Bernstein’s Theorem and the Gauss Map
Throughout, manifolds and maps are smooth unless stated otherwise. On a Riemannian manifold, \(\nabla\) denotes the Levi–Civita connection. The needed local constructions start with coordinate charts.
Bernstein’s theorem gives a rigidity result for minimal graphs over the whole plane: every such graph is a plane. The argument combines a curvature estimate with cutoff functions whose Dirichlet energies tend to zero. We say a graph \(\Sigma\) is entire if it is defined over all of \(\mathbb R^2\). Here is one of the classical results that we wish to prove.
Theorem 1 (Bernstein). Let \(u : \mathbb R^2 \to \mathbb R\) be an entire solution to the minimal surface equation. Then, \(u(x,y) = ax + by + c\) for some constants \(a,b,c\in \mathbb R\).
We use two geometric facts: the upward Gauss map takes values in a hemisphere, and minimal graphs have a uniform quadratic bound for their area inside ambient balls. The first gives a curvature estimate; the second makes logarithmic cutoff energies small. Both facts and the needed estimates are proved below. The Gauss-map coordinate calculations are in the final section of this note, rather than the unrelated projective-space appendix.
Lemma 2. Let \(u\in C^2(\Omega)\) solve the minimal surface equation and let \(\Sigma=\operatorname{Graph}(u)\). For every compactly supported Lipschitz function \(\eta\) on \(\Sigma\), with support away from its boundary,
Here \(B\), also written \(\operatorname{II}\), is the vector-valued second fundamental form. No bound on \(|\nabla u|\) is required.
Proof. Orient \(\Sigma\) by its upward normal \(N\). It takes values in the open upper hemisphere \(\{(X,Y,Z)\in\mathbb S^2:Z>0\}\). Let \(\omega\) be the standard outward-oriented sphere area form. A bounded primitive, rather than contractibility alone, is essential. An explicit choice is
In polar coordinates on the sphere, \(Z=\cos\theta\), this is \((1-\cos\theta)d\phi\), so \(d\alpha=\sin\theta\,d\theta\wedge d\phi=\omega\), and
The Cartesian formula is smooth at the north pole, removing the polar-coordinate singularity.
The Gauss equation and minimality give \(K=-|B|^2/2\), and the signed Jacobian formula gives \(N^*\omega=K\,dA\), with these orientations. Hence
For a smooth compactly supported \(\eta\), Stokes and Cauchy–Schwarz therefore imply
If the first integral is zero there is nothing to prove; otherwise divide by its square root and square. This gives the stated constant \(16\). For a primitive bounded by \(C_\alpha\), the constant would be \(16C_\alpha^2\), not \(16C_\alpha\).
Lipschitz tests follow by smooth approximation in \(W^{1,2}\) on a fixed compact support. For a \(C^2\) graph, \(N\) is \(C^1\); the pullback/exterior-derivative identity used here holds weakly, by approximating \(N\) in \(C^1\) on compact coordinate patches. Thus the proof does not require unmentioned third derivatives of \(u\). ◻
Remark 3. Lemma 2 tells us that we can bound the total curvature of the graph in terms of the energy of a cutoff function \(\eta\). We will use the following strategy: it is possible to choose a sequence of cutoff functions, \(\eta\), such that this sequence converges to one, but has energy converging to zero. This will be referred to as the logarithm cutoff trick.
Remark 4 (Logarithmic cutoffs). On \(\mathbb R^2\), set
The cutoff is Lipschitz and \(|\nabla\eta_N|=1/(Nr)\) on its transition annulus. Therefore
while \(\eta_N\to1\) on every fixed compact set.
More generally, suppose a surface has an exhaustion \(r\) with \(|\nabla_\Sigma r|\leq1\) almost everywhere, compact sublevels, and \(A(s)=\operatorname{Area}\{r<s\}\leq C_0s^2\). For integer \(N\), divide the transition annulus into \(e^{j-1}<r<e^j\), \(j=N+1,\ldots,2N\). Then
Quadratic growth refers to the area of balls, not the length or area of their boundary. The argument applies to intrinsic distance when intrinsic balls have this bound, and to extrinsic distance for the graphs considered below. It requires no pointwise bound on the lengths of distance spheres.
Area bound for a minimal graph. Suppose \(\Omega\) contains the planar disk of radius \(b\), and take \(0<s<b\). The closed ambient ball \(\overline B_s(0)\subset\mathbb R^3\) projects compactly into \(\Omega\). Extend \(N=(-u_x,-u_y,1)/\sqrt{1+|\nabla u|^2}\) vertically as before; then \(\operatorname{div}N=0\). For a regular radius \(s\), split \(B_s\) into the regions below and above the graph. Their spherical boundary portions, denoted \(S_-\) and \(S_+\), need not be connected. Their common internal boundary is \(\Sigma\cap B_s\), with outward normal \(N\) for the lower region and \(-N\) for the upper region.
Applying the divergence theorem separately gives
Because \(|N|=|\nu|=1\), this area is no larger than either spherical portion's area. Those two areas add to \(4\pi s^2\), so
Approximation by regular radii extends the estimate to all \(s<b\). This is the missing growth input for the cutoff argument; it uses neither a gradient bound nor Bernstein's conclusion. Compare Colding–Minicozzi, §1.2, equation (1.21), p. 5.
