The maximal curves and heat flow in fully affine geometry

TL;DR

This paper explores fully affine maximal curves in two-dimensional space, revealing their abundance beyond paraboloids, and studies the associated heat flow, including solitons, inequalities, and convergence properties.

Contribution

It introduces explicit fully affine maximal curves, generalizes curve theory to higher dimensions, and analyzes the fully affine heat flow's solitons and long-term behavior.

Findings

Fully affine maximal curves include explicit forms like $y=x^\alpha$ and $y=x\log x$.

The paper generalizes curve theory to higher dimensions with GA(n) group.

The fully affine heat flow's solitons are classified, and convergence to ellipses is proved.

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.