An upper bound on the growth of minimal graphs

1. Problem Statement

The paper addresses the asymptotic behavior of solutions to the minimal surface equation over unbounded, simply connected domains in the complex plane. Specifically, it investigates the growth rate of a non-trivial solution $u(z)$ satisfying the boundary value problem:
$$\begin{cases} \text{div}\left( \frac{\nabla u}{\sqrt{1 + |\nabla u|^2}} \right) = 0 & \text{in } D \\ u > 0 & \text{in } D \\ u = 0 & \text{on } \partial D \end{cases}$$
where $D$ is an unbounded, simply connected domain.

The central question is to determine the maximum possible growth of the function $M(r, u) = \max_{|z|=r, z \in D} u(z)$ as $r \to \infty$. Previous results had established lower bounds (logarithmic growth) and upper bounds for specific sub-cases (e.g., domains contained in a half-plane), but a general upper bound for arbitrary simply connected domains was a conjecture.

2. Methodology

Weitsman employs a geometric function theory approach, utilizing isothermal coordinates and harmonic mappings to transform the nonlinear partial differential equation into a problem involving analytic functions.

A. Parametrization via Weierstrass Representation

The surface is parametrized using the right half-plane $H = \{ \zeta = \sigma + i\tau : \sigma > 0 \}$. The mapping is defined by:
$$f(\zeta) = x(\zeta) + iy(\zeta) = h(\zeta) + g(\zeta)$$
where $h$ and $g$ are analytic functions in $H$ satisfying the univalence condition $|h'(\zeta)| > |g'(\zeta)|$. The height function $u$ corresponds to a harmonic function $U(\zeta)$ in $H$.

B. Normalization and Conjugate Functions

Using the boundary condition $u=0$ on $\partial D$, the author normalizes the conjugate harmonic function $V(\zeta)$ (associated with the minimal surface parametrization) to be $V(\zeta) = 2\Im(\zeta)$. Consequently, the height function becomes:
$$U(\zeta) = 2\Re(\zeta) = 2\sigma$$
This establishes a direct link between the geometric height $u$ and the real part of the parameter $\zeta$. The analytic dilatation is defined as $a(\zeta) = g'(\zeta)/h'(\zeta) = -1/h'(\zeta)^2$.

C. Growth Analysis of $h'(\zeta)$

The core of the proof relies on analyzing the growth of the analytic function $h'(\zeta)$.

1. Herglotz Representation: Since $\log|h'(\zeta)|$ is a positive harmonic function in $H$, it is represented via a Poisson-type integral (Lemma 1).
2. Sectorial Estimates: The author restricts attention to a sector $S_{\pi/4}$ within the right half-plane. Using the integral representation, it is shown that $\log|h'(\zeta)|$ grows at least linearly with $1/|\zeta|$ in this sector, implying $|h'(\zeta)| > 1 + k/|\zeta|$.

D. Application of Clunie and Sheil-Small Theorem

The proof utilizes a theorem by Clunie and Sheil-Small regarding the class of univalent harmonic mappings ($S^0_H$).

• By constructing a normalized mapping $F_2(z)$ from the original mapping $f$, the author invokes Theorem E: If $F \in S^0_H$, the image $F(\mathbb{U})$ contains a disk of radius $1/16$.
• This geometric property is used to establish Lemma 2: A small increase in the parameter $\zeta$ (specifically within a disk of radius proportional to $|\zeta|$) guarantees a uniform, bounded increase in the image domain $|f(\zeta)|$.

E. Iterative Expansion Argument

The final step involves an iterative process:

1. Start with an arc $\gamma_0$ in the $\zeta$-plane.
2. Map it to the physical domain to get $\Gamma_0$.
3. Expand $\Gamma_0$ outward by a fixed relative amount.
4. Use the "covering property" (Lemma 2) to show that the preimage of this expansion requires only a bounded increase in the real part of $\zeta$ ($\Re \zeta$).
5. Repeat this process $n$ times. The analysis shows that to increase the radius $r$ in the physical domain linearly (or quadratically in the iteration count), the parameter $\Re \zeta$ grows linearly.

3. Key Contributions

• Resolution of a Conjecture: The paper proves the conjecture (previously stated in [13]) that for any simply connected unbounded domain, the growth of minimal graphs is at most exponential.
• Generalization: Unlike previous results that required the domain to be contained in a half-plane, this result holds for general simply connected domains bounded by a single smooth unbounded Jordan arc.
• Sharpness: The author confirms that the exponential bound is optimal (sharp) by referencing the horizontal catenoid, a classic minimal surface solution that exhibits exactly exponential growth.

4. Main Results

Theorem 1: Let $D$ be a simply connected domain and $u$ satisfy the minimal surface equation (1.1). There exists a constant $C > 0$ such that for sufficiently large $r$:
$$M(r, u) \le e^{Cr}$$
where $M(r, u) = \max_{|z|=r, z \in D} u(z)$.

Corollary on Sharpness: The bound cannot be improved to polynomial growth, as demonstrated by the upper half of the horizontal catenoid, which satisfies $M(r, u) \sim e^{Cr}$.

5. Significance

• Theoretical Completeness: This work closes a significant gap in the theory of minimal surfaces by providing a universal upper bound for the growth of solutions over simply connected domains, complementing existing lower bound results (Theorem A: logarithmic growth).
• Methodological Innovation: The paper demonstrates a powerful technique of combining harmonic mapping theory (specifically the distortion properties of univalent harmonic maps) with the geometry of minimal surfaces. The use of the Clunie-Sheil-Small theorem to control the expansion of the image domain is a novel application in this context.
• Implications for Asymptotics: The result clarifies the asymptotic landscape of minimal graphs. It establishes that while minimal graphs can grow very fast (exponentially), they cannot grow arbitrarily fast (e.g., super-exponentially) under the constraint of being a graph over a simply connected domain with zero boundary values. This helps in classifying the possible behaviors of minimal surfaces at infinity.