$L^{2}$-bounds for drilling short geodesics in convex co-compact hyperbolic 3-manifolds

Let $M$ be the interior of $\bar{M}$. Then a $\left(\mathsf{PSL}_{2}(\mathbb{C}),{\bar{\mathbb{H}}^{3}}\right)$-structure is a hyperbolic structure and the bundle has a natural decomposition and metric structure that we now describe. Each fiber $E_{p}$ is the space of germs of infinitesimal isometries. In particular, $s\in E_{p}$ is a vector field in a neighborhood of $p$ so $s(p)$ is a vector in $T_{p}M$. As $E$ is a complex bundle we can multiply $s$ by $i$ and then $(is)(p)$ is another vector in $T_{p}M$. Then the map from $E$ to $TM\oplus TM$ given by $s\mapsto(s(p),(is)(p))$ is bundle isomorphism. In fact, the map $s\mapsto s(p)+i(is)(p)$ is a complex vector bundle isomorphism from $E$ to the the complexification $T^{\mathbb{C}}M$ of the tangent bundle. This isomorphism from $E$ to $TM\oplus TM$ gives a decomposition of sections of $E$ into real and imaginary parts.

If $v$ is a vector in $T_{p}M$, we define $\hat{v}\in E_{p}$ such that under our isomorphism from $E_{p}$ to $T_{p}M\oplus T_{p}M$ we have $\hat{v}\mapsto(v,0)$. Then $\hat{v}$ is the infinitesimal translation with axis through $v$ and $i\hat{v}$ is an infinitesimal rotation about $v$. Note that $\hat{v}$ is real and $i\hat{v}$ is imaginary, as one would expect.

As $E$ is isomorphic to $TM\oplus TM$, the dual bundle $E^{*}$ is isomorphic to $T^{*}M\oplus T^{*}M$. The hyperbolic metric on $M$ determines an isomorphism from $TM$ to $T^{*}M$ and therefore an isomorphism from $E$ to $E^{*}$. Note that this isomorphism is $\mathbb{R}$-linear but is $\mathbb{C}$-anti-linear with respect to the complex structures on $E$ and $E^{*}$. For sections $s$ of $E$ we let $s^{\sharp}$ be the dual section of $E^{*}$. When going from $E^{*}$ to $E$ we replace the $\sharp$ with a $\flat$.

As $E_{p}$ is a complex vector space we have $E_{p}=E_{p}\otimes\mathbb{C}$ and more generally for alternating tensors with values in $E_{p}$ we have

$$\Lambda^{k}(T_{p}M;E_{p})=E_{p}\otimes\Lambda^{k}(T_{p}M)=E_{p}\otimes\Lambda^{k}\left(T^{\mathbb{C}}_{p}M\right).$$

In particular, every $E$-valued form is locally the sum of terms $\phi s\omega$ where $\phi$ is a complex valued function, $s$ is a section of $E$, and $\omega$ is a $\mathbb{R}$-valued form. The $\sharp$ (and $\flat$) operators extend to $E$-valued forms and we have $(\phi s\omega)^{\sharp}=\bar{\phi}s^{\sharp}\omega$. We also linearly extend the Hodge star operator from real forms to $E$-valued forms so that $\star(\phi s\omega)=\phi s(\star\omega)$. This extends to a linear map from $\Omega^{k}(M;E)$ to $\Omega^{3-k}(M;E)$. We the define the inner product

$$(\alpha,\beta)=\int_{M}\alpha\wedge(\star\beta)^{\sharp}.$$

Note that the wedge product of an $E$-valued form and an $E^{*}$-valued form is real form so this is a real inner product. We also let $\|\alpha\|^{2}=(\alpha,\alpha)$ be the $L^{2}$-norm of an $E$-valued form.

If $\alpha$ is either an $E$-valued or $\mathbb{C}$-valued form we define the pointwise norm $|\alpha|$ by $|\alpha|^{2}=\star(\alpha\wedge(\star\alpha)^{\sharp})$. Then $\|\alpha\|$ is the usual $L^{2}$-norm of the function $|\alpha|$.

