Fischer-Marsden conjecture for hypersurfaces of the complex hyperbolic space

Abstract

In this paper, we want to solve the Fischer-Marsden conjecture on real hypersurfaces in the complex hyperbolic space CHn=SU1,n/S(U1Un). First we prove that it is true on hypersurfaces with isometric Reeb flow in CHn. Next it is also true on a certain class of contact hypersurfaces in CHn. That is, we have shown that there does not exist a non-trivial solution (g,ν) of Fischer-Marsden equation on contact real hypersurfaces with radius r∈[r1,∞), where coth⁡(2r1)=2n−3n−2, in the complex hyperbolic space CHn. Consequently, the Fischer-Marsden conjecture is true on these two kinds of real hypersurfaces in the complex hyperbolic space CHn.