完全仿射几何中的极大曲线与热流
The maximal curves and heat flow in fully affine geometry
AI总结：
本文研究完全仿射几何中的极大曲线，通过第二变分公式证明其丰富性，并推广曲线理论，给出热流孤子分类及收敛性结果。
摘要：
在欧几里得几何中，两点之间的最短距离是直线。Chern 在1977年提出猜想：二维欧几里得空间 $\mathbb{R}^2$ 上光滑、局部一致凸函数的仿射极大图必定是抛物面。2000年，Trudinger 和 Wang 完成了该猜想在仿射几何中的证明。（注意：在这些文献中，“仿射几何”指的是“等仿射几何”。）一个自然的问题随之产生：双曲线是否是 $\mathbb{R}^2$ 中的完全仿射极大曲线？本文利用曲线的演化方程，得到了 $\mathbb{R}^2$ 中完全仿射极值曲线的第二变分公式，并证明了 $\mathbb{R}^2$ 中的完全仿射极大曲线远比预想的丰富，其中包括显式曲线 $y=x^\alpha~\left(\alpha\\;\text{为常数且}\\;\alpha\notin\{0,1,\frac{1}{2},2\}\right)$ 和 $y=x\log x$。同时，我们将曲线的基本理论推广到配备 $\text{GA}(n)=\text{GL}(n)\ltimes\mathbb{R}^n$ 的高维情形。此外，在完全仿射平面几何中，我们研究了一个等周不等式，并提供了完全仿射热流的孤子完整分类。我们还研究了该完全仿射热流的局部存在性、唯一性和长期行为。我们证明了闭嵌入曲线在完全仿射热流演化下将收敛于椭圆。
英文摘要：
In Euclidean geometry, the shortest distance between two points is a straight line. Chern made a conjecture in 1977 that an affine maximal graph of a smooth, locally uniformly convex function on two-dimensional Euclidean space $\mathbb{R}^2$ must be a paraboloid. In 2000, Trudinger and Wang completed the proof of this conjecture in affine geometry. (Caution: in these literatures, the term "affine geometry" refers to "equi-affine geometry".) A natural problem arises: Whether the hyperbola is the fully affine maximal curve in $\mathbb{R}^2$? In this paper, by utilizing the evolution equations for curves, we obtain the second variational formula for fully affine extremal curves in $\mathbb{R}^2$, and show the fully affine maximal curves in $\mathbb{R}^2$ are much more abundant and include the explicit curves $y=x^α~\left(α\;\text{is a constant and}\;α\notin\{0,1,\frac{1}{2},2\}\right)$ and $y=x\log x$. At the same time, we generalize the fundamental theory of curves in higher dimensions, equipped with $\text{GA}(n)=\text{GL}(n)\ltimes\mathbb{R}^n$. Moreover, in fully affine plane geometry, an isoperimetric inequality is investigated, and a complete classification of the solitons for fully affine heat flow is provided. We also study the local existence, uniqueness, and long-term behavior of this fully affine heat flow. A closed embedded curve will converge to an ellipse when evolving according to the fully affine heat flow is proved.
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