If $d^{*}$ is the flat connection for $E^{*}$ then define the operator $\partial$ on $E$ by

$$\partial\omega=\left(d^{*}(\omega^{\sharp})\right)^{\beta}.$$

Then the formal adjoint $\delta$ for $d$ satisfies the formula

$$\delta=\star\partial\star.$$

Note that if $s$ is a flat section ($ds=0$) then the real and imaginary parts, $\operatorname{Re}s$ and $\operatorname{Im}s$, will not be flat. That is $d$ will not preserve our bundle decomposition. Instead we define operators $D$ and $T$ such that $d=D+T$ where $D$ preserves the bundle decomposition and $T$ permutes it. That is for a real section $s$ we have that $Ds$ is a real $E$-valued 1-form while $Ts$ is imaginary. We have formulas for both $D$ and $T$. If $v$ is a vector field then

$$D\hat{v}(w)=\widehat{\nabla_{w}v}$$

where $\nabla$ is the Riemannian connection for the hyperbolic metric and

$$T\hat{v}(w)=[\hat{v},\hat{w}]$$

where $[,]$ is the Lie bracket. Note that the operator $T$ is purely algebraic.

We also have

$$\partial=D-T.$$

This is a manifestation of the fact that the $\sharp$-operator is $\mathbb{C}$-anti-linear.

The Laplacian for $E$-valued 1-forms is $\Delta=d\delta+\delta d$ and $\omega\in\Omega^{k}(E)$ is harmonic if $\Delta\omega=0$. If $M$ is compact then this is equivalent $\omega$ be closed and co-closed. However, our manifolds will be non-compact so we will define $\omega$ to be a Hodge form if it is closed, co-closed and the real and imaginary parts are symmetric and traceless.

We now make a few computations that will be very useful later and will also serve as an example of how to do computations in the bundle. We will work in the upper half space model of ${{\mathbb{H}}^{3}}=\mathbb{C}\times\mathbb{R}^{+}$ with $\left\{{\frac{\partial}{\partial x}},{\frac{\partial}{\partial y}},\frac{\partial}{\partial t}\right\}$ and $\{dx,dy,dt\}$ the usual basis and dual basis at each $T_{p}{{\mathbb{H}}^{3}}$. We also let

$$\frac{\partial}{\partial z}=\frac{1}{2}\left({\frac{\partial}{\partial x}}-i{\frac{\partial}{\partial y}}\right)\mbox{ and }\frac{\partial}{\partial{\overline{z}}}=\frac{1}{2}\left({\frac{\partial}{\partial x}}+i{\frac{\partial}{\partial y}}\right)$$

be tangent vectors in the complexified tangent space with dual 1-forms $dz=dx+idy$ and $d{\overline{z}}=dx-idy$. We can then write any $E$-valued 1-form on ${{\mathbb{H}}^{3}}$ as a sum of $dz,d{\overline{z}}$ and $dt$ terms.

The Lie algebra $\mathfrak{sl}_{2}(\mathbb{C})$ can be identified with traceless 2-by-2 matrices in $\mathbb{C}$, projective vector fields on $\widehat{{\mathbb{C}}}$ and infinitesimal isometries of ${{\mathbb{H}}^{3}}$. The reader can check the correspondence given in the following lemma:

To calculate $Ts$ we note that

$$Ts=\left[s,\hat{\frac{\partial}{\partial z}}\right]dz+\left[s,\hat{\frac{\partial}{\partial{\overline{z}}}}\right]d{\overline{z}}+\left[s,\hat{\frac{\partial}{\partial t}}\right]dt.$$

Proof: We can assume that ${\mathfrak{p}}=\lambda z^{2}\frac{\partial}{\partial z}$ and $p=(0,1)$ in the upper half space model. Then $e_{n}=\hat{\frac{\partial}{\partial t}}$ and $\omega_{n}=dt$. To calculate $T{\mathfrak{p}}$ we use Lemma 2.2 to write ${\mathfrak{p}},\hat{\frac{\partial}{\partial z}},\hat{\frac{\partial}{\partial{\overline{z}}}}$ and $\hat{\frac{\partial}{\partial t}}$ as matrices and calculate the Lie brackets using matrix multiplication and get

$$T{\mathfrak{p}}=\frac{\lambda}{2}\hat{\frac{\partial}{\partial t}}\otimes dz+{\mathfrak{p}}\otimes dt$$

so $\omega=\lambda dz$. We can then compute to see that $|\omega|=|{\mathfrak{p}}|$. $\Box$

