Related papers: An integral formula on the Heisenberg group
We consider Ricci flow on two classes of nilpotent Lie groups that generalize the three-dimensional Heisenberg group: the higher-dimensional classical Heisenberg groups, and the groups of real unitriangular matrices. Each group is known to…
We introduce a family of $n$-dimensional Hamiltonian systems which, contain, as special reductions, several superintegrable systems as the Tremblay-Turbiner-Winternitz system, a generalized Kepler potential and the anisotropic harmonic…
The existence of the Hamiltonians of the renormalized point interactions in two and three dimensional Riemannian manifolds and that of a relativistic extension of this model in two dimensions are proven. Although it is much more difficult,…
An $integral$ of a group $G$ is a group $H$ whose commutator subgroup is isomorphic to $G$. This paper continues the investigation on integrals of groups started in the work arXiv:1803.10179. We study: (1) A sufficient condition for a bound…
We propose a systematic method for analyzing Rohrlich-type divisor sums for arbitrary congruence subgroups $\Gamma_0(N)$. Our main theorem unifies various results from the literature, and its significance is illustrated through the…
We prove the coarea formula for Lipschitz maps from the subriemannian $n$th Heisenberg group $\mathbb H_n$ to $\mathbb R^{2n}$. Our result is new even when $n=1$ and provides the simplest vector-valued instance of the coarea formula in…
For every odd prime $p$ and every integer $n\geq 12$ there is a Heisenberg group of order $p^{5n/4+O(1)}$ that has $p^{n^2/24+O(n)}$ pairwise nonisomorphic quotients of order $p^{n}$. Yet, these quotients are virtually indistinguishable.…
On H-type sub-Riemannian manifolds we establish sub-Hessian and sub-Laplacian comparison theorems which are uniform for a family of approximating Riemannian metrics converging to the sub-Riemannian one. We also prove a sharp sub-Riemannian…
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…
In this paper we consider certain classes of generalized double Eisenstein series by simple differential calculations of trigonometric functions. In particular, we give four new transformation formula for some double Eisenstein series. We…
We study the class of 3-dimensional nonlinear 2-hessian equations mentioned in the text. We perform preliminary group classification on 2-hessian equation. In fact, we find additional equivalence transformation on the space (x,y,z,u,f),…
Given a finite subgroup G of SL(2,C) we define an additive 2-category H^G whose Grothendieck group is isomorphic to an integral form of the Heisenberg algebra. We construct an action of H^G on derived categories of coherent sheaves on…
The classification of Riemannian manifolds by the holonomy group of their Levi-Civita connection picks out many interesting classes of structures, several of which are solutions to the Einstein equations. The classification has two parts.…
In this short note, we determine the spectrum of the Heisenberg oscillator which is the operator defined as $L + |z|^2$ on the 3-dimensional Heisenberg group $H_1=\mathbb C \times \mathbb R$ where $L$ stands for the positive sublaplacian…
We prove a family of Hardy-Rellich and Poincar\'e identities and inequalities on the hyperbolic space having, as particular cases, improved Hardy-Rellich, Rellich and second order Poincar\'e inequalities. All remainder terms provided…
We find a volume form on moduli space of double punctured Riemann surfaces whose integral satisfies the Painlev\'e I recursion relations of the genus expansion of the specific heat of 2D gravity. This allows us to express the asymptotic…
The Gauss-Bonnet Formula is a significant achievement in 19th century differential geometry for the case of surfaces and the 20th century cumulative work of H. Hopf, W. Fenchel, C. B. Allendoerfer, A. Weil and S.S. Chern for…
In this article we collect results obtained by the authors jointly with other authors and we discuss old and new ideas. In particular we discuss singularities of the exponential map, completeness and homogeneity for Riemannian Hilbert…
Let $\mathbb{H}^{n}$ be the $(2n+1)$-dimensional Heisenberg group, and let $K$ be a compact subgroup of U(n), such that $(K,\mathbb{H}^{n})$ is a Gelfand pair. Also assume that the $K$-action on $\mathbb{C}^n$ is polar. We prove a…