  Sr$_2$RuO$_4$ is a leading candidate for chiral $p$-wave superconductivity.
The detailed mechanism of superconductivity in this material is still the
subject of intense investigations. Since superconductivity is sensitive to the
topology of the Fermi surface (the contour of zero-energy quasi-particle
excitations in the momentum space in the normal state), changing this topology
can provide a strong test of theory. Recent experiments tuned the Fermi surface
topology efficiently by applying planar anisotropic strain. Using functional
renormalization group theory, we study the superconductivity and competing
orders in Sr$_2$RuO$_4$ under strain. We find a rapid initial increase in the
superconducting transition temperature $T_c$, which can be associated with the
evolution of the Fermi surface toward a Lifshitz reconstruction under
increasing strain. Before the Lifshitz reconstruction is reached, however, the
system switches from the superconducting state to a spin density wave state.
The theory agrees well with recent strain experiments showing an enhancement of
$T_c$ followed by an intriguing sudden drop.
