Bernstein’s Theorem and Generalizations

Theorem 1.9.1

Let \(u: \mathbb{R}^2 \to \mathbb{R}\) be an entire solution of the minimal surface equation. Then \(u\) is an affine function, i.e., \(u(x_1,x_2) = ax_1 + bx_2 + c\) for some constants \(a, b, c \in \mathbb{R}\).

This result was generalized to higher dimensions:

Theorem 1.9.2

Let \(u: \mathbb{R}^n \to \mathbb{R}\) be an entire solution of the minimal surface equation.

If \(n \leq 7\), then \(u\) must be affine.
For \(n \geq 8\), there exist non-affine entire solutions (as shown by a counterexample due to Bombieri–De Giorgi–Giusti [BDGG69]).

Proof. Let

\[ \Sigma=\{(x,u(x)):x\in \mathbb{R}^2\}\subset \mathbb{R}^3 \]

be the graph of an entire solution of the minimal surface equation. By the calibration argument above, \(\Sigma\) is area-minimizing. In particular, \(\Sigma\) is stable, so for every \(\varphi\in C_c^\infty(\Sigma)\),

\[ \int_\Sigma |A|^2\varphi^2\,d\Sigma \leq \int_\Sigma |\nabla^\Sigma \varphi|^2\,d\Sigma, \]

since \(\mathrm{Ric}_{\mathbb{R}^3}=0\).

We first establish a quadratic area bound. For a.e. \(R\gt{}0\), the intersection

\[ \Gamma_R:=\Sigma\cap \partial B_R \]

is a smooth \(1\)-cycle in the sphere \(\partial B_R=S_R\). Since \(H_1(S_R)=0\), the curve \(\Gamma_R\) bounds a region \(D_R\subset S_R\). Replacing \(D_R\) by its complement if necessary, we may assume

\[ |D_R|\leq \frac{1}{2}|S_R|=2\pi R^2. \]

Since \(\Sigma\) is area-minimizing and \(\partial(\Sigma\cap B_R)=\Gamma_R=\partial D_R\), we obtain

\[ |\Sigma\cap B_R|\leq |D_R|\leq 2\pi R^2 \]

for a.e. \(R\gt{}0\).

Now choose the logarithmic cutoff

\[ \eta_R(x)= \begin{cases} 1, & |x|\leq R,\\[4pt] \dfrac{k+\log R-\log |x|}{k}, & R\lt{}|x|\lt{}e^{k}R,\\[8pt] 0, & |x|\geq e^{k}R, \end{cases} \]

Here \(|x|\) denotes the Euclidean distance to the origin in \(\mathbb{R}^3\). Since \(|\nabla^\Sigma |x||\leq 1\), on the annulus \(R\lt{}|x|\lt{}e^{k}R\) we have

\[ |\nabla^\Sigma \eta_R|\leq \frac{1}{k|x|}. \]

Applying stability with \(\varphi=\eta_R\) gives

\[ \int_\Sigma |A|^2\eta_R^2\,d\Sigma \leq \int_\Sigma |\nabla^\Sigma \eta_R|^2\,d\Sigma. \]

To estimate the right-hand side, decompose

\[ B_{e^{k}R}\setminus B_R=\bigcup_{i=0}^{k-1}\bigl(B_{e^{i+1}R}\setminus B_{e^iR}\bigr), \]

Using the area bound,

\[ \begin{aligned} \int_\Sigma |\nabla^\Sigma \eta_R|^2\,d\Sigma &\leq \frac{1}{k^{2}} \sum_{i=0}^{k-1}\int_{\Sigma\cap (B_{e^{i+1}R}\setminus B_{e^iR})} \frac{1}{|x|^2}\,d\Sigma \\ &\leq \frac{1}{k^2} \sum_{i=0}^{k-1}\frac{1}{(e^iR)^2}\, |\Sigma\cap B_{e^{i+1}R}| \\ &\leq \frac{1}{k^2} \sum_{i=0}^{k-1}\frac{1}{(e^iR)^2}\,2\pi (e^{i+1}R)^2 \\ &\leq \frac{C}{k}. \end{aligned} \]

Hence

\[ \int_{\Sigma\cap B_R} |A|^2\,d\Sigma \leq \frac{C}{k}. \]

Letting \(k\to \infty\), we conclude that \(A\equiv 0\) on \(\Sigma \cap B_R\). Since this holds for a.e. \(R\gt{}0\), we have \(A\equiv 0\) on \(\Sigma\). Therefore \(\Sigma\) is totally geodesic, hence a plane in \(\mathbb{R}^3\).

Since \(\Sigma\) is a graph over \(\mathbb{R}^2\), that plane cannot be vertical. Thus

\[ u(x_1,x_2)=ax_1+bx_2+c \]

for some constants \(a,b,c\in\mathbb{R}\). This proves the theorem. ◻

The key point in dimension \(2\) is that stability plus the quadratic area growth

\[ |\Sigma\cap B_R|\leq C R^2 \]

allows the logarithmic cutoff to force

\[ \int_\Sigma |A|^2\,d\Sigma=0. \]

This argument is specific to two dimensions and yields the classical stability proof of Bernstein’s theorem.

Bernstein’s Theorem and Generalizations

Gaoming Wang

Assistant Professor