The maximal curves and heat flow in general-affine geometry

Abstract

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 R2 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 R2? In this paper, by utilizing the evolution equations for curves, we obtain the second variational formula for general-affine extremal curves in R2, and show the general-affine maximal curves in R2 are much more abundant and include the explicit curves y=xα(αis a constant andα∉{0,1,12,2}) and y=xlog⁡x. At the same time, we generalize the fundamental theory of curves in higher dimensions, equipped with GA(n)=GL(n)⋉Rn. 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.