The maximal curves and heat flow in fully affine geometry

Abstract

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 $y=x^\alpha ~\left(\alpha\;\text{is a constant and}\;\alpha\notin\{0,1,\frac{1}{2},2\}\right)$ 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.