[Paper Review] Entire Graphs Evolving by Powers of the Mean Curvature

Martin Franzen|arXiv (Cornell University)|Dec 19, 2011

Geometric Analysis and Curvature Flows10 references3 citations

TL;DR

This paper establishes the longtime existence of strictly convex entire graphs evolving under the geometric flow where the normal velocity is a positive power $\rho > 0$ of the mean curvature. Using barrier methods, Harnack inequalities, and local $C^2$-estimates under a $\nu$-condition on the initial graph, the authors prove the existence of a solution $u \in C^{2,1}_{\text{loc}}(\mathbb{R}^n \times (0,\infty)) \cap C^0_{\text{loc}}(\mathbb{R}^n \times [0,\infty))$ that remains strictly mean convex for all time, extending results for $\rho=1$ to general powers.

ABSTRACT

We study convex entire graphs evolving with normal velocity equal to a positive power of the mean curvature. Under mild assumptions we prove longtime existence.

Motivation & Objective

• To establish the existence of strictly convex entire solutions to the geometric flow $\partial_t X = -H^\rho \nu$ for $\rho > 0$.
• To extend known results on mean curvature flow ($\rho=1$) and $\rho$-powers of mean curvature to entire, non-compact convex graphs.
• To prove longtime existence under mild regularity and asymptotic normal behavior assumptions on the initial data.
• To develop local $C^2$-estimates and lower bounds via Harnack inequalities for the parabolic PDE governing the graph evolution.

Proposed method

• The authors use a graphical formulation of the flow, representing the hypersurface as a graph $u: \mathbb{R}^n \to \mathbb{R}$, and derive the parabolic PDE for $u$ using the induced metric and second fundamental form.
• They apply the maximum principle to a carefully chosen test function to derive local $C^2$-estimates on the Hessian $D^2u$.
• A $\nu$-condition is imposed on the initial data: the normal vector varies continuously at infinity, ensuring uniform control on the asymptotic behavior of the graph.
• The Harnack inequality is applied to the solution $\psi(t) = u(\varphi(t),t)$, where $\varphi(t)$ is the point of minimal gradient, to derive positive lower bounds on the velocity $\dot{u}$.
• The existence of a solution is established via approximation by compact convex hypersurfaces $M^k$, which are evolved under the same flow and shown to converge smoothly to a global solution.
• Schauder theory and Krylov-Safonov estimates are used to upgrade regularity, and the Arzelà-Ascoli theorem ensures convergence of a subsequence to a global solution.

Experimental results

Research questions

• RQ1 Under what conditions does a strictly convex entire graph evolve under $\partial_t X = -H^\rho \nu$ for $\rho > 0$ without developing singularities in finite time?
• RQ2 Can the longtime existence result for $\rho=1$ (mean curvature flow) be extended to general powers $\rho > 0$ for entire convex graphs?
• RQ3 What role does the $\nu$-condition—on the asymptotic behavior of the normal vector—play in ensuring uniform $C^1$-bounds and long-term existence?
• RQ4 How can local $C^2$-estimates be established for non-compact graphs using the maximum principle with a novel test function?
• RQ5 Can Harnack-type estimates be used to derive positive lower bounds on the normal velocity, ensuring uniform parabolicity of the PDE?

Key findings

• A strictly convex entire solution $u$ to the flow $\partial_t X = -H^\rho \nu$ exists for all time $t \in [0,\infty)$, with $u \in C^{2,1}_{\text{loc}}(\mathbb{R}^n \times (0,\infty)) \cap C^0_{\text{loc}}(\mathbb{R}^n \times [0,\infty))$.
• The solution remains strictly mean convex for all $t > 0$, ensuring the flow does not degenerate.
• The $\nu$-condition—on the uniform continuity of the normal vector at infinity—ensures locally uniform $C^1$-estimates and is essential for the construction.
• Local $C^2$-estimates are obtained via the maximum principle applied to a test function involving the Hessian and velocity, enabling control on the second derivatives.
• A Harnack inequality is used to derive a positive lower bound on the normal velocity $\dot{u}$, which implies strict parabolicity of the PDE and allows application of Schauder theory.
• The solution is obtained as a smooth limit of approximating solutions on compact domains, with convergence guaranteed by Arzelà-Ascoli and regularity estimates.