  We study the distribution of domain areas, areas enclosed by domain
boundaries (''hulls''), and perimeters for curvature-driven two-dimensional
coarsening, employing a combination of exact analysis and numerical studies,
for various initial conditions. We show that the number of hulls per unit area,
$n_h(A,t) dA$, with enclosed area in the interval $(A,A+dA)$, is described, for
a disordered initial condition, by the scaling function $n_h(A,t) = 2c_h/(A +
\lambda_h t)^2$, where $c_h=1/8\pi\sqrt{3} \approx 0.023$ is a universal
constant and $\lambda_h$ is a material parameter. For a critical initial
condition, the same form is obtained, with the same $\lambda_h$ but with $c_h$
replaced by $c_h/2$. For the distribution of domain areas, we argue that the
corresponding scaling function has, for random initial conditions, the form
$n_d(A,t) = 2c_d (\lambda_d t)^{\tau'-2}/(A + \lambda_d t)^{\tau'}$, where
$c_d=c_h + {\cal O}(c_h^2)$, $\lambda_d=\lambda_h + {\cal O}(c_h)$, and $\tau'
= 187/91 \approx 2.055$. For critical initial conditions, one replaces $c_d$ by
$c_d/2$ (possibly with corrections of ${\cal O}(c_h^2)$) and the exponent is
$\tau = 379/187 \approx 2.027$. These results are extended to describe the
number density of the length of hulls and domain walls surrounding connected
clusters of aligned spins. These predictions are supported by extensive
numerical simulations. We also study numerically the geometric properties of
the boundaries and areas.