We will be interested in complex projective structures $\Sigma$ that have a fixed underlying conformal structure $X$. The space $P(X)$ of such projective structures has a natural affine structure as the space $Q(X)$ of holomorphic quadratic differentials on $X$. That is the difference of $\Sigma_{0}$ and $\Sigma_{1}$ in is quadratic differential in $Q(X)$ defined as follows. Let $(U,\psi_{0})$ and $(U,\psi_{1})$ be charts for $\Sigma_{0}$ and $\Sigma_{1}$. Then $\psi_{1}\circ\psi_{0}^{-1}$ is a conformal map for an open neighborhood in $\widehat{{\mathbb{C}}}$ to $\widehat{{\mathbb{C}}}$. The Schwarzian derivative is a holomorphic function $\phi(\Sigma_{0},\Sigma_{1})$ on $\psi_{0}(U)$ and it determines a holomorphic quadratic differential $\Phi(\Sigma_{0},\Sigma_{1})$. Properties of the Schwarzian derivative imply that if $\Sigma_{2}\in P(X)$ is a third projective structure then

$$\Phi(\Sigma_{0},\Sigma_{2})=\Phi(\Sigma_{0},\Sigma_{1})+\Phi(\Sigma_{1},\Sigma_{2}).$$

This gives a canonical identification of the tangent space $T_{\Sigma}P(X)$ with $Q(X)$.

If $\hat{g}$ is a conformal metric on $X$ and $\Phi\in Q(X)$ then the ratio $|\Phi|/\hat{g}$ is a positive function on $S$. More concretely in a local chart $\Phi$ can be written as $\phi dz^{2}$ and the conformal metric can be written as $\hat{g}=\rho{g_{\rm euc}}$, where $\rho$ is a positive function and ${g_{\rm euc}}$ is the Euclidean metric. Then the ratio $|\phi|/\rho$, defined in the chart, is a well defined function on the surface. We denote this function by $\|\Phi(z)\|_{\hat{g}}$ and let $\|\Phi\|_{\hat{g},p}$ be the $L^{p}$-norm of this function with respect to the $\hat{g}$ metric. We will mostly be interested in the hyperbolic metric but much of what we do will work in a more general setting. When we are using the hyperbolic metric we will drop the metric $\hat{g}$ from our notation.

For every conformal structure $X$ there is unique Fuchsian projective structure $\Sigma_{F}\in P(X)$. We let $\|\Sigma\|_{\hat{g},\infty}=\|\Phi(\Sigma,\Sigma_{F})\|_{\hat{g},\infty}$.

If $\Sigma_{t}$ is a smooth path in $P(X)$ then its tangent vectors $\Phi_{t}$ lie in $Q(X)=T_{\Sigma_{t}}P(X)$. To prove our main result, Theorem 1.3, we bound the distances between the endpoints of a path $\Sigma_{t}$ by bounding the norms of the derivative $\Phi_{t}$.

We also describe how a quadratic differential $\Phi\in Q(X)=T_{\Sigma}P(X)$ determines a cohomology class in $H^{1}(\Sigma;E(\Sigma))$. This was originally introduced by the second author in [Brm].

Define a section ${\mathfrak{p}}$ of $E(\mathbb{C})$ by

$${\mathfrak{p}}(z)=(w-z)^{2}\frac{\partial}{\partial w}=\frac{1}{2}\begin{bmatrix}-z&z^{2}\\ -1&z\end{bmatrix}.$$

Then if $\Phi$ is represented in a projective chart by $\phi dz^{2}$ we defined an $E$-valued 1-form in the chart by

$${\mathfrak{p}}(z)\phi(z)dz.$$

One can then check that this gives a well defined $E$-valued 1-form $\omega_{\Phi}$ on $\Sigma$. As both ${\mathfrak{p}}$ and $\phi$ are holomorphic, $\omega_{\Phi}$ is closed and therefore determines a cohomology class in $H^{1}(\Sigma;E(\Sigma))$.

Let $v$ be a vector field on an open neighborhood $U$ in ${\bar{\mathbb{H}}^{3}}$ that is conformal on $U\cap\widehat{{\mathbb{C}}}$. We then define a section $s$ of $E(U)=U\times\mathfrak{sl}_{2}(\mathbb{C})$ as follows. For $x\in U\cap{{\mathbb{H}}^{3}}$ let $s(x)$ be the unique infinitesimal isometry that agrees with $v$ at $x$ and whose curl agrees with the curl of $v$ at $x$. On $U\cap\widehat{{\mathbb{C}}}$, the vector field is (the real part) of the product of a holomorphic function $f$ and $\frac{\partial}{\partial z}$. For each $z\in U\cap\widehat{{\mathbb{C}}}$ let $f_{z}$ be the complex quadratic polynomial whose 2-jet agrees with $f$ at $z$. Then $s(z)=f_{z}\frac{\partial}{\partial z}$.

