arXiv research

It is shown that two Levi-Tanaka and infinitesimal CR automorphism algebras, associated with a totally nondegenerate model of CR dimension one are isomorphic. As a result, the model surfaces are maximally homogeneous and standard. This gives an affirmative answer in CR dimension one to a certain question formulated by …

Researchers create normal forms for CR manifolds in complex space.

In this paper, we prove that every real analytic totally nondegenerate model CR manifold of length >= 3 has rigidity. This result was actually conjectured before by Valerii Beloshapka as the so-called "maximum conjecture". It follows that the transformation Lie group of all CR automorphisms associated with each of the …

We construct the first examples of complete, properly embedded minimal surfaces in $\mathbb{H}^2 \times \mathbb{R}$ with finite total curvature and positive genus. These are constructed by gluing copies of horizontal catenoids or other nondegenerate summands. We also establish that every horizontal catenoid is nondegen…

A CR manifold $M$, with CR distribution $\mathcal D^{10}\subset T^\mathbb C M$, is called {\it totally nondegenerate of depth $μ$} if: (a) the complex tangent space $T^\mathbb C M$ is generated by all complex vector fields that might be determined by iterated Lie brackets between at most $μ$ fields in $\mathcal D^{10} …

Study nondegenerate fibrations of Euclidean spaces and their relation to sphere fibrations.

A well known Conjecture due to Beloshapka asserts that all totally nondegenerate polynomial models with the length $l\geq 3$ of their Levi-Tanaka algebra are {\em rigid}, that is, any point preserving automorphism of them is completely determined by the restriction of its differential at the fixed point onto the comple…

Motivated by the ideas and methods used by Naitoh in the consideration of parallel totally real submanifolds in complex space forms, the author of the present paper successfully makes use of the so called Jordan triple and (restricted) structure Lie algebra associated with a given Jordan algebra to establish a one-to-o…

Applying Elie Cartan's classical method, we show that the biholomorphic equivalence problem to a totally nondegenerate Beloshapka's model of CR dimension one and codimension $k> 1$, whence of real dimension $2+k$, is reducible to some absolute parallelism, namely to an {e}-structure on a certain prolonged manifold of r…

We study higher rank Cartan actions on compact manifolds preserving an ergodic measure with full support. In particular, we classify actions by $\R ^k$ with $k \geq 3$ whose one-parameter groups act transitively as well as nondegenerate totally nonsymplectic $\Zk$-actions for $k \geq 3$.

Classifies homogeneous CR hypersurfaces in low dimensions with maximal symmetry.

New metrics produce discrete zero sets for nondegenerate harmonic forms.

Models of 2-nondegenerate CR hypersurfaces in C^N are characterized and their defining equations simplified.

We study CR hypersurfaces in C^4 with constant rank Levi form and find their defining equations.

The study finds nondegenerate harmonic 1-forms using symmetry conditions.

The Arnold conjecture is proven for integers using Floer theory.

New Lie groups generalize H-type groups with nondegenerate centers.

Characterizes CR manifolds in complex flag manifolds.

Study on CR structures in 7D, proving maximal symmetry dimension.

A smooth fibration of $\mathbb{R}^3$ by oriented lines is given by a smooth unit vector field $V$ on $\mathbb{R}^3$, for which all of the integral curves are oriented lines. Such a fibration is called skew if no two fibers are parallel, and it is called nondegenerate if $\nabla V$ vanishes only in the direction of $V$.…

We study the generalized Kähler-Ricci flow on complex surfaces with nondegenerate Poisson structure, proving long time existence and convergence of the flow to a weak hyperKähler structure.

GN algorithm solves batched bandit for nondegenerate functions near-optimally.

Study finds maximal symmetry groups for CR structures with specific properties.

Uniqueness of nondegenerate blowups for planar networks shown.

The study describes Nijenhuis operators with specific properties.

We extend the notion of a fundamental negatively $\mathbb Z$-graded Lie algebra $\mathfrak{m}_x=\bigoplus_{p\leq -1}\mathfrak{m}_x^p$ associated to any point of a Levi nondegenerate CR manifold to the class of $k$-nondegenerate CR manifolds $(M,\mathcal D,\mathcal J)$ for all $k\geq 2$ and call this invariant the core …

Study neck pinches in Lagrangian flows, proving stability and introducing new singularities.

