This is based on joint work with Song Sun.
As an analogue of Frankel conjecture (Mori, Siu-Yau theorem) in Kahler geometry, we
classify compact Sasaki manifolds with positive curvature by deforming metrics.
Roughly speaking, such Sasaki structure is a standard Sasaki structure on (odd
dimensional) spheres. Our theorem gives a new proof of Frankel conjecture as a
special case. We have also similar results as in Kahler setting for nonnegative
curvature.