双曲型曲率流及其不变解

The study of the curvature flow has been the subject of much research because of its wide range of applications and mathematical beauty. The traditional curvature flow is the evolution of curves or hypersurface, which is related to nonlinear parabolic equations. It is precisely the hyperbolic analogue of the curvature flow that we study in this project. An important feature of these curvature equations to the hyperbolic case, it is thus highly desirable to retain this feature. Based on this, we start with the nonparametric curve flow case and apply the prolongation formula to investigate which corresponding hyperbolic equations' are invariant under the Euclidean and affine groups. We then move on to look at convex hypersurfaces and find the related curvature flows. Some related hyperbolic curvature flows are introduced. We will present a study of these flows. A rather satisfactory family of equations was proposed which can be naturally interpreted from the point of view of support functions or graphs. Short time existence will be established and finite time singularity will be characterized for these equations. The traditional curvature flow is a parabolic equation, and many details of its analysis are based on the parabolic maximum principle, which is not available to us in the hyperbolic curvature flow problem. The equation under investigation here has some remarkable special features not shared by most other hyperbolic equations: we have the advantage of a large group of symmetries. In this project, we will give a systematic investigation on group invariant solution for these curvature flow. Moreover, stability of these group invariant solutions will be considered. In addition, we will use these symmetries and the evolution equations of support function, arc-length and area to study the long time behavior of these equations. This research will be helpful to understand hyperbolic curvature flows.

曲率流的研究由于其鲜明的应用背景和丰富的数学内涵，得到了研究者的广泛重视。传统的曲率流模型是抛物型曲率流。本项目拟将研究双曲型曲率流。申请人注意到描述曲率流的偏微分方程满足一个重要的性质：方程在几何运动群下保持不变。根据这个性质，本项目首先对双曲型偏微分方程进行分类，寻找欧氏不变和仿射不变的双曲方程并建立其与双曲型曲率流之间的等价关系。我们将主要研究描述双曲型曲率流的双曲型方程的适定性和不变解，着重研究解的几何性质和长时间行为。抛物型曲率流在具体分析时可利用抛物方程的极值原理，而这一点并不适用于双曲型曲率流。但是，我们所研究的方程具有其他双曲方程没有的几何特征：拥有丰富的对称群。申请人将系统的研究双曲型曲率流的群不变解及其渐近稳定性，充分利用方程的几何不变性、不变解以及几何量的演化方程，研究方程的几何性质如凹凸性、长时间行为等。本项目的研究有助于深化人们对双曲型曲率流的理解和认识。

本项目主要研究了仿射几何和欧氏几何中不变曲率流问题。首先研究了一类仿射几何中的双曲型不变流。我们通过对一般双曲型方程的分类，得到仿射不变的曲率流。在建立与曲率流等价的双曲型偏微分方程后，我们研究解的局部存在性和长时间存在性。另外根据周长和面积的演化方程，给出了解有限时间爆破的结果。并进一步研究了几何流的不变解。另外本项目研究了欧氏几何中非局部的平面曲率流问题。分别对不同的初始曲线，得到了平面曲线在保长度和保面积的曲率流驱动下最后收敛的结果。对曲率做了各种估计，得到了不同闭浸入曲线的大时间性态。项目另外研究了曲线收缩流特殊解的稳定性问题，并推广到非函数形式的初始曲线情况。项目还研究了中心仿射几何不变几何流问题，研究了解的各种性质。