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The mathematics of a 4-dimensional renormalizable generally covariant lagrangian model (with first order derivatives) is reviewed. The lorentzian CR manifolds are totally real submanifolds of 4(complex)-dimensional complex manifolds…
The Chern-Simons functionals built from various connections determined by the initial data $h_{\mu\nu}$, $\chi_{\mu\nu}$ on a 3-manifold $\Sigma$ are investigated. First it is shown that for asymptotically flat data sets the logarithmic…
Invariant Lagrangians yield invariant Euler-Lagrange equations, and it was discussed in the literature how to compute those using various local methods. The focus of this paper is on global algebraic differential invariants. In this case…
Let M be a simply-connected complete Kahler manifold whose sectional curvature is bounded between two negative numbers. In this paper we prove the existence of non-constant bounded holomorphic functions on M if the complex dimension of M is…
The requirements of conformal invariance for two and three point functions for general dimension $d$ on flat space are investigated. A compact group theoretic construction of the three point function for arbitrary spin fields is presented…
We prove that on any closed Riemannian three-manifold $(M,g)$ the time-dependent Euler equations are non-mixing on the space of smooth volume-preserving vector fields endowed with the $C^1$-topology, for any fixed helicity and large enough…
Our aim is to study invariant hypersurfaces immersed in the Euclidean space $\mathbb{R}^{n+1}$, whose mean curvature is given as a linear function in the unit sphere $\mathbb{S}^n$ depending on its Gauss map. These hypersurfaces are closely…
Given a three dimensional pseudo-Einstein CR manifold $(M,T^{1,0}M,\theta)$, we establish an expression for the difference of determinants of the Paneitz type operators $A_{\theta}$, related to the problem of prescribing the $Q'$-curvature,…
This paper provides a mathematical perspective on fragile topology phenomena in condensed matter physics. In dimension $d \leq 3$, vanishing Chern classes of bundles of Bloch eigenfunctions characterize operators with exponentially…
Two questions on the topology of compact energy surfaces of natural two degrees of freedom Hamiltonian systems in a magnetic field are discussed. We show that the topology of this 3-manifold (if it is not a unit tangent bundle) is uniquely…
We determine the equilibria of a rigid loop in the plane, subject to the constraints of fixed length and fixed enclosed area. Rigidity is characterized by an energy functional quadratic in the curvature of the loop. We find that the area…
One primary objective in submanifold geometry is to discover fascinating and significant classical examples of $H_1$. In this paper which relies on the theory we established in [Adv. Math. 405 (2022), 08514, 50 pages, arXiv:2101.11780] and…
We examine the equilibrium conditions of a curve in space when a local energy penalty is associated with its extrinsic geometrical state characterized by its curvature and torsion. To do this we tailor the theory of deformations to the…
In this article, we consider metrically thin singularities E of the solutions of the tangential Cauchy-Riemann operators on a C^{2,a}-smooth embedded Cauchy-Riemann generic manifold M (CR functions on M - E) and more generally, we consider…
We give a geometric derivation of Branson's Q-curvature in terms of the ambient metric associated with conformal structures; it naturally follows from the ambient metric construction of conformally invariant operators and can be applied to…
In this paper, we consider the classical variational problem in the Galilean space. we develop the Euler-Lagrange equations for a elastic line on an oriented surface in the Galilean 3-dimensional space $G_3$. Using the varia- tion method,…
We study analytic torsion and eta like invariants on CR contact manifolds of any dimension admitting a circle transverse action, and equipped with a unitary representation. We show that, when defined using the spectrum of relevant operators…
We derive the conformal Ward identities for the correlation functions of the Stress--Energy tensor in probabilistic Liouville Conformal Field Theory on compact Riemann surfaces by varying the correlation functions with respect to the…
Two sharp comparison results are derived for three-dimensional complete noncompact manifolds with scalar curvature bounded from below. The first one concerns the Green's function. When the scalar curvature is nonnegative, it states that the…
We prove that the dynamical system defined by the hydrodynamical Euler equation on any closed Riemannian 3-manifold $M$ is not mixing in the $C^k$ topology ($k > 4$ and non-integer) for any prescribed value of helicity and sufficiently…