Generalized Birkhoff theorem and its applications in mimetic gravity
  There is undetermined potential function $V(\phi)$ in the action of mimetic
gravity which should be resolved through physical means. In general
relativity(GR), the static spherically symmetric(SSS) solution to the Einstein
equation is a benchmark and its deformation also plays a crucial role in
mimetic gravity. The equation of motion is provided with high nonlinearity, but
we can reduce primal nonlinearity to a frequent Riccati form in the SSS case of
mimetic gravity. In other words, we obtain an expression of solution to the
functional differential equation of motion with any potential function.
Remarkably, we proved rigorously that there is a zero point of first order for
the metric function $\beta(r)$ if another metric function $\alpha(r)$ possesses
a pole of first order within mimetic gravity. The zero point theorem may be
regarded as the generalization of Birkhoff theorem $\alpha\beta=1$ in GR. As a
corollary, we show that there is a modified black hole solution for any given
$V(\phi)$, which can pass the test of solar system. As another corollary, the
zero point theorem provides a dynamical mechanism for the maximum size of
galaxies. Especially, there are two analytic solutions which provide good fits
to the rotation curves of galaxies without the demand for particle dark matter.
