that we can also state the lower bound as follows: $$\lambda_1(\Omega,A)\geq\dfrac{4\pi^2}{\tilde K_{\Omega}}\cdot d(\Phi^A,{\bf Z})^2,$$ where $\tilde K_{\Omega}$ is an invariant depending only $\Omega$: $$\tilde K_{\Omega}=\inf_{\psi\in{\cal F}_{\Omega}}K_{\Omega,\psi}L_{\psi}^2\quad\text{and}\quad L_{\psi}=\sup_{r\in {\rm range}(\psi)}{\lvert{\psi^{-1}(r)}\rvert}.$$ It is is not always easy to estimate $K$. In Section \[estimate K\] we will show how to estimate $K$ in terms of the metric tensor. Note that $K\geq 1$; we will see that in many interesting situations (for example, for revolution cylinders, or for smooth embedded tubes around a closed curve) one has in fact $K=1$.

Doubly connected planar domains
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We now estimate the constant $K$ above when $\Omega$ is an annular region in the plane, bounded by the inner curve $\Sigma_1$ and the outer curve $\Sigma_2$.

[$\bullet\quad$]{}We assume that the inner curve $\Sigma_1$ is convex.

From each point $x\in\Sigma_1$, consider the ray $\gamma_x(t)=x+tN_x$, where $N_x$ is the exterior normal to $\Sigma_1$ at $x$ and $t\geq 0$. Let $Q(x)$ be the first intersection of $\gamma_x(t)$ with $\Sigma_2$, and let $$r(x)=d(x,Q(x)).$$ We say that $\Omega$ is [*starlike with respect to $\Sigma_1$*]{} if the map $x\to Q(x)$ is a bijection between $\Sigma_1$ and $\Sigma_2$; equivalently, if given any point $y\in\Sigma_2$, the geodesic segment which minimizes distance from $y$ to $\Sigma_1$ is 