Multiplier Stiffness Theorem On PU(3,1)

The Mostow rigid theorem,published in 1973,is an epoch-making contribution to the theory of Lie groups.Roughly speaking,the theorem proves that the two manifolds in a symmetric space of topological homeomorphism are isometric isomorphic under certain appropriate conditions.The Mostow rigid theorem and its proof established the connection between topology,differential geometry,conformal geometry,Lie groups,harmonic analysis and ergodic theory.After that,Mostow’s rigidity theorem has been generalized in many directions.For example,Tukia studied the problem of boundary rigidity:given two nonelementary Mobius groups on a n dimensional sphere and compatible mapping on their invariant subsets,when is the compatible mapping a Mobius transformation?Tukia gave the following two sufficient conditions:the compatible map is differentiable at a certain cone limit point,or the function of the non-elementary group on the product space of invariant subsets is ergodic.Some domestic scholars have promoted his work.For example,Chen Min reduced the sufficiency condition of "compatibilities map at a typical limit point to be differentiable" to"the content map has a Jacobian with a positive rank at a typical limit point".In 1991,Tukia proved that the homomorphism between non-elementary M?bius transformation groups on Rn is a constant stretch if and only if it preserves multipliers.In this paper,we generalize it to PU（3,1）and prove it using an algebraic method.In our proof,we mainly use the algebraic relationship between the multiplier and the traces of the matrix,and the problems are classified and discussed by using the existence of limit.