Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 26J (Differential Geometry)

Choose a smooth unit normal $N$. For $v,w\in T_p\Sigma$, the first fundamental form and second fundamental form arewhere the shape operator is $S_p=-d{N}_p$. The principal curvatures are the eigenvalues ${\kappa }_1,{\kappa }_2$ of $S_p$; the mean curvature and Gaussian curvature areThe surface is minimal when $H=0$ everywhere.

Write $h_x={\partial }_{x}h$ and $h_y={\partial }_{y}h$. Sincethe first fundamental form in the coordinate basis is

Put $W=\sqrt{1+h_x^2+h_y^2}$. An upward-pointing Gauss map isFor tangent vectors $v,w$, differentiation ofgivesThe Weingarten map is self-adjoint and $S=-dN$, so the right side is $-2I{I}^S(v,w)$. HenceAlternatively differentiating $⟨N,{\varphi }_i⟩=0$ givesThereforeThis is the fundamental forms of a graph surface.

The inverse first-fundamental-form matrix isThus $2H=\mathrm{tr}((I^S{)}^{-1}I{I}^S)$ vanishes exactly whenThis is the minimal surface equation for a graph.

The area element is $\sqrt{\det I^S}dxdy=Wdxdy$, soFor the Lagrangian $L(p,q)=\sqrt{1+p^2+q^2}$, stationarity under every compactly supported variation gives the Euler-Lagrange equationMultiplication by $W^3$ expands this intorecovering part (d).