The Harmonic Measures of Lucy Garnett

Question

The Harmonic Measures of Lucy Garnett, Alberto Candel, Advances in Mathematics $176$ ($2003$) $187-247$.

Page $208$, Proposition $5.2$ I have actually two questions:

1. what does he mean by saying "$h(•,z)$ is $\Delta^*-$ harmonic on $D×\{z\}$"?
2. what does he mean by saying " $ vol_z(x)$ is the volume density of the plaque $D×\{z\}$"?

Answer 1

1. It means that $\Delta^* h(\cdot, z) = 0$.
2. It means that $ vol_z $ is the induced Riemannian volume on $ D \times z $.