Mean Curvature Flow Closes Open Ends of Noncompact Surfaces of Rotation

Abstract

We discuss the motion of noncompact axisymmetric hypersurfaces Γt evolved by mean curvature flow. Our study provides a class of hypersurfaces that share the same quenching time with that of the shrinking cylinder evolved by the flow and prove that they tend to a smooth hypersurface having no pinching neck and having closed ends at infinity of the axis of rotation as the quenching time is approached. Moreover, they are completely characterized by a condition on initial hypersurface.

On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow

“…This is a continuation of our study [4] on motion of noncompact axisymmetric n-dimensional hypersurface Γ t moved by its mean curvature. Let Γ t be given by a rotation of the graph of a function y = u(x, t) (defined on x ∈ R) around the x-axis (cf [1,2]).…”

“…Let Γ t be given by a rotation of the graph of a function y = u(x, t) (defined on x ∈ R) around the x-axis (cf [1,2]). In our previous paper [4], among other results, we have proved that if u(x, 0) → m := inf x∈R u(x, 0) > 0 as |x| → ∞, then Γ t closes open ends at the time T (m), where T (m) is the quenching (pinching) time of the regular cylinder with radius m. (Moreover, there is no neck-pinch in R at t = T (m).) These results imply that lim x→∞ u(x, T (m)) = 0 or lim x→−∞ u(x, T (m)) = 0, but it does not provide the convergence rate.…”

“…The Cauchy problem (1)-( 2) has a unique positive classical solution with the conditions (3)-( 4) to the initial data (cf [4]). However, the solution quenches in finite time.…”

On the flow of non-axisymmetric perturbations of cylinders via surface diffusion

“…A proof of well-posedness is carried out in section 3 of the current article. For other recent results regarding geometric evolution equations in unbounded settings, refer to Giga, Seki, Umeda [21,22], wherein the mean curvature flow is shown to close open ends of non-compact surfaces of revolution given appropriate decay rates at the ends of the initial surface. The current setting differs from that of Giga et.…”

Mean curvature flow without singularities

“…T. Colding and W. Minicozzi [7] consider generic initial data that develop only singularities that look spherical or cylindrical. In the rotationally symmetric case, Y. Giga, Y. Seki and N. Umeda consider mean curvature flow that changes topology at infinity [17,18].…”