Entire downward solitons to the scalar curvature flow in Minkowski space

Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space $\mathbb R^{n+1}$ with speed $f r^α K$, where $K$ is the Gauss curvature, $r$ is the distance from the hypersurface to the origin, and $f$ is a positive and smooth function. If $α\ge n+1$, we prove that the flow exists for all time and converges smoothly after normalisation to a soliton, which is a sphere centred at the origin if $f \equiv 1$. Our argument provides a parabolic proof in the smooth category for the classical Aleksandrov problem, and resolves the dual q-Minkowski problem introduced by…

An Anisotropic shrinking flow and L_p Minkowski problem

We consider a shrinking flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_n^{beta}, where u is the support function of the hypersurface, alpha, beta are two constants, and beta>0, sigma_n is the n-th symmetric polynomial of the principle curvature radii of the hypersurface. We prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a soliton which is a solution of an elliptic equation, when the constants alpha, beta belong to a suitable range, provid…

A class of anisotropic expanding curvature flows

We consider an expanding flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_k^{beta}, where u is the support function of the hypersurface, alpha, beta are two constants, and beta>0, sigma_k is the k-th symmetric polynomial of the principle curvature radii of the hypersurface, k is an integer and 1<= k<= n. We prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a soliton which is a solution of a elliptic equation, when the constants alpha, beta bel…

Conditional stable soliton resolution for a semi-linear Skyrme equation

We study a semi-linear version of the Skyrme system due to Adkins and Nappi. The objects in this system are maps from $(1+3)$-dimensional Minkowski space into the $3$-sphere and 1-forms on $\mathbb{R}^{1+3}$, coupled via a Lagrangian action. Under a co-rotational symmetry reduction we establish the existence, uniqueness, and unconditional asymptotic stability of a family of stationary solutions $Q_n$, indexed by the topological degree $n \in \mathbb{N} \cup \{0\}$ of the underlying map. We also prove that an arbitrarily large equivariant perturbation of $Q_n$ leads to a globally defined soluti…

对称空间中的1+1波映射

1+1 wave maps into symmetric spaces

我们解释了如何应用可积系统的技术来构建$2k$ -孤子同宿波映射从周期Minkowski空间$S^1\times R^1$到一个紧凑的李群，更一般地到一个紧凑的对称空间。本文给出了紧李群$G$上的-1流方程的解与到$G$上的波映射之间的对应关系。我们使用Bäcklund变换构造显式的$2k$ -孤子呼吸解的-1流方程，并表明，相应的波映射是周期的和同宿的。紧对称空间$G/K$可以通过Cartan嵌入嵌入为$G$的全测地子流形。我们规定了与$G$相关的流量方程的约束条件，以确保$G$中的相应波图实际上位于$G/K$中。例如，当$G/K=SU(2)/SO(2)=S^2$时，与SU（2）相关联的约束-1-流方程具有sine-Gordon方程（SGE）作为子方程，并且SGE的经典呼吸子解是2-孤子呼吸子。因此，我们的结果推广了Shatah和Strauss的结果，即SGE的经典呼吸子解产生到$S^2$的周期同宿波映射.当群$G$是非紧群时，$G$上的双不变度量是伪黎曼度量，光滑解的Bäcklund变换往往是奇异的.利用Bäcklund变换证明了存在具有常数边界条件和有限能量的光滑初始数据，使得从$R^{1,1}$到伪黎曼流形$SL(2,R)$的波映射的Cauchy问题在有限时间内发展出奇性.

双曲型零平均曲率方程的三维悬链面稳定性

Stability of the 3-dimensional catenoid for the hyperbolic vanishing mean curvature equation

本文证明三维悬链面在双曲型零平均曲率方程中经平移与推进调制后，对余维一初值扰动渐近稳定，并以 Darboux 变换、Price 律估计和精细剖面克服慢衰减困难。

对称性缺失条件下双曲消失平均曲率方程中悬链面的稳定性

Stability of the Catenoid for the Hyperbolic Vanishing Mean Curvature Equation Outside Symmetry

本文证明了 n≥5 维悬链面作为闵可夫斯基空间中双曲消失平均曲率方程稳态解的非线性渐近稳定性，通过新的轮廓构造、调制分析及向量场方法适配，处理了无对称性假设下余维一初值扰动的拟线性波动方程稳定性问题。