Inverse Mean Curvature Flow in Entire Graphs

• Inverse Mean Curvature Flow is a geometric evolution equation where hypersurfaces move outward at speed equal to the reciprocal of their mean curvature, impacting curvature analysis in Rⁿ⁺¹.
• Its formulation for entire graphs uses a fully nonlinear PDE that exhibits ultra-fast diffusion, ensuring global smoothness for star-shaped data and finite extinction for conical profiles.
• Advanced methods like parabolic comparison, barrier techniques, and energy estimates secure preservation of geometry and facilitate detailed long-time existence and convergence analysis.

Inverse mean curvature flow (IMCF) is a geometric evolution equation for hypersurfaces under which each point moves in the outward normal direction at a speed equal to the reciprocal of its mean curvature. IMCF is central in geometric analysis, with connections to isoperimetric inequalities, minimal surfaces, general relativity, and geometric flows. This article presents the canonical IMCF theory for entire graphs in $\mathbb{R}^{n+1}$, with particular emphasis on the analytic structure of the flow, detailed existence and convergence theory for star-shaped and conical initial data, the ultra-fast diffusion regime, and its geometric implications (Daskalopoulos et al., 2017).

1. IMCF for Entire Graphs: Formulation and Evolution Equations

Let $M_t$ be a family of smooth, strictly mean-convex hypersurfaces in $\mathbb{R}^{n+1}$ parametrized as entire graphs

$$M_t = \left\{ (x, u(x,t)) : x \in \mathbb{R}^n \right\},$$

where $u(x,t)$ is a scalar height function. The IMCF is defined by

$$\frac{\partial F}{\partial t}(z,t) = H^{-1}(z,t)\,\nu(z,t), \qquad H>0,$$

with $F(\cdot,t): M^n \to \mathbb{R}^{n+1}$, $H$ the mean curvature, and $\nu$ the outward unit normal.

For graphs, the flow admits a fully nonlinear scalar PDE: $$\partial_t u = -\sqrt{1 + |D u|^2}\ \left[ \mathrm{div}\left( \frac{D u}{\sqrt{1 + |D u|^2}} \right) \right]^{-1}.$$ Here, the mean curvature reads

$$H(x,t) = \mathrm{div}\left(\frac{D u}{\sqrt{1 + |D u|^2}}\right),$$

and the unit normal

$\nu = \frac{1}{\sqrt{1+|D u|^2}}\ (-D u,\ 1).$

This form highlights both the nonlinearity and the degeneracy at points of vanishing curvature.

2. Long-Time Existence and Characterization of Solutions

2.1 Superlinear Star-Shaped Initial Graphs

If the initial data $u_0(x)$ is $C^2$, strictly star-shaped about some fixed point $\bar{x}_0$, and exhibits superlinear growth as $|x| \to \infty$ (i.e., $u_0(x) = |x|^q$, $q > 1$), the IMCF admits a global $C^\infty$ solution for all $t \geq 0$. The key is a uniform lower bound on the support function $$f = H\langle F-\bar{x}_0,\nu\rangle \geq \delta > 0$$, which is preserved under the evolution

$$\partial_t f = -\frac{2}{H^3} \nabla H\cdot\nabla f.$$

Approximations via compact, star-shaped truncations and monotonicity arguments yield a globally smooth limit graph, preserving convexity if present in the initial data.

2.2 Asymptotically Conical Convex Graphs: Critical Phenomenon

For entire convex graphs initially asymptotic to a cone $u_0(x) \sim \alpha_0|x|$ as $|x| \to \infty$, the IMCF exhibits extinction in finite time. The exact solutions for cones evolve as

$$u^i(x,t) = \alpha(t)\,|x| + (\kappa \text{ or } 0), \qquad \alpha'(t) = -\frac{1}{n-1}\left(\alpha + \frac{1}{\alpha}\right),$$

and flatten at time

$$T(\alpha_0) = \frac{n-1}{2} \ln\left(1 + \alpha_0^2\right).$$

The comparison principle traps the solution between the upper and lower cones. As $t\to T$, the entire graph converges in $C^{1,\alpha}_\mathrm{loc}$ to a horizontal plane.

