arXiv research

Study solves overdetermined problems for rotationally invariant Poisson equations in model manifolds.

In this paper we study the gradient Ricci shrinking soliton equation on rotationally symmetric manifolds of dimension three and higher and prove that the only complete examples of such metrics on $S^n$, $\R{n}$ and $\R{}\times S^{n-1}$ are, respectively, the round, flat, and standard cylindrical metrics.

In the present paper, we find a system of non-linear ODEs that gives rotationally invariant solutions to the Kapustin-Witten equations in 4-dimensional Euclidean space. We explicitly solve these ODEs in some special cases and find decaying rational solutions, which provide solutions to the Kapustin-Witten equations. Th…

Invariant reduction preserves Poisson structures in PDEs.

Using the characterization of last multipliers as solutions of the Liouville's transport equation, new results are given in this approach of ODE by providing several new characterizations, e.g. in terms of Witten and Marsden differentials or adjoint vector field. Applications to Hamiltonian vector fields on Poisson man…

The paper studies $G_2$-Poisson equations on 7-spheres and classifies invariant solutions.

Rotationally invariant Ricci flows are constructed and shown to converge to spacetimes.

The paper classifies rotationally symmetric extremal Kähler metrics on complex manifolds.

We present a contact transformation of the generalized Hunter--Saxton equation to the Euler--Poisson equation with special values of the Ovsiannikov invariants. We also find the general solution for the generalized Hunter--Saxton equation.

In this paper we study sets in the $n$-dimensional Heisenberg group $\hhn$ which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in $\hhn$. We define a notion of mean curvature for hypersurfaces and we show that the boundary of a…

Describes reconstructing Poisson structures from Lie group actions.

A new Poisson bracket defined on Poisson structures with applications to fixed points and cohomology.

Rotationally symmetric solutions persist after mean curvature flow starts from a double cone.

We investigate the properties of the Cheeger sets of rotationally invariant, bounded domains $Ω\subset \mathbb{R}^n$. For a rotationally invariant Cheeger set $C$, the free boundary $\partial C \cap Ω$ consists of pieces of Delaunay surfaces, which are rotationally invariant surfaces of constant mean curvature. We show…

We study a second order differential equation corresponding to rotationally symmetric $F$-harmonic maps between certain noncompact manifolds. We show unique continuation and Liouville's type theorems for positive solutions. Asymptotic properties and the existence of bounded positive solutions are investigated.

Researchers transform equations and define integral operators on a ball.

New AMP algorithms for rotationally invariant models with reduced complexity.

Study invariant Poisson structures on homogeneous manifolds, algebraically and geometrically.

New comparison theorems for rotationally symmetric self-shrinkers help in proving the uniqueness of the Angenent torus.

Symmetric Poisson structures linked to geodesic foliations and Jordan algebras.

We describe first integrals of geostrophic equations, which are similar to the enstrophy invariants of the Euler equation for an ideal incompressible fluid. We explain the geometry behind this similarity, give several equivalent definitions of the Poisson structure on the space of smooth densities on a symplectic manif…

Motivated by the rich theory of harmonic maps from a 2-sphere, we study biharmonic maps from a 2-sphere in this paper. We first derive biharmonic equation for rotationally symmetric maps between rotationally symmetric 2-manifolds. We then apply the equation to obtain a classification of biharmonic maps in a family of r…

We study noncommutative bundles and Riemannian geometry at the semiclassical level of first order in a deformation parameter $λ$, using a functorial approach. The data for quantisation of the cotangent bundle is known to be a Poisson structure and Poisson preconnection and we now show that this data defines to a functo…

We study a second order ordinary differential equation corresponding to rotationally symmetric $p$-harmonic maps. We show unique continuation and Liouville's type theorems for positive solutions. We discuss the existence of bounded positive entire solutions. Asymptotic properties of the positive solutions are investiga…

Upper bounds on Laplacian eigenvalues on manifolds with non-negative curvature.

Study on solutions to conformally invariant fourth order equations, classifying their properties.

New bialgebra structures for relative Poisson algebras are introduced.

We give a full classification of complete rotationally invariant surfaces with constant Gauss curvature in Berger spheres: they are either Clifford tori, which are flat, or spheres of Gauss curvature $K \geq K_0$ for a positive constant $K_0$, which we determine explicitly and depends on the geometry of the ambient Ber…

We study invariant Nijenhuis $(1,1)$-tensors on a homogeneous space $G/K$ of a reductive Lie group $G$ from the point of view of integrability of a Hamiltonian system of differential equations with the $G$-invariant Hamiltonian function on the cotangent bundle $T^*(G/K)$. Such a tensor induces an invariant Poisson tens…

New Ricci flow solutions found with rotational symmetry and cone-like singularities.

New methods prove existence of rotating shapes moving in space.

In this paper, we determine the maximally stable, rotationally invariant domains on the catenoids $\cC_a$ (minimal surfaces invariant by rotations) in the Heisenberg group with a left-invariant metric. We show that these catenoids have Morse index at least 3 and we bound the index from above in terms of the parameter $…

New AMP algorithms improve multi-layer signal reconstruction.

The paper constructs many ancient solutions to the Yamabe flow on spheres.

We study a formulation of the standard Poisson sigma model in which the target space Poisson manifold carries the Hamilton action of some finite dimensional Lie algebra. We show that the structure of the action and the properties of the gauge invariant observables can be understood in terms of the associated target spa…

When a Riemannian manifold $(M,g)$ is rotationally symmetric, the critical order of the lower bound of radial curvatures for the absence of eigenvalues of the Laplacian is equal to $-\frac{1}{r}$, where $r$ stands for the distance to the center point. In this paper, we shall perturb the Riemannian metric around a rota…

Reduces field theories using Poisson-Poincaré method.

New algorithm for signal estimation in noisy matrix models.

New proof for rotationally symmetric gradient Ricci solitons in 2-4 dimensions.

Reduces field theories on principal bundles by a subgroup, deriving reduced equations.

In this paper we relate the geometric Poisson brackets on the Grassmannian of 2-planes in R^4 and on the (2,2) Moebius sphere. We show that, when written in terms of local moving frames, the geometric Poisson bracket on the Moebius sphere does not restrict to the space of differential invariants of Schwarzian type. But…

Proves local solvability for $G_2$-structures with Poisson equations.

New ansatz for generalized Kähler surfaces derived from hyperKähler ansatz.

Study fourth order Schrödinger equation on Cartan-Hadamard manifolds, proving existence, scattering, and blow-up results.

Let $u$ denote a solution to a rotationally invariant Hessian equation $F(D^2u)=0$ on a bounded simply connected domain $Ω\subset R^2$, with constant Dirichlet and Neumann data on $\partial Ω$. In this paper we prove that if $u$ is real analytic and not identically zero, then $u$ is radial and $Ω$ is a disk. The fully …

New method extends invariant reduction to rescaled geometric structures.

New algorithms improve rank one signal estimation from noisy data.

Study exact Lie bialgebras from flat Lie groups, classifying them.