Related papers: Flows of co-closed $G_{2}$-structures
In this paper, it is shown that (with no additional assumptions) on a compact 7-dimensional manifold which admits a $G_2$-structure soliton solutions to the Laplacian flow of R. Bryant can only be shrinking or steady. We also show that the…
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…
In this paper, we consider the generalized Laplace operator equipped with the G-dynamics operator of type I, the Dirichlet and Neumann eigenvalue problems are extended to associate with the G-dynamics of type I, it is proved that the…
2-group global symmetries are a particular example of how higher-form and conventional global symmetries can fuse together into a larger structure. We construct a theory of hydrodynamics describing the finite-temperature realization of a…
In this paper, we observe a set of functionals of metrics which are all decrease under the Calabi flow and have uniform lower bound along the flow, which give rise to a set of integral estimates on the curvature flow. Using these estimates,…
By a perturbation approach, we construct geometric solitons with various vortex structures(vortex pairs, vortex rings) for some geometric flows(Wave maps, Shr\"odinger flows) from Minkowski spaces to ${\mathbb S}^2\subset\R^3$.
A simple scheme for incompressible, constant density flows is presented, which avoids odd-even decoupling for the Laplacian on a collocated grids. Energy stability is implied by maintaining strict energy conservation. Momentum is conserved.…
The small-scale statistical properties of velocity circulation in classical homogeneous and isotropic turbulent flows are assessed through a modeling framework that brings together the multiplicative cascade and the structural descriptions…
We study systems of multiple localized closed string tachyons and the phenomena associated with their condensation, in C3/ZN nonsupersymmetric noncompact orbifold singularities using gauged linear sigma model constructions, following…
Based on the compactness of the moduli of non-collapsed Calabi-Yau spaces with mild singularities, we set up a structure theory for polarized K\"ahler Ricci flows with proper geometric bounds. Our theory is a generalization of the structure…
Geometric Hydrodynamics has flourished ever since the celebrated 1966 paper of V. Arnold. In this paper we present a collection of open problems along with several new constructions in fluid dynamics and a concise survey of recent…
Recently, J. Streets and G. Tian introduced a natural way to evolve an almost-K\"ahler manifold called the symplectic curvature flow, in which the metric, the symplectic structure and the almost-complex structure are all evolving. We study…
In this paper we consider new geometric flow equations, called D-flow, which describe the variation of space-time geometries under the change of the number of dimensions. The D-flow is originating from the non-trivial dependence of the…
We prove that the geodesic flow on closed surfaces displays a hyperbolic set if the shadowing property holds C2-robustly on the metric. Similar results are obtained when considering even feeble properties like the weak shadowing and the…
We revisit the study of $G_2$-structures with special torsion, and isolated singularities. Many of the known examples with conical singularities admit additional symmetries, and we describe circle-invariant $G_2$-structures in this context.…
The eventual concavity properties are useful to characterize geometric properties of the final state of solutions to parabolic equations. In this paper we give characterizations of the eventual concavity properties of the heat flow for…
We introduce and study a conformal heat flow of harmonic maps defined by an evolution equation for a pair consisting of a map and a conformal factor of metric on the two-dimensional domain. This flow is designed to postpone finite time…
Given a symplectic class $[\omega]$ on a four torus $T^4$ (or a $K3$ surface), a folklore problem in symplectic geometry is whether symplectic forms in $[\omega]$ are isotropic to each other. We introduce a family of nonlinear Hodge heat…
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…
In this paper, we survey known results on closed self-shrinkers for mean curvature flow and discuss techniques used in recent constructions of closed self-shrinkers with classical rotational symmetry. We also propose new existence and…