论文标题
完全仿射几何形状中的最大曲线和热流
The maximal curves and heat flow in fully affine geometry
论文作者
论文摘要
在欧几里得几何形状中，两个点之间的最短距离是一条直线。 Chern在1977年提出了一个猜想，即在二维Euclidean Space上的光滑，局部均匀凸功能的仿射最大图$ \ Mathbb {r}^2 $必须是抛物面。在2000年，特鲁丁（Trudinger）和王（Wang）在仿射几何形状中完成了这种猜想的证明。 （注意：在这些文献中，“仿射几何学”一词是指“等级几何学”。）出现一个自然问题：双曲线是否是$ \ mathbb {r}^2 $中的完全仿射最大曲线？在本文中，通过利用曲线的进化方程式，我们获得了$ \ mathbb {r}^2 $中完全播种极端曲线的第二个变异公式，并显示出$ \ m athbb {r}^2 $中的完全仿射最大曲线，并包括$ y = x^al = al = al = al = al = al y y y y = al = al = al = al = al = all and} \;α\ notin \ {0,1，\ frac {1} {2}，2 \} \ right）$和$ y = x \ log x $。同时，我们将曲线的基本理论推广到更高的维度，配备了$ \ text {ga}（n）= \ text {gl}（n）\ ltimes \ mathbb {r}^n $。此外，在完全仿射平面的几何形状中，研究了等等的不等式，并提供了完全仿射热流的孤子子的完整分类。我们还研究了这种完全仿射热流的局部存在，独特性和长期行为。封闭的嵌入式曲线将在证明完全贴热流量时会收敛到椭圆形。
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.
