Higher Dimensional Holonomy Map for Rules Submanifolds in Graded Manifolds

AbstractThe deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and to reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [29], and we provide a characterization for singularities as well as a deformability criterion

Geometric properties of 2-dimensional minimal surfaces in a sub-Riemannian manifold which models the Visual Cortex

In this paper we study the notion of degree forsubmanifolds embedded in an equiregular sub-Riemannian manifold and we provide the definition of their associated area functional. In this setting we prove that the Hausdorff dimension of a submanifold coincides with its degree, as stated by Gromov. Using these general definitions we compute the first variation for surfaces embedded in low dimensional manifolds and we obtain the partial differential equation associated to minimal surfaces. These minimal surfaces have several applications in the neurogeometry of the visual cortex

Il problema di Bernstein subfinsleriano in H1

This is a note based on the paper [32] written in collaboration with M. Ritoré. The purpose of this note is to present and discuss the Bernstein type problems in the sub-Finsler Heisenberg group H1. We give a general idea of the state of the art and we prove that a complete, stable, (X,Y)-Lipschitz surface is a vertical plane.Queste note sono basate sull'articolo [32] scritto in collaborazione con M. Ritoré. Lo scopo di queste note è quello di presentare e discutere alcuni problemi di tipo Bernstein nel gruppo di Heisenberg H1 subfinsleriano. Forniamo un'idea generale dello stato dell'arte e proviamo che una superficie (X,Y)-lipschitziana è un piano verticale

Variations for submanifolds of fixed degree

The aim of this PhD thesis is to study the area functional for submanifolds immersed in an equiregular graded manifold. This setting, extends the sub-Riemannian one, removing the bracket generating condition. However, even in the sub-Riemannian setting only sub-manifolds of dimension or codimension one have been extensively studied. We will study the general case and observe that in higher codimension new phenomena arise, which can not show up in the Riemannian case. In particular, we will prove the existence of isolated surfaces, which do not admit degree preserving variation: a phenomena observed by now only for curves, related to the notion of abnormal geodesics

Mitogen-induced oscillations of membrane potential and Ca2+ in human fibroblasts

AbstractUsing the whole-cell technique, we have measured recurring hyperpolarizations induced by fetal calf serum and bradykinin in human fibroblasts. By coupling fura-2 microfluorimetry to electrophysiology, we have also measured directly cytosolic Ca2+ and found that Ca2+ oscillations occur in synchrony with membrane currents. Mitogen stimulation of cells in which intracellular K+ had been replaced with Cs+ resulted in the abolishment of the outward current. We conclude then that the mitogen-induced recurring hyperpolarizations in human fibroblasts are due to the opening of Ca2+-activated K+ channels

Regularity of Lipschitz boundaries with prescribed sub-Finsler mean curvature in the Heisenberg group H^1

Given a strictly convex set K\subset\rr^2 of class $C^2$ we consider its associated sub-Finsler $K$-perimeter |\ptl E|_K in \hh^1 and the prescribed mean curvature functional |\ptl E|_K-\int_E f associated to a function $f$. Given a critical set for this functional, we prove that where the boundary of $E$ is Euclidean lipschitz and intrinsic regular, the characteristic curves are of class $C^2$. Moreover, this regularity is optimal. The result holds in particular when the boundary of $E$ is of class $C^1

Monge solutions for discontinuous Hamilton-Jacobi equations in Carnot groups

In this paper we study Monge solutions to stationary Hamilton-Jacobi equations associated to discontinuous Hamiltonians in the framework of Carnot groups. After showing the equivalence between Monge and viscosity solutions in the continuous setting, we prove existence and uniqueness for the Dirichlet problem, together with a comparison principle and a stability result

Variational formulas for submanifolds of fixed degree

We consider in this paper an area functional defined on submanifolds of fixed degree immersed into a graded manifold equipped with a Riemannian metric. Since the expression of this area depends on the degree, not all variations are admissible. It turns out that the associated variational vector fields must satisfy a system of partial differential equations of first order on the submanifold. Moreover, given a vector field solution of this system, we provide a sufficient condition that guarantees the possibility of deforming the original submanifold by variations preserving its degree. As in the case of singular curves in sub-Riemannian geometry, there are examples of isolated surfaces that cannot be deformed in any direction. When the deformability condition holds we compute the Euler-Lagrange equations. The resulting mean curvature operator can be of third order.Horizon 2020 Project ref. 777822: GHAIAMEC-Feder grants MTM2017-84851-C2-1-P and PID2020-118180GB-I00Junta de Andalucía grants A-FQM-441-UGR18 and P20-00164Research Unit MNat SOMM17/6109 and PRIN 2015 “Variational and perturbative aspects of nonlinear differential problems”Universidad de Granada/CBU