Hilbert’s Theorem, via moving frames

For all $P\in\mathbb{D}$, there is a positive number $\epsilon_{P}$, such that $\phi$ is an embedding, when restricted to the open disk $U_{P}$ of radius $\epsilon_{P}$ in the hyperbolic metric. The tangent space to the surface $\phi(U_{P})=:\bar{U}_{P}$ at $\phi(P)=:\bar{P}$ can inherit an orientation from $T_{P}(\mathbb{D})$, via $(\phi_{*})_{P}$, so there is a preferred unit normal vector at $\bar{P}$, which can be continuously (indeed, smoothly) extended to a unit normal vector field $\bar{e}_{3}:\bar{U}_{P}\rightarrow S^{2}$.

At every point $Q\in U_{P}$, the map $(-d\bar{e}_{3})_{\bar{Q}}:T_{\bar{Q}}(\bar{U}_{P})\rightarrow T_{\bar{e}_{3}(\bar{Q})}S^{2}$, is a self-adjoint linear transformation of $T_{\bar{Q}}(\bar{U}_{P})=\phi_{*}(T_{Q}(\mathbb{D}))$ [dC16, page 142]; note that $T_{\bar{e}_{3}(\bar{Q})}S^{2}$ and $T_{\bar{Q}}(\bar{U}_{P})$ are both names for the orthogonal complement in $\mathbb{R}^{3}$ of $\operatorname{span}(\bar{e}_{3}(\bar{Q}))$. The eigenvalues are equal to the principal curvatures (let $\bar{\kappa}_{1}(\bar{Q})$ denote the positive eigenvalue, and $\bar{\kappa}_{2}(\bar{Q})$ the negative eigenvalue), and the corresponding (one-dimensional, orthogonal) eigenspaces $\bar{L}_{1}(\bar{Q}),\bar{L}_{2}(\bar{Q})$ give the principal directions [dC16, page 146]. At this stage, we are using the fact that the Gaussian curvature (product of the principal curvatures) is negative at every point of $\bar{U}_{P}$, but not (yet) the fact that it is constant and equal to $-1$.

Via $(\phi|_{U_{P}})_{*}^{-1}$, we can transfer these eigenspaces back to form two smooth line fields on each $U_{P}$. Whenever $R\in U_{P}\cap U_{Q}$, the line fields for $U_{P}$ and $U_{Q}$ will agree at $R$, since they both describe the extrinsic geometry of $\phi(U_{P}\cap U_{Q})$ at $\phi(R)$. Hence we can merge all of the “local” line fields into two “global” line fields $L_{1},L_{2}$ on $\mathbb{D}$. There is a two-sheeted covering space of $\mathbb{D}$, consisting of all the unit vectors in $L_{1}$, and since $\mathbb{D}$ is simply-connected, this covering space is not connected. In other words, there are two unit vector fields $\pm e_{1}$ on $\mathbb{D}$ which everywhere point along $L_{1}$. We define $e_{2}$ to be the result of rotating $e_{1}$ a quarter-turn counter-clockwise (i.e., $+\pi/2$ radians).

This allows us to define $\bar{e}_{1}:=\phi_{*}(e_{1})$ and $\bar{e}_{2}:=\phi_{*}(e_{2})$ on each $\bar{U}_{P}$, so we now have a (positively oriented) moving frame $\{\bar{e}_{1},\bar{e}_{2},\bar{e}_{3}\}$ on $\bar{U}_{P}$. As in [dC94, page 82], we can extend this moving frame to a small open subset $W_{P}$ of $\mathbb{R}^{3}$ by displacing frames normally by a sufficiently small amount (choose $\nu_{P}>0$ such that $U_{P}\times(-\nu_{P},\nu_{P})\rightarrow W_{P}$ by $(Q,t)\mapsto\bar{Q}+t\cdot[\bar{e}_{3}]_{\bar{Q}}$ is a diffeomorphism). See Figure 2 for an illustration.

