  Inspired by work of Ejiri-Micallef on closed minimal surfaces, we compare the
energy index and the area index of a free-boundary minimal surface of a
Riemannian manifold with boundary, and show that the area index is controlled
from above by the area and the topology of the surface. Combining these results
with work of Fraser-Li, we conclude that the area index of a free-boundary
minimal surface in a convex domain of Euclidean three-space, is bounded from
above by a linear function of its genus and its number of boundary components.
We also prove index bounds for submanifolds of higher dimension.
