A (possible) error in Ahlfors' text (analytic continuation along arcs)

Question

In Ahlfors' complex analysis text, page 289 he discusses analytic continuation along arcs. I will start with some background

Let $\mathbf{f}$ be a global analytic function, with corresponding Riemann surface $\mathfrak{S_0} (\mathbf{f})$ (which is a connected a component of the sheaf $\mathfrak{S}$ of germs of analytic functions in the complex plane). Ahlfors defines an analytic continuation, along an arc $\gamma:[a,b] \to \mathbb C$ to be a continuous function $\overline{\gamma}:[a,b] \to \mathfrak{S_0}(\mathbf{f})$, such that $\pi \circ \bar{\gamma}=\gamma$, where $\pi$ is the standard projection map.

Later in the same page, he investigates the case where analytic continuation is impossible along an arc $\gamma:[a,b] \to \mathbb{C}$, starting at the initial germ $\mathbf{f}_{\zeta(a)}$ (I think that's a typo, and should read $\mathbf{f}_{\gamma(a)}$ instead). He notes that analytic continuation is possible if we restrict ourselves to subarcs $\gamma \big|_{[a,t_0]}$ for small enough $t_0$. Next, he considers the supremum of the set (which I'll call $E$) of all such numbers $t_0$, and denotes it $\tau$.

The following claims are made about $\tau$, which I'm having trouble agreeing with:

• $a<\tau<b$
• continuation will be possible for $t_0 < \tau$, impossible for $t_0 \geq \tau$.

Regarding the first claim, I agree that $a<\tau$, since $E$ contains numbers greater than $a$ - however, I can't see why $\tau$ is strictly lesser than $b$.

Regarding the second claim, I agree that analytic continuation is possible for $t_0<\tau$ since these are members of $E$. I agree also that analytic continuation is impossible for $t_0>\tau$, by the properties of the supremum - however, why is analytic continuation never possible for the point $t_0=\tau$ itself?

Please help me settle this. Thanks!

Answer 1

Regarding $a < \tau < b$, yes, it is absolutely possible that $\tau = b$. That's a mistake, it should be $a < \tau \leqslant b$.

As for

however, why is analytic continuation never possible for the point $t_0 = \tau$ itself

if you have $\overline{\gamma}(t) \in \mathfrak{S}_0$, then that is a germ of an analytic function, and that means there is an anaytic function in a neighbourhood $U$ of $\gamma(t)$ whose germ in $\gamma(t)$ is $\overline{\gamma}(t)$, and that function allows extending $\overline{\gamma}$ a little bit beyond $t$, since $\gamma([t,t+\varepsilon]) \subset U$ for small enough $\varepsilon > 0$.

Concerning the $\mathbf{f}_{\zeta(a)}$ issue, I think Ahlfors uses $\zeta(a)$ to denote a generic point above $\gamma(a)$. Note that the surface may have several points above any point of the plane, so one needs to specify which point of $\mathfrak{S}_0$ one is talking about. It would be desirable if that was made entirely clear.