  A fourth-order dispersive flow equation for closed curves on the canonical
two-dimensional unit sphere arises in some contexts in physics and fluid
mechanics. In this paper, a geometric generalization of the sphere-valued model
is considered, where the solutions are supposed to take values in compact
Riemann surfaces. As a main results, time-local existence and the uniqueness of
a solution to the initial value problem is established under the assumption
that the sectional curvature of the Riemann surface is constant. The analytic
difficulty comes from the so-called loss of derivatives and the absence of the
local smoothing effect. The proof is based on the geometric energy method
combined with a kind of gauge transformation to eliminate the loss of
derivatives. Specifically, to show the uniqueness of the solution, the detailed
geometric analysis of the solvable structure for the equation is presented.
