*Published Paper*

**Inserted:** 28 sep 2016

**Last Updated:** 28 sep 2016

**Journal:** New York J. Math.

**Volume:** 22

**Pages:** 1111-1133

**Year:** 2016

**Links:**
New York Journal of Mathematics,
arXiv:1405.1211

**Abstract:**

We study the geodesic equation for the Dirichlet (gradient) metric in the space of Kähler potentials. We first solve the initial value problem for the geodesic equation of the *combination metric*, including the gradient metric. We then discuss a comparison theorem between it and the Calabi metric. As geometric motivation of the combination metric, we find that the Ebin metric restricted to the space of type II deformations of a Sasakian structure is the sum of the Calabi metric and the gradient metric.

**Tags:**
SIR2014-AnHyC