_4$, such that the sections of $C$ by hyperplanes tangent to $\mathbb B_4$ have constant area $A>0$, then $C$ is a…

We consider the problem of minimising the $k$-th eigenvalue of the Laplacian with some prescribed boundary condition over collections of convex domains of prescribed perimeter or diameter. It is known that these minimisation problems are…

We consider hyperbolic 3-manifolds with either non-empty compact geodesic boundary, or some toric cusps, or both. For any such M we analyze what portion of the volume of M can be recovered by inserting in M boundary collars and cusp…

The maximal hyperplane section of the $l_\infty^n$-ball, i.e. of the $n$-cube, is the one perpendicular to 1/sqrt 2 (1,1,0, ... ,0), as shown by Ball. Eskenazis, Nayar and Tkocz extended this result to the $l_p^n$-balls for very large $p…

The classical Szego-Weinberger inequality states that among bounded planar domains of given area, the first nonzero Neumann eigenvalue is maximized by a disk. Recently, it was shown by Girouard, Nadirashvili and Polterovich that, for simply…

We show, using the Kobayashi and Caratheodory metrics on special holomorphic disks in the universal Teichmuller space, that a wide class of holomorphic functionals on the space of univalent functions in the disk is maximized by the Koebe…

In this paper, we prove some isoperimetric bounds for lower order eigenvalues of the Wentzell-Laplace operator on bounded domains of a Euclidean space or a Hadamard manifold, of the Laplacian on closed hypersurfaces of a Euclidean space or…

We prove an upper bound for the volume-normalized second nonzero eigenvalue of the Laplace operator on closed Riemannian manifold, in terms of the conformal volume. This bound provides effective upper bound for a large class of manifolds,…

We show that the ball does not maximize the first nonzero Steklov eigenvalue among all contractible domains of fixed boundary volume in $\mathbb{R}^n$ when $n \geq 3$. This is in contrast to the situation when $n=2$, where a result of…