Holomorphic quadratic differentials dual to Fenchel–Nielsen coordinates

Abstract

We discuss bases of the space of holomorphic quadratic differentials that are dual to the differentials of Fenchel–Nielsen coordinates and hence appear naturally when considering functions on the set of hyperbolic metrics which are invariant under pull-back by diffeomorphisms, such as eigenvalues of the Laplacian. The precise estimates derived in the current paper form the basis for the proof of the sharp eigenvalue estimates on degenerating surfaces obtained by the authors in another paper.

Appendix A. Appendix

We will need the following “Collar lemma” throughout the paper.

Lemma A.1

(Keen–Randol [6]) Let (M, g) be a closed oriented hyperbolic surface and let $\sigma$ be a simple closed geodesic of length $\ell$. Then, there is a neighbourhood $\mathcal {C}(\sigma )$ around $\sigma$, a so-called collar, which is isometric to $\big ((-\,X(\ell ),X(\ell ))\times S^1, \rho ^2(s)(\mathrm{d}s^2+\mathrm{d}\theta ^2)\big )$ where

$$\begin{aligned} \rho (s)=\rho _\ell (s)=\frac{\ell }{2\pi \cos \left( \frac{\ell s}{2\pi }\right) } \quad \text {and}\quad X(\ell )=\frac{2\pi }{\ell }\left( \frac{\pi }{2} -\arctan \left( \sinh \left( \frac{\ell }{2}\right) \right) \right) .\end{aligned}$$

(A.1)

On collars, we will always use the complex variable $z=s+\mathrm {i}\theta$.

We will use in particular the following properties of hyperbolic collars, and refer to [1] as well as the appendices of [8,9,10] and the references therein for more information:

The width of a collar, i.e. the distance $w_{\ell }\mathrel {\mathrm {:=}}\int _{-X(\ell )}^{X(\ell )} \rho (s) \mathrm{d}s$ between the two boundary curves, is related to the length $\ell$ of the central geodesic by

$$\begin{aligned} \sinh \tfrac{w_{\ell }}{2} \sinh \tfrac{\ell }{2}=1. \end{aligned}$$

(A.2)

The injectivity radius of points on the boundary curves of a collar is at least $\mathrm{arsinh}(1)$ and as the injectivity radii and conformal factors $\rho$ are of comparable size at points with bounded (euclidean) distance, we hence have that

$$\begin{aligned} \pi \rho (s)\ge {{\mathrm{inj}}}_g(s,\theta )\ge c_\Lambda >0 \quad \text {for all } \left| s\right| \in [X(\ell )-\Lambda ,X(\ell )) \end{aligned}$$

(A.3)

with $c_\Lambda >0$ depending only on $\Lambda$, compare e.g. [8, (A.7)–(A.9)].

In our analysis of holomorphic quadratic differentials, we use repeatedly that on a collar $\mathcal {C}(\sigma )$ around a geodesic of length $\ell \in (0,2\mathrm{arsinh}(1))$ we have

$$\begin{aligned} \begin{aligned} |\mathrm{d}z^2|_g&=2\rho ^{-2}; \quad \Vert \mathrm{d}z^2\Vert _{L^1(\mathcal {C}(\sigma ))}=8\pi X(\ell );\\ \qquad \Vert \mathrm{d}z^2\Vert _{L^\infty (\mathcal {C}(\sigma ))}&=\frac{8\pi ^2}{\ell ^2}; \quad \Vert \mathrm{d}z^2\Vert _{L^2(\mathcal {C}(\sigma ))}^2=\frac{32\pi ^5}{\ell ^3} -\frac{16\pi ^4}{3}+O(\ell ^2), \end{aligned} \end{aligned}$$

(A.4)

where norms on $\mathcal {C}(\sigma )$ are always computed with respect to $g=\rho ^2(\mathrm{d}s^2+\mathrm{d}\theta ^2)$.

We also remark that for every ${\bar{L}}$, there exists a constant $c_1=c_1({\bar{L}})>0$ so that if $\ell <{\bar{L}}$ then

$$\begin{aligned} \Vert \mathrm{d}z^2\Vert _{L^2(\mathcal {C}(\sigma ))}\ge c_1, \end{aligned}$$

(A.5)

while an upper bound of the form

$$\begin{aligned} \Vert \mathrm{d}z^2\Vert _{L^2(\mathcal {C}(\sigma ))}\le C\ell ^{-3/2} \end{aligned}$$

(A.6)

holds true for a universal constant C.

As the principal part is orthogonal to the collar decay part, we may combine the above estimates with

$$\begin{aligned} \left| b_0(\Upsilon ,\mathcal {C}(\sigma ))\right| \cdot \Vert {\mathrm{d}z^2}\Vert _{L^2(\mathcal {C}(\sigma ))}\le \left\| \Upsilon \right\| _{L^2(\mathcal {C}(\sigma ))}\le \left\| \Upsilon \right\| _{L^2(M,g)} \end{aligned}$$

(A.7)

to obtain a trivial upper bound for the coefficient of the principal part on collars of

$$\begin{aligned} \left| b_0(\Upsilon ,\mathcal {C}(\sigma ))\right| \le C \ell ^{3/2} \left\| \Upsilon \right\| _{L^2(M,g)} \end{aligned}$$

(A.8)

so in particular

$$\begin{aligned} \left| b_0(\Upsilon ,\mathcal {C}(\sigma ))\right| \le C({\bar{L}}) \left\| \Upsilon \right\| _{L^2(\mathcal {C}(\sigma ))}\le C({\bar{L}}) \left\| \Upsilon \right\| _{L^2(M,g)} \end{aligned}$$

(A.9)

for collars around geodesics of bounded length $L_g(\sigma )\le {\bar{L}}$.