Estimate on the size of flat balls where the Weyl tensor vanishes

Question

Let $(M,g)$ be a Riemannian manifold and suppose that the Weyl tensor of $g$ vanishes at a point $p \in M$. Can one estimate the size of the largest geodesic ball around $p$ that we can make $g$ flat on through conformal deformation in terms of geometric data, i.e., $g$ and the usual curvature invariants? Is this just a function of the injectivity radius?

Also: suppose $g$ is locally conformally Einstein. Can we produce similiar estimates on the largest geodesic ball around a given point that we can make $g$ Einstein on?

Answer 1

First, presumably you want to assume that the Weyl tensor vanishes not just at one point but everywhere, since at any point where the Weyl tensor does not vanish, there will be no conformal factor of the metric that is flat.

Second, I doubt what you want can be characterized using local pointwise invariants such as curvature. The largest geodesic ball for there is a flat metric conformal to the original metric is essentially a global concept. And it doesn't seem easy to me to describe this ball. For one thing, it does not need to be diffeomorphic to a ball; just think about the flat torus.

Answer 2

I believe that the answer to the first question is the injectivity radius if the manifold is locally conformally flat. The injectivity radius provides us with an estimate on the size of a chart that contains the point $p$. In this chart the metric has to take the form $h dx^i dx^j$, where $h$ is a smooth, positive function, if the Weyl tensor vanishes everywhere. It follows that we can make the metric flat through conformal deformation on the whole of the interior of the chart.