A hyperbolic metric and stability conditions on K3 surfaces with \rho=1

In this article we introduce a hyperbolic metric on the (normalized) space of stability conditions on projective K3 surfaces $X$ with Picard rank $\rho (X) =1$. And we show that all walls are geodesic in the normalized space with respect to the hyperbolic metric. Furthermore we demonstrate how the hyperbolic metric is helpful for us by discussing mainly three topics. We first make a study of so called Bridgeland's conjecture. In the second topic we prove a famous Orlov's theorem without the global Torelli theorem. In the third topic we give an explicit example of stable complexes in large volume limits by using the hyperbolic metric. Though Bridgeland's conjecture may be well-known for algebraic geometers, we would like to start from the review of it.