Since $d\bar{e}_{1},d\bar{e}_{2},d\bar{e}_{3}$ are linear transformations on $\mathbb{R}^{3}$ at every point of $W_{P}$ (in other words, they are $\mathbb{R}^{3}$-valued $1$-forms, as defined in [Spi79a, page 546], while $\bar{e}_{1},\bar{e}_{2},\bar{e}_{3}$ are $\mathbb{R}^{3}$-valued $0$-forms), we can define the connection $1$-forms $\bar{\omega}^{k}_{j}\ (1\leq j,k\leq 3)$ by

Now, on each $U_{P}$, we can define the $1$-forms $\theta^{i}:=\phi^{*}\bar{\theta}^{i}$, $1\leq i\leq 3$ (noting that $\bar{\theta}^{3}$ pulls back to the zero $1$-form) and $\omega^{i}_{j}:=\phi^{*}\bar{\omega}^{i}_{j}$, $1\leq i,j\leq 3$. The reader can easily check that $\theta^{1},\theta^{2}$ are dual to $e_{1},e_{2}$.

Since $\{e_{1},e_{2}\}$ are globally defined on $\mathbb{D}$, the moving frames $\{\bar{e}_{1},\bar{e}_{2},\bar{e}_{3}\}$ at $\bar{R}\in\bar{U}_{P}\cap\bar{U}_{Q}$, arising respectively from $\phi|_{U_{P}}$ and $\phi|_{U_{Q}}$, will agree. Hence the $1$-forms $\theta^{i}$ and $\omega^{i}_{j}$, defined above, will agree on $U_{P}\cap U_{Q}$. Consequently, as we did above with the line fields, we merge the “local” $\theta$’s and $\omega$’s into “global” smooth $1$-forms that are defined on $\mathbb{D}$.

Furthermore, since the “unbarred” versions of equations (3) and (4) can be justified by basic facts about pullbacks and wedge products in each $U_{P}$, Proposition 1 implies the following equations on $\mathbb{D}$ (or more precisely, in the exterior algebra of $C^{\infty}$ differential forms on $\mathbb{D}$):

Also, by definition, in each $\bar{U}_{P}$,

We now turn to the proofs of two key lemmas, which relate $\omega_{1}^{2}$ to the principal curvatures and to the angle between the asymptotic directions and $e_{1}$.

For all $P\in\mathbb{D}$, the normal curvatures at $\phi(P)$ can be described by $\operatorname{II}(v):=\langle-d\bar{e}_{3}(\phi_{*}(v)),\phi_{*}(v)\rangle$, defined in terms of the Euclidean inner product for vectors in $\mathbb{R}^{3}$ [dC94, page 87]. In essence, this is the second fundamental form for the immersed surface near $\phi(P)$, pulled back to a quadratic form on $T_{P}(\mathbb{D})$. Using Euler’s theorem [dC16, page 147] [MP77, page 129], and (for the first time) making use of the fact that Gaussian curvature $K:=\kappa_{1}\kappa_{2}$ equals $-1$ everywhere on $\mathbb{D}$, we have normal curvature zero when

We now turn our attention to the properties of the asymptotic vector fields $E_{1},E_{2}$.

We next show that the vector fields $E_{1}$ and $E_{2}$ on $\mathbb{D}$ each generate a global flow (i.e., each integral curve is defined for all $t\in\mathbb{R}$). We will use the following special case of [Cod61, Chapter 6, §6, Theorem 2].

Note that $\epsilon$ does not depend on $L$, the Lipschitz constant.

Since Proposition 3 has shown that $F$ is a coordinate mapping for $\mathbb{D}$, we can define coordinate functions $u^{1},u^{2}$ on $\mathbb{D}$ such that $u^{1}=x^{1}\circ F,\ u^{2}=x^{2}\circ F$ and

Adding and subtracting the equations in (16), we have