Long time behavior of Riemannian mean curvature flow of graphs
The authors consider the mean curvature flow of graphs \(u\) over convex domains \(\Omega\) in \({\mathbb{R}}^n\) with respect to the conformal metric \(h \delta_{ij}\) where \(h \in {\mathcal{C}}^2(\Omega) \) such that \(h \geq {\text{constant}} >0\). Imposing a zero boundary condition, they prove that the solutions tend to \(0\) exponentially fast as \(t \rightarrow \infty\). The problem has been used to model image recognition. An initial function \(u_0\), which contains the dependence of the observer for a given image \(I_0\), is evolved by the mean curvature flow with respect to a Riemannian metric \(h\delta_{ij}\) induced by \(I_0\). The reconstructed image is the normalization \(u / \sup u\) of the solution. It is further proved in this paper that the quotient \(u / \sup u\) tends to the first eigenfunction of the associated linearized problem.
scientific article; zbMATH DE number 1866568