Proposition 5. Let \(u\in C^2(\Omega)\) solve the minimal surface equation, let \(R>0,\ \kappa>1\), and assume the open planar disk \(D_{\kappa R}(0)\) is contained in \(\Omega\). For ambient balls in \(\mathbb R^3\),
The constant is universal; the graph need not pass through the ambient origin.
Proof. Put \(a=\sqrt\kappa R\), choose \(a<b<\kappa R\), and let \(r=|x|\) on \(\Sigma=\operatorname{Graph}(u)\). Define
Its support is compact in the graph because the closed planar disk of radius \(b\) lies inside \(\Omega\), and \(u\) is continuous there. Writing \(L=\log(b/a)\), we have
There is no extra factor \(R\) when differentiating \(\log(r/R)\).
The annular estimate in Remark 4 already gives the required decay when \(\kappa\) is large. To justify the displayed bound for every \(\kappa>1\), including thin annuli, use minimality once more. Since \(\Delta_\Sigma r^2=4\),
Let \(v=\log(r/a)\) on \(a<r<b\). Integration by parts, initially for regular radii, gives
The inner boundary term is zero because \(v=0\) at \(r=a\). The flux identity in the equality is the divergence calculation \(2A(b)=b\int_{\partial B_b\cap\Sigma}|\nabla_\Sigma r|\) from monotonicity. Approximation by regular radii gives the estimate for arbitrary \(a,b\).
Consequently \(\int_\Sigma|\nabla_\Sigma\eta|^2\leq4\pi/L\). Lemma 2 yields
Let \(b\uparrow\kappa R\); then \(\log(b/a)\to\frac12\log\kappa\), proving the assertion. This limit keeps every cutoff compactly supported even when the domain is assumed to contain only the open disk \(D_{\kappa R}\). ◻
Apply the curvature estimate on larger and larger balls. The cutoff energy tends to zero, so the total squared second fundamental form must vanish on every fixed compact region.
Proof of Theorem 1. Since the domain is the whole plane, Proposition 5 applies with \(R=1\) and every \(\kappa>1\):
Fix any finite ambient radius \(s\). For all \(\sqrt\kappa>s\), nonnegativity gives
Sending \(\kappa\to\infty\) forces the integral on that fixed region to be zero. Since \(|B|^2\) is continuous and \(s\) was arbitrary, \(B\equiv0\) everywhere. For a graph with upward normal, \(\langle B(\partial_iF,\partial_jF),N\rangle=u_{ij}/\sqrt{1+|\nabla u|^2}\), so all second partial derivatives of \(u\) vanish. Its gradient is constant on connected \(\mathbb R^2\), giving \(u(x,y)=ax+by+c\). ◻
1. The Gauss Map and Local Coordinates on Minimal Graphs
For a connected oriented smooth surface \(\Sigma^2\subset\mathbb R^3\), the chosen unit normal defines a smooth Gauss map \(N:\Sigma\to\mathbb S^2\). Reversing orientation changes it to \(-N\); a nonorientable surface has only local such choices. Since both \(T_x\Sigma\) and \(T_{N(x)}\mathbb S^2\) equal \(N(x)^\perp\) as vector subspaces of \(\mathbb R^3\), the differential identifies with our shape operator:
For a minimal surface, the symmetric operator \(dN\) has eigenvalues \(\kappa,-\kappa\). Thus
Away from flat points, the Gauss map is conformal and reverses orientation with the indicated surface/sphere orientations; at flat points its differential is zero, so the appropriate global statement is weak conformality. The signed area identity is \(N^*\omega=K\,dA\). Its unsigned area counted with multiplicity is
whenever these integrals are finite (and also as nonnegative extended integrals). The area of the set \(N(\Sigma)\) alone can be smaller when the map has multiplicity.
For a graph \(F(x,y)=(x,y,u(x,y))\), put \(p=\nabla u\) and \(W=\sqrt{1+|p|^2}\). Its coordinate metric is
A vector perpendicular to \(p\) has eigenvalue \(1\), and \(p\) has eigenvalue \(1+|p|^2=W^2\). If \(p=0\), both eigenvalues are one. This direct eigenspace argument is necessary: elementary row operations do not generally preserve eigenvalues. The determinant is \(W^2\), and
The scalar normal components of the vector-valued second fundamental form are
Our shape operator matrix is \(-g^{-1}b\). Its trace vanishes exactly when
the nondivergence form of the minimal surface equation. The Gauss equation gives
Equivalently, \(\langle N_x,F_x\rangle=-u_{xx}/W\), \(\langle N_x,F_y\rangle=-u_{xy}/W\), and \(\langle N_y,F_y\rangle=-u_{yy}/W\); these identities give the same determinant.
Finally
The eigenvalues of \(g^{-1}\) lie between \(W^{-2}\) and \(1\). Diagonalize \(g^{-1}\) by a Euclidean orthogonal matrix; each squared entry of the transformed Hessian is multiplied by a product of two eigenvalues, between \(W^{-4}\) and \(1\). Thus
The final, weaker upper bound is the one recorded in the source; the preceding inequality gives its sharper constant.
Reading: Carmo (1992), Riemannian geometry.