Related papers: On algebraic solitons for geometric evolution equa…
In this paper, we compute canonical connections and Kobayashi-Nomizu connections and their curvature on three-dimensional Lorentzian Lie groups with some product structure. We define algebraic Ricci solitons associated to canonical…
For all left-invariant Riemannian metrics on three-dimensional unimodular Lie groups, there exist particular left-invariant orthonormal frames, so-called Milnor frames. In this paper, for any left-invariant Riemannian metrics on any Lie…
In this paper, we consider a translating soliton for the mean curvature flow starting from a graph of a function on a domain in a unit sphere which is constant along each leaf of isoparametric foliation. First, we show that such a function…
Lamb has identified a certain class of moving space curves with soliton equations. We show that there are two other classes of curve evolution that may be so identified. Hence three distinct classes of curve evolution are associated with a…
We consider Lie groups equipped with a left-invariant cyclic Lorentzian metric. As in the Riemannian case, in terms of homogeneous structures, such metrics can be considered as different as possible from bi-invariant metrics. We show that…
It is well known that $\mathbb{C}H^n$ has the structure of solvable Lie group with left invariant metric of constant holomorphic sectional curvature. In this paper we give the full classification of all possible left invariant Riemannian…
We study modular and integral flow polynomials of graphs by means of subgroup arrangements and lattice polytopes. We introduce an Eulerian equivalence relation on orientations, flow arrangements, and flow polytopes; and we apply the theory…
In this article we investigate a system of geometric evolution equations describing a curvature driven motion of a family of 3D curves in the normal and binormal directions. Evolving curves may be subject of mutual interactions having both…
In this work, we propose a new evolving geometric flow (called translating mean curvature flow) for the translating solitons of hypersurfaces in $R^{n+1}$. We study the basic properties, such as positivity preserving property, of the…
There are five unimodular simply connected three dimensional unimodular non abelian Lie groups: the nilpotent Lie group $\mathrm{Nil}$, the special unitary group $\mathrm{SU}(2)$, the universal covering group…
We consider the existence of cohomogeneity one solitons for the isometric flow of $G_2$-structures on the following classes of torsion-free $G_2$-manifolds: the Euclidean $R^7$ with its standard $G_2$-structure, metric cylinders over…
In this paper, we compute the Bott connections and their curvature on three-dimensional Lorentzian Lie groups with three different distributions, and we classify affine Ricci solitons associated to the Bott connection on three-dimensional…
Normal geodesic flows flows of Carnot-Caratheodory are discussed from the point of view of the theory of Hamiltonian systems. The geodesic flows corresponding to left-invariant metrics and left- and -right-invariant rank 2 distributions on…
We study some properties of mean curvature flow solitons in general Riemannian manifolds and in warped products, with emphasis on constant curvature and Schwarzschild type spaces. We focus on splitting and rigidity results under various…
This paper investigates curve flows on the light-cone in the 3-dimensional Minkowski space. We derive the Harnack inequality for the heat flow and present a detailed classification of space-periodic solitons for a third-order curvature…
The approach to nonholonomic Ricci flows and geometric evolution of regular Lagrange systems [S. Vacaru: J. Math. Phys. \textbf{49} (2008) 043504 \& Rep. Math. Phys. \textbf{63} (2009) 95] is extended to include geometric mechanics and…
This thesis presents three results in geometric analysis. We first analyze the curve-shortening flow on figure eight curves in the plane. Afterwards, we examine the point-wise curvature preserving flow on space curves. Lastly, we present an…
The mitosis process of an eukaryotic cell can be represented by the structure constants of an evolution algebra. Any isotopism of the latter corresponds to a mutation of genotypes of the former. This paper uses Computational Algebraic…
In this paper, we define the corresponding submanifolds to left-invariant Riemannian metrics on Lie groups, and study the following question: does a distinguished left-invariant Riemannian metric on a Lie group correspond to a distinguished…
Universal bi-Hamiltonian hierarchies of group-invariant (multicomponent) soliton equations are derived from non-stretching geometric curve flows $\map(t,x)$ in Riemannian symmetric spaces $M=G/H$, including compact semisimple Lie groups…