Fundamental forms of a graph surface

For $X(x,y)=(x,y,h(x,y))$ and $W=\sqrt{1+h_x^2+h_y^2}$,The upward unit normal is $(-h_x,-h_y,1)/W$.

For the graph of $h$,

Suppose two graph surfaces $z=f(x,y)$ and $z=g(x,y)$ are tangent at the origin, are oriented upward, and $g\ge f$ nearby. Then $g-f$ has a local minimum, so its Hessian matrix is positive semidefinite. At the common horizontal tangent plane,

The same contact does not order Gaussian curvature, because determinant is not monotone under addition of a positive-semidefinite matrix when the original Hessian is indefinite or negative definite. For example,satisfy $g\ge f$ and are tangent at zero, but $K_g(0)=1<4=K_f(0)$.

After a rigid motion, write the plane as $z=0$ and the surface locally as $z=h(x,y)$. Along the curve of tangency, $h=0$ and $\nabla h=0$. Differentiating $\nabla h(\gamma (s))=0$ shows that the Hessian annihilates the nonzero tangent ${\gamma }^{\prime }(s)$, so its determinant and therefore the Gaussian curvature vanish.

This equation characterizes a minimal surface presented as a graph.

The graph of $h$ is minimal exactly whenEquivalently, it is the Euler--Lagrange equationfor the area functional $\int \sqrt{1+|\nabla h{|}^2}$.

This functional measures the surface area of the graph of a function. On a bounded domain it is finite for $u\in H^1$. Its directional derivative is ${\partial }_{v}F(u)=\int \nabla u\cdot \nabla v/\sqrt{1+|\nabla u{|}^2}$, by the dominated convergence theorem, since the pointwise difference quotients are bounded by $|\nabla v|$. The resulting dual-valued operator is monotone: the scalar integrand $p\mapsto \sqrt{1+|p{|}^2}$ is convex and its gradient is $p/\sqrt{1+|p{|}^2}$. It is not a nonlinear coercive operator on the full $H^1$ space, since it vanishes on constant functions whose $H^1$ norms can tend to infinity.

Every entire classical minimal graph over ${\mathbb{R}}^n$ is affine for $1\le n\le 7$. The dimension restriction is essential. This theorem applies to the minimal surface equation for a graph, without assuming a global bound on its gradient.

// Target: analysis.bigb

The flux $F(p)=p/\sqrt{1+|p{|}^2}$ of the minimal surface equation for a graph has a positive definite symmetric matrix derivative. Its eigenvalues are $(1+|p{|}^2{)}^{-3/2}$ in the direction of $p$ and $(1+|p{|}^2{)}^{-1/2}$ orthogonally; at $p=0$ every eigenvalue is one. On $|p|\le M$ it satisfiesThus an averaged linearization of a nonlinear divergence-form equation is uniformly elliptic where the relevant gradients are bounded. Pointwise positivity alone does not supply a uniform bound on an unbounded gradient range.

For a $C^2$ solution $u$ of the minimal surface equation for a graph on the closure of a bounded domain,Write $a^{ij}(Du)u_{ij}=0$ with $a^{ij}(p)={\delta }_{ij}-p_i{p}_j/(1+|p{|}^2)$. Its differentiated operator $\mathcal{L}=a^{ij}{D}_{ij}+b^{\ell }{D}_{\ell }$, where $b^{\ell }=(\partial a^{ij}/\partial p_{\ell })u_{ij}$, satisfiesThe weak maximum principle for elliptic operators gives the result. Interior Schauder estimates justify differentiating a solution initially assumed only $C^2$.

If a smooth minimal surface equation for a graph solution on ${\mathbb{R}}^n\setminus \{0\}$ is homogeneous of degree one, with $n\ge 2$, then its graph extends to a plane through zero. Its gradient is homogeneous of degree zero, so $|Du{|}^2$ attains a global maximum on the unit sphere. The strong maximum principle for elliptic operators makes it constant. The differentiated equation in the gradient maximum principle for a minimal graph then forces $D^{2}u=0$, and degree-one homogeneity removes the affine constant. This argument applies to cones that are graphs of smooth functions away from their vertex; it is not a claim that every minimal surface cone is flat.