Zhuang-dan Daniel Guan (UCR)

Abstract: Using the confirmation of the Yamabi conjecture, one could search the Einstein metrics by searching the critical metrics of the scalar curvature of Riemannian metrics with constant scalar curvatures. This effort lead to the Fischer-Marsden equation. In 2013, Paul Cernea obtained some structure of the Riemannian manifolds with multiple solutions to the Fischer-Marsden equation. Let W be the space of the solutions, he also proved that the dimension k of W is less than n+1 and it is a standard sphere if the dimension is n+1. In 2014, we finished the classification for the compact homogeneous case and also proved that in general the factor of the isometric group generated by the Killing fields coming from the solutions has orbits of spheres of dimension k-1 or the fixed points. This also implies Cernea's earlier result. A paper appeared in International Journal of Mathematics last year.

In 2015, we proved that if k=n, then M is again a sphere. That is, k can not be n. In the proof, we used warp product calculations.

We would continue our earlier work and classify the cohomogeneity one case.

We shall give some background, a proof and some examples.