Motion by weighted mean curvature is affine invariant

Abstract

Suppose curves are moving by curvature in a plane, but one embeds the plane in $R^3$ and looks at the plane from an angle. Then circles shrinking to a round point would appear to be ellipses shrinking to an ``elliptical point,'' and the surface energy would appear to be anisotropic as would the mobility. The result of this paper is that if one uses the apparent surface energy and the apparent mobility, then the motion by weighted curvature with mobility in the apparent plane is the same as motion by curvature in the original plane but then viewed from the angle. This result applies not only to the isotropic case but to arbitrary surface energy functions and mobilities in the plane, to surfaces in 3-space, and (in the case that the surface energy function is twice differentiable) to the case of motion viewed through distorted lenses (i.e., diffeomorphisms) as well. This result is to be contrasted with an earlier result which states that for area-preserving affine transformations of the plane where the energy and mobility are NOT also transformed, motion by curvature to the power 1/3 (rather than 1) is invariant.