The Geometric Heat Equation

In this section, we show how the geometric heat equation arises from specific evolution equations. First, from a curvature deformation in 2D and a similar deformation on 3D surfaces in Euclidean geometry. Then, we explore the corresponding affine geometric flows. To do so, we introduce the basic differential geometry tools needed. There are important differences between the two geometries but the flows turn out to be quite similar and all have the desired property of smoothing along edges and not across them. In this sense, they are all cases of the geometric heat equation. The next section will investigate in more depth the key mathematical properties of the flows.