3. Ultra-Fast Diffusion Regime and A Priori Estimates

IMCF for graphs manifests as a fully nonlinear, ultra-fast diffusion process ($m=-1$). The evolution for $v = H^{-1}$ is: $$\partial_t v \leq a_{ij}(x,t)\,D_{ij}v + b_i(x,t)\,D_iv + c(x,t)\,v,$$ with $a_{ij} \sim (1+|x|^2)$, $b_i \sim |x|$, and $c$ bounded. This structure guarantees instantaneous regularization: any weak positivity of $H$ near infinity immediately propagates inward.

Weighted $L^p$ estimates for $v$ use a rescaling,

$$w(t) = \hat{\gamma}(t)\,v, \quad \hat{\gamma}'(t) = \frac{2}{n-1}\,\hat{\gamma} - 2,$$

and a delicate energy argument involving Hardy-type and local Sobolev inequalities, plus Moser iteration in spacetime cylinders, leading to uniform $L^\infty$ control and consequently a positive lower bound on $H$ up to any time $< T$.

4. Parabolic Comparison, Barrier Methods, and Preservation of Geometry

IMCF's analytic degeneracy at $H=0$ is controlled via parabolic maximum principles, allowing barrier techniques on unbounded domains to compare sub- and supersolutions and to maintain graphical and convex properties. Conical barriers guarantee convergence and monotonicity properties even as initial data grow at infinity.

The evolution of support function $f$ and curvature quantities ensures the preservation of strict star-shapedness and convexity for entire graphs.

5. Summary of Key Formulas

Formula Type	Description	Formula
Graph mean curvature	$H$ of entire graph $(x,u(x,t))$	$$H = \mathrm{div}\left(\frac{D u}{\sqrt{1+|D u|^2}}\right)$$
IMCF-graph evolution PDE	Evolution for $u$	$\partial_t u = -\sqrt{1+|Du|^2}\,H^{-1}$
Cone slope evolution ODE	For conical solutions $u(x,t) = \alpha(t)|x| + c$	$$\alpha'(t) = -\frac{1}{n-1}\left(\alpha + \frac{1}{\alpha}\right)$$
Extinction time for cones	Time when conical solutions become flat	$T = \frac{n-1}{2} \ln(1+\alpha_0^2)$
Ultra-fast diffusion for $v$	$v=H^{-1}$ evolution	$$\partial_t v - \nabla\cdot\left( \frac{1}{H^2} \nabla v \right) \leq -\frac{2}{H^2 v} |\nabla v|^2$$
Support function evolution	$f=H\langle F,\nu\rangle$	$$\partial_t f = \frac{1}{H^2}\,\Delta f + \frac{|A|^2}{H^2}f$$

A detailed discussion and additional technical evolution formulas are presented in (Daskalopoulos et al., 2017).

6. Geometric Consequences and Intuition

The ultra-fast diffusion property implies that decay of curvature at spatial infinity instantly impacts the entire graph. The extinction of conical solutions in finite time shows that conical initial data act as a critical threshold: cones flatten to planes in finite time, while superlinear growth forces global existence for all $t$. The contrast with the compact/closed hypersurface case is sharp: entire graphs can exhibit collapse to a flat plane, whereas compact flows expand without bound.

7. Extensions and Related Directions

This IMCF theory for entire graphs forms the foundational setting for analysis of IMCF in noncompact and unbounded domains. It is instrumental for understanding weak solutions, singularity formation, and the classification of long-time behavior under geometric flows. Ultra-fast diffusion phenomena established here have direct analogs in nonlinear heat flows and fast-diffusion equations, with broad geometric and analytic applications (Daskalopoulos et al., 2017).