Irina Markina

Geometric analysis on quaternion ℍ-type groups

We construct some examples of ℍ-types Carnot groups related to quaternion numbers and study their geometric properties. We involve the Hamiltonian formalism to obtain the equations of geodesics and calculate the cardinality of geodesics joining two different points on these groups. We prove Kepler’s law and give a nice geometric interpretation of the length of geodesies.

On the geometry of Hele-Shaw flows with small surface tension

We discuss the Hele‐Shaw problem in the plane in two basic cases. The first one deals with the classical situation of injection through a unique source in a finite region. The second one is concerned with the free boundary extending to the point at infinity. Starting with the earlier works of L. A. Galin [9] and P. Ya. Polubarinova-Kochina [19, 20], various aspects of the planar Hele‐Shaw viscous flows with vanishing surface tension were investigated by a number of scientists. It is known [31] that in the zero-surface-tension Hele‐Shaw problem with an initial region with an analytic boundary the classical solution exists locally in time. Recently [25], it became clear that the model could be interpreted as a particular case of the abstract Cauchy problem, and thus, the classical solvability (locally in time) may be proved using the nonlinear abstract Cauchy‐Kovalevskaya Theorem.

Hopf fibration: Geodesics and distances

Abstract Here we study geodesics connecting two given points on odd-dimensional spheres respecting the Hopf fibration. This geodesic boundary value problem is completely solved in the case of three-dimensional sphere and some partial results are obtained in the general case. The Carnot–Caratheodory distance is calculated. We also present some motivations related to quantum mechanics.

On coincidence of p-module of a family of curves and p-capacity on the Carnot group

The notion of the extremal length and the module of families of curves has been studied extensively and has given rise to a lot of applications to complex analysis and the potential theory. In particular, the coincidence of the p-module and the p-capacity plays an mportant role. We consider this problem on the Carnot group. The Carnot group G is a simply connected nilpotent Lie group equipped vith an appropriate family of dilations. Let omega be a bounded domain on G and Ko, K1 be disjoint non-empty compact sets in the closure of omega. We consider two quantities, associated with this geometrical structure...

Sub-Riemannian geometry of parallelizable spheres

The first aim of the present paper is to compare various subRiemannian structures over the three dimensional sphere S originating from different constructions. Namely, we describe the sub-Riemannian geometry of S arising through its right action as a Lie group over itself, the one inherited from the natural complex structure of the open unit ball in C and the geometry that appears when it is considered as a principal S−bundle via the Hopf map. The main result of this comparison is that in fact those three structures coincide. We present two bracket generating distributions for the seven dimensional sphere S of step 2 with ranks 6 and 4. The second one yields to a sub-Riemannian structure for S that is not widely present in the literature until now. One of the distributions can be obtained by considering the CR geometry of S inherited from the natural complex structure of the open unit ball in C. The other one originates from the quaternionic analogous of the Hopf map.

On the Hodge-type decomposition and cohomology groups of k-Cauchy–Fueter complexes over domains in the quaternionic space

Abstract The k -Cauchy–Fueter operator D 0 ( k ) on one dimensional quaternionic space H is the Euclidean version of spin k / 2 massless field operator on the Minkowski space in physics. The k -Cauchy–Fueter equation for k ≥ 2 is overdetermined and its compatibility condition is given by the k -Cauchy–Fueter complex. In quaternionic analysis, these complexes play the role of Dolbeault complex in several complex variables. We prove that a natural boundary value problem associated to this complex is regular. Then by using the theory of regular boundary value problems, we show the Hodge-type orthogonal decomposition, and the fact that the non-homogeneous k -Cauchy–Fueter equation D 0 ( k ) u = f on a smooth domain Ω in H is solvable if and only if f satisfies the compatibility condition and is orthogonal to the set ℋ ( k ) 1 ( Ω ) of Hodge-type elements. This set is isomorphic to the first cohomology group of the k -Cauchy–Fueter complex over Ω , which is finite dimensional, while the second cohomology group is always trivial.

Generalized Hamilton—Jacobi Equation and Heat Kernel on Step Two Nilpotent Lie Groups

We study geometrically invariant formulas for heat kernels of subelliptic differential operators on two step nilpotent Lie groups and for the Grusin operator in ℝ2. We deduce a general form of the solution to the Hamilton—Jacobi equation and its generalized form in ℝn × ℝm. Using our results, we obtain explicit formulas of the heat kernels for these differential operators.

Rigidity of 2-Step Carnot Groups

In the present paper, we study the rigidity of 2-step Carnot groups, or equivalently, of graded 2-step nilpotent Lie algebras. We prove the alternative that depending on bi-dimensions of the algebra, the Lie algebra structure makes it either always of infinite type or generically rigid, and we specify the bi-dimensions for each of the choices. Explicit criteria for rigidity of pseudo H- and J-type algebras are given. In particular, we establish the relation of the so-called

Rolling Against a Sphere: The Non-transitive Case

J^2