Let $\left\{\left(U^{\alpha},\psi^{\alpha}_{t}\right)\right\}$ be a 1-parameter family of $\left(\mathsf{PSL}_{2}(\mathbb{C}),{\bar{\mathbb{H}}^{3}}\right)$-structures on a 3-manifold with boundary $\bar{M}$ with the conformal boundary fixed. For each $x\in U^{\alpha}$, the time zero derivative of the path $\psi^{\alpha}_{t}(x)$ is vector field $v^{\alpha}$ on $\psi^{\alpha}_{0}(U^{\alpha})$ that is conformal on $\psi^{\alpha}_{0}(U^{\alpha})\cap\widehat{{\mathbb{C}}}$. This determines a section $s^{\alpha}$ of $\bar{E}(\psi^{\alpha}_{0}(U^{\alpha}))$ and $\omega^{\alpha}=ds^{\alpha}$ is an $E$-valued 1-form on $\psi^{\alpha}_{0}(U^{\alpha})$. While the sections $s^{\alpha}$ will not necessarily agree on overlapping charts, the $E$-valued 1-forms $\omega^{\alpha}$ will agree and determine an $E$-valued 1-form $\omega$ on $\bar{M}$. As locally $\omega$ is $d$ of a section, $\omega$ is closed and therefore represents an element of $H^{1}(\bar{M};\bar{E})$.

Since the conformal boundary is some fixed conformal structure $X$, the $\left(\mathsf{PSL}_{2}(\mathbb{C}),{\bar{\mathbb{H}}^{3}}\right)$-structures on $\bar{M}$ determine a family of projective structures $\Sigma_{t}\in P(X)$. The time zero derivative of $\Sigma_{t}$ will be a holomorphic quadratic differential $\Phi\in Q(X)=T_{\Sigma_{0}}P(X)$.

If $(U,\psi)$ is a projective chart for $\Sigma$ then we can extend it to a chart $(U\times[0,1],\Psi)$ where $\Psi$ is the continuous extension of $\psi$ to a map to ${\bar{\mathbb{H}}^{3}}$ that is an isometry on $U\times(0,1]$. We say that the chart is adapted to $z_{0}\in U$ if $\Psi(z_{0},t)=(0,t)$ where the coordinates on the right are in the upper half space model for ${{\mathbb{H}}^{3}}$. We can always construct a chart adapted to $z_{0}$ by taking any projective chart $(U,\psi)$ with $z_{0}\in U$ and post-composing with an element of $\mathsf{PSL}_{2}(\mathbb{C})$.

In a projective chart the metric at infinity $\hat{g}$ is scalar function times the Euclidean metric ${g_{\rm euc}}$. If the chart is adapted to $z_{0}$ then we can calculate the value of this function.

Proof: Define a function $\rho\colon\Psi(U\times[1,0))\to\mathbb{R}^{+}$ with $\rho\circ\Psi(z,t)=4t^{2}$. If $g_{{\mathbb{H}}^{3}}$ is the metric for the upper half space model of ${{\mathbb{H}}^{3}}$ then $\rho\cdot g_{{\mathbb{H}}^{3}}$ extends continuously to the metric $\hat{g}$ on $\psi(U)$. Since $\Psi(z_{0},t)=(0,t)$ we have that $\rho(0,t)=4t^{2}$ and therefore $\rho g_{{{\mathbb{H}}^{3}}}$ extends continuously to $4{g_{\rm euc}}$ at $0$. $\Box$

On a chart $(U,\psi)$ for $\Sigma$ we have the usual coordinate vector fields ${\frac{\partial}{\partial x}}$ and ${\frac{\partial}{\partial y}}$ along with the vector fields in $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial{\overline{z}}}$ in the complexified tangent bundle. On a chart $(U\times[0,1],\Psi)$ for the end ${\mathcal{E}}$ these coordinate vector fields, along with $\frac{\partial}{\partial t}$ are a basis but, unlike in the upper half space model for ${{\mathbb{H}}^{3}}$, the vector fields ${\frac{\partial}{\partial x}}$ and ${\frac{\partial}{\partial y}}$ may not be orthogonal or of the same length as the operators $({\operatorname{Id}}+t^{2}\hat{B})$ are not conformal. In particular, the complex 1-form $dz$ will not be $\mathbb{C}$-linear on the complex structure on $S$ induced by the metric $g_{t}$. However, we can write down $\mathbb{C}$-linear and $\mathbb{C}$-anti-linear forms in terms of the Beltrami differential of the endomorphisms that define $g_{t}$.

