  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.