Classifies a specific type of Lie groups related to Einstein geometry.

We use Cartan's method of moving frames to compute a complete set of local invariants for nondegenerate, 2-dimensional centroaffine surfaces in $\mathbb{R}^5 \setminus \{0\}$ with nondegenerate centroaffine metric. We then give a complete classification of all homogeneous centroaffine surfaces in this class.

Normal forms and invariants for nondegenerate hypersurfaces in C^2.

Mean curvature flow is not a gradient flow on two nondegenerate metric spaces.

The study generalizes a specific geometric correspondence to higher dimensions.

The paper introduces Morse theory for Lie groupoids and proves inequalities.

For an almost complex structure $J$ in dimension 6 with nondegenerate Nijenhuis tensor $N_J$, the automorphism group $G=Aut(J)$ of maximal dimension is the exceptional Lie group $G_2$. In this paper we establish that the sub-maximal dimension of automorphism groups of almost complex structures with nondegenerate $N_J$,…

The study identifies two sources of invariants in 2--nondegenerate CR geometries.

This paper contains some more results on the topology of a nondegenerate action of $\mathbb{R}^n$ on a compact connected $n$-manifold $M$ when the action is totally hyperbolic (i.e. its toric degree is zero). We study the $\mathbb{R}$-action generated by a fixed vector of $\mathbb{R}^n$, that provides some results on t…

For Hamiltonian flows we establish the existence of periodic orbits on a sequence of level sets approaching a Bott-nondegenerate symplectic extremum of the Hamiltonian. As a consequence, we show that a charge on a compact manifold with a nondegenerate (i.e. symplectic) magnetic field has periodic orbits on a sequence o…

Study nondegenerate singularities in mean curvature flow.

3D contact forms have supporting decompositions, leading to entropy results.

We introduce geometric flows on a compact almost complex manifold, with the aim to flow a nondegenerate two form to a symplectic two form. We discuss mainly two flows, $d^*d$-flow and $d^*d$-Ricci flow. Among others, we prove the uniqueness and short time existence for smooth initial data. We also discuss the extension…

In this note, we show that a nontrivial, compact, degenerate or nondegenerate, gradient Einstein-type manifold of constant scalar curvature is isometric to the standard sphere with a well defined potential function. Moreover, under some geometric assumptions, the noncompact case is also treated. In this case, the main …

We extend the notions of CR GJMS operators and Q-curvature to the case of partially integrable CR structures. The total integral of the CR Q-curvature turns out to be a global invariant of compact nondegenerate partially integrable CR manifolds equipped with an orientation of the bundle of contact forms, which is nontr…

A parametric curve $γ$ of class $C^n$ on the $n$-sphere is said to be nondegenerate (or locally convex) when $\det\left(γ(t),γ'(t),\cdots,γ^{(n)}(t)\right)>0$ for all values of the parameter $t$. We orthogonalize this ordered basis to obtain the Frenet frame $\mathfrak{F}_γ$ of $γ$ assuming values in the orthogonal gro…

We define a nondegenerate Monge-Ampère structure on a 6-dimensional manifold as a pair $(Ω,ω)$, such that $Ω$ is a symplectic form and $ω$ is a 3-differential form which satisfies $ω\wedgeΩ=0$ and which is nondegenerate in the sense of Hitchin. We associate with such a pair a generalized almost (pseudo) Calabi-Yau stru…

The paper explores CR structures and their leaf spaces in semi-Riemannian manifolds.

New insights into $(2,3,5)$-distributions via Legendrian curves.

In this article, we solve the equivalence problem for 2--nondegenerate CR geometries that have (at every point) a homogeneous space $G/H$ as a maximally symmetric model for $G$ simple real Lie group of CR automorphisms. This completes the classification of real submanifolds in complex space that are maximally symmetric…

We apply E. Cartan's method of equivalence to classify 7-dimensional, 2-nondegenerate CR manifolds $M$ up to local CR equivalence in the case that the cubic form of $M$ satisfies a certain symmetry property with respect to the Levi form of $M$. The solution to the equivalence problem is given by a parallelism on a prin…