  We derive several new results for Renyi entropy, $S_n$, across generic
entangling surfaces. We establish a perturbative expansion of the Renyi
entropy, valid in generic quantum field theories, in deformations of a given
density matrix. When applied to even-dimensional conformal field theories,
these results lead to new constraints on the $n$-dependence, independent of any
perturbative expansion. In 4d CFTs, we show that the $n$-dependence of the
universal part of the ground state Renyi entropy for entangling surfaces with
vanishing extrinsic curvature contribution is in fact fully determined by the
Renyi entropy across a sphere in flat space. Using holography, we thus provide
the first computations of Renyi entropy across non-spherical entangling
surfaces in strongly coupled 4d CFTs. Furthermore, we address the possibility
that in a wide class of 4d CFTs, the flat space spherical Renyi entropy also
fixes the $n$-dependence of the extrinsic curvature contribution, and hence
that of arbitrary entangling surfaces. Our results have intriguing implications
for the structure of generic modular Hamiltonians.
