Research On The Bartnik Mass Of High-Dimensional Constant Mean Curvature Manifold

General relativity describes the macroscopic physical world.Its research is of great sig-nificance to the understanding of the whole universe.It is still an active research field of mathematical physics.It has always been concerned about the problem of quasi-local mass.A well-recognized and reasonable definition.Some of the quasi-local masses now have impor-tant applications for geometrical physics.Bartnik gives a definition of quasi-local mass[2,3]by ADM mass[1],that is,a smooth asymptotically flat 3-dimensional Riemannian manifold can be tolerated.Expanding the lower bound of ADM mass.Cabrera,Cederbaum,Mc-cormick,and Miao gave an upper bound on the Bartnik quasi-local mass[6]associated with Hawking mass[11].This paper considers high-dimensional manifolds with non-negative and constant mean curvature boundaries.For the compact manifold with boundary,using the high-dimensional positive mass theorem given by R.Schoen and S.T.Yau[19],we discuss the influ-ence of the non-negative curvature of the region on the boundary geometry,and establish some inequalities for the manifold with inner and outer boundaries.Even for compact manifolds without inner boundaries,some inequalities involving boundary geometry can be obtained.This paper discusses the admissible extension of the Bartnik data with constant mean curvature in a high-dimensional asymptotically flat Riemannian manifold,which yields an upper bound for the Bartnik quasi-local mass.