The maximal curves and heat flow in general-affine geometry

In Euclidean geometry, the shortest distance between two points is a straight line. Chern made a conjecture (cf. [11]) in 1977 that an affine maximal graph of a smooth and locally uniformly convex function on two-dimensional Euclidean space $R^2$ must be a paraboloid. In 2000, Trudinger and Wang completed the proof of this conjecture in affine geometry (cf. [47]). (Caution: in these literatures, the term “affine geometry” refers to “equi-affine geometry”.) A natural problem arises: Whether the hyperbola is a general-affine maximal curve in $R^2$? In this paper, by utilizing the evolution equations for curves, we obtain the second variational formula for general-affine extremal curves in $R^2$, and show the general-affine maximal curves in $R^2$ are much more abundant and include the explicit curves $y=x^{\alpha }(\alpha \text{is a constant and}\alpha \notin \{0,1,\frac{1}{2},2\left\}\right)$ and $y=x\log x$. At the same time, we generalize the fundamental theory of curves in higher dimensions, equipped with $\text{GA}\left(n\right)=\text{GL}\left(n\right)⋉R^n$. Moreover, in general-affine plane geometry, an isoperimetric inequality is investigated, and a complete classification of the solitons for general-affine heat flow is provided. We also study the local existence, uniqueness, and long-term behavior of this general-affine heat flow. A closed embedded curve will converge to an ellipse when evolving according to the general-affine heat flow is proved.