Related papers: $S^1$-invariant Laplacian flow

A first principle explanation of the origin of intermittency and nonlinear structure formation in the Lagrangian velocity increments of a turbulent flow is presented in the context of a scale invariant analytical formalism that is being…

This paper initiates a classification programme of flows of $\mathrm{SU}(2)$-structures on $4$-manifolds which have short-time existence and uniqueness. Our approach adapts a representation-theoretic method originally due to Bryant in the…

We develop an abstract theory of flows of geometric $H$-structures, i.e., flows of tensor fields defining $H$-reductions of the frame bundle, for a closed and connected subgroup $H\subset SO(n)$, on any connected and oriented $n$-manifold…

The curvature and the reduced curvature are basic differential invariants of the pair (Hamiltonian system, Lagrange distribution) on the symplectic manifold. It is shown that the negativity of the reduced curvature implies the hyperbolicity…

We consider the Laplacian "co-flow" of $G_2$-structures: $\frac{d}{dt} \psi = - \Delta_d \psi$ where $\psi$ is the dual 4-form of a $G_2$-structure $\phi$ and $\Delta_d$ is the Hodge Laplacian on forms. This flow preserves the condition of…

Starting from the vortex filament flow introduced in 1906 by Da Rios, there is a hierarchy of commuting geometric flows on space curves. The traditional approach relates those flows to the nonlinear Schr\"odinger hierarchy satisfied by the…

Let $\varphi(t), t\in [0,T]$ be a smooth solution to the Laplacian flow for closed G_2 structures on a compact 7-manifold $M$. We show that for each fixed positive time $t\in (0,T]$, $(M,\varphi(t),g(t))$ is real analytic, where $g(t)$ is…

We prove short time existence and uniqueness of solutions to the Laplacian flow for closed $G_2$ structures on a compact manifold $M^7$. The result was claimed in \cite{BryantG2}, but its proof has never appeared.

We prove the local existence of unique smooth solutions of the Donaldson geometric flow on the space of symplectic forms on a closed smooth four-manifold, representing a fixed cohomology class. It is a semiflow on the Besov space…

In this article, we study geometric and analytical features of complete noncompact $\rho$-Einstein solitons, which are self-similar solutions of the Ricci-Bourguignon flow. We study the spectrum of the drifted Laplacian operator for…

We study the evolution of the Whitney sphere along the Lagrangian mean curvature flow. We show that equivariant Lagrangian spheres in $\mathbb{C}^n$ satisfying mild geometric assumptions collapse to a point in finite time and the tangent…

We study the asymptotic behavior of the pluriclosed flow in the case of left-invariant Hermitian structures on Lie groups. We prove that solutions on 2-step nilpotent Lie groups and on almost-abelian Lie groups converge, after a suitable…

We derive, under a technical assumption, the first variation formula for the eigenvalues of the Laplacian on a closed manifold evolving by the Ricci flow and give some applications.

We study flows of $G_2$-structures guided by the principle of dimensional reduction: natural geometric flows in $G_2$-geometry reduce to natural flows in complex geometry. Our main examples are the $G_2$-Laplacian coflow, which lifts the…

We introduce a notion of stability for non-autonomous Hamiltonian flows on two-dimensional annular surfaces. This notion of stability is designed to capture the sustained twisting of particle trajectories. The main Theorem is applied to…

This paper surveys results found by the authors in the previous papers (see for example, A. Duyunova, V. Lychagin, S. Tychkov, Differential invariants for spherical layer flows of a viscid fluid, Journal of Geometry and Physics, 130,…

We study the relationship between the classical Hamilton flow and the quantum Schr\"odinger evolution where the Hamiltonian is a degree-2 complex-valued polynomial. When the flow obeys a strict positivity condition equivalent to compactness…

We investigate left-invariant Hitchin and hypo flows on $5$-, $6$- and $7$-dimensional Lie groups. They provide Riemannian cohomogeneity-one manifolds of one dimension higher with holonomy contained in $SU(3)$, $G_2$ and $Spin(7)$,…

We study a first variation formula for the eigenvalues of the Laplacian evolving under the Ricci flow in a simple example of a noncommutative matrix geometry, namely a finite dimensional representation of a noncommutative torus. In order to…

In this paper we introduce and study a new kind of hyperbolic geometric flows --dissipative hyperbolic geometric flow. This kind of flow is defined by a system of quasilinear wave equations with dissipative terms. Some interesting exact…