Transverse rational curves and the LeBrun--Salamon conjecture

Source abstract

We prove that every compact connected positive quaternionic--Kähler manifold of real dimension $4n$ with $n\ge2$ is isometric to a Wolf space after rescaling the metric by a positive constant. This proves the LeBrun--Salamon conjecture. The proof uses transverse rational curves and the contact structure of the twistor space.