Some metric properties of spaces of stability conditions

We show that the metric on a full component of Stab(C) for which the central charges factor through a finite rank quotient of the Grothendieck group K(C) is complete. As an example, we compute the metric on the space of numerical stability conditions on a smooth complex projective curve of genus greater than 1, and show that the quotient Stab(C)/C by the natural action of the complex numbers is isometric to the upper half-plane equipped with half the hyperbolic metric. We also make two observations about the way in which the heart changes as we move through the space of stability conditions. First, hearts of stability conditions in the same component of the space of stability conditions are related by finite sequences of tilts. Secondly, if each of a convergent sequence of stability conditions has the same heart, then the heart of the limiting stability condition is obtained from this by a right tilt.