We begin with a computation on a single vector space. The usual Euclidean metric ${g_{\rm euc}}$ on $\mathbb{R}^{2}$ has a unique $\mathbb{C}$-linear extension to $\mathbb{R}^{2}\otimes\mathbb{C}$. Then $dz$ and $d{\overline{z}}$ are the usual dual basis for $\mathbb{R}^{2}\otimes\mathbb{C}$. While they are both $\mathbb{C}$-linear on $\mathbb{R}^{2}\otimes\mathbb{C}$, when restricted to $\mathbb{R}^{2}$, with the complex structure induced by ${g_{\rm euc}}$, $dz$ is $\mathbb{C}$-linear while $d{\overline{z}}$ is $\mathbb{C}$-anti-linear. A linear isomorphism $A\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ has a unique $\mathbb{C}$-linear extension to $\mathbb{R}^{2}\otimes\mathbb{C}$. The Beltrami differential for $A$ is

$$\mu=A_{\overline{z}}/A_{z}$$

where $A_{z}$ and $A_{\overline{z}}$ are complex numbers with

$$A^{*}(dz)=A_{z}dz+A_{\overline{z}}d{\overline{z}}.$$

We have the following:

Proof: As $dz$ and $d{\overline{z}}$ are $\mathbb{C}$-linear and $\mathbb{C}$-anti-linear for the complex structure on $\mathbb{R}^{2}$ induced by ${g_{\rm euc}}$, $A^{*}(dz)$ and $A^{*}(d{\overline{z}})$ are $\mathbb{C}$-linear and $\mathbb{C}$-anti-linear for the complex structure induced by $g$. As $A$ is $\mathbb{C}$-linear on $\mathbb{R}^{2}\otimes\mathbb{C}$ and

$$A^{*}(dz)=A_{z}dz+A_{{\overline{z}}}d{\overline{z}}$$

for complex numbers $A_{z}$ and $A_{\overline{z}}$ we have

$$A^{*}(d{\overline{z}})=\bar{A_{{\overline{z}}}}dz+\bar{A_{z}}d{\overline{z}}.$$

Dividing $A^{*}(dz)$ by $A_{z}$ and $A^{*}(d{\overline{z}})$ by $\bar{A_{z}}$ we define $dw$ and $d\bar{w}$ by

$$\begin{bmatrix}dw\\ d\bar{w}\end{bmatrix}=\begin{bmatrix}1&\mu\\ \bar{\mu}&1\end{bmatrix}\begin{bmatrix}dz\\ d{\overline{z}}\end{bmatrix}$$

so that $dw$ is $\mathbb{C}$-linear and $d\bar{w}$ is $\mathbb{C}$-anti-linear on the complex structure induced by $g$. Inverting gives our formula for $dz$ and $d{\overline{z}}$ in terms of $dw$ and $d\bar{w}$.

As $dw$ and $d\bar{w}$ are $\mathbb{C}$-linear and $\mathbb{C}$-anti-linear with respect to $g$ we have

$$\star_{g}dw=-idw,\qquad\star_{g}d\overline{w}=id\overline{w}$$

or

$$\star_{g}\begin{bmatrix}dw\\ d\bar{w}\end{bmatrix}=\begin{bmatrix}-i&0\\ 0&i\end{bmatrix}\begin{bmatrix}dw\\ d\bar{w}\end{bmatrix}.$$

We then have

$$\star_{g}\begin{bmatrix}dz\\ d\bar{z}\end{bmatrix}=\left(\frac{1}{1-|\mu|^{2}}\begin{bmatrix}1&-\mu\\ -\bar{\mu}&1\end{bmatrix}\right)\begin{bmatrix}-i&0\\ 0&i\end{bmatrix}\begin{bmatrix}1&\mu\\ \bar{\mu}&1\end{bmatrix}\begin{bmatrix}dz\\ d\bar{z}\end{bmatrix}.$$

Multiplying, we obtain the stated formulas. For the norm $|dw|_{g}$ we note that define $dW=A^{*}dz=A_{z}dw$. Then

$$|A_{z}dw|_{g}=|dW|_{g}=|dz|_{g_{euc}}.$$

$\Box$

We can apply the above to the metrics $g_{t}$. The Beltrami differential for endomorphisms $\frac{1}{2t}({\operatorname{Id}}+t^{2}\hat{B})$ can be written as $t^{2}\mu_{t}$ where

$$\mu_{t}=\frac{\hat{B}_{\overline{z}}}{1+t^{2}\hat{B}_{z}}.$$

We obtain the following immediate corollary.