The Gravitational Energy Of Several Geometric Problems

This PhD dissertation is concerned with some geometric problems related to gravitational energy.It is nontrivial to define the gravitational energy. The definition of the total energy was given by Arnowitt-Deser-Misner [Phys. Rev. 122, 997-1006(1961) ; et al.] from the Hamiltonian point of view in early 1960's. Physicists believe, with some justification, that the total energy for a nontrivial isolated gravitational system must be positive. This is the famous positive energy conjecture in general relativity.Motivated by superstring theory, we consider the spaces which asymptotically approach the product of a flat space with a Calabi-Yau manifold. Dai [Commun. Math. Phys. 244, 335-345(2004) ; J. Math. Phys. 46, 042505(2005) ] first defined the total energy and also established a positive energy theorem for such spaces. Inspired by Dai's work and Zhang's work on total angular momentum [Commun. Math. Phys. 206, 137-155(1999) ], under Calabi-Yau compactification, the author extends the positive energy theorem to the case of nonsymmetric initial data set (Theorem 2．2) .It is natural to extend the positive energy theorem to asymptotically AdS space-time. Chru(?)ciel et al. [Adv. Theor. Math. Phys. 19, 697-754(2001) ; Pacific J. Math. 212, 231-264(2003) ; et al.] first studied the total energy for asymptotically hyperbolic manifolds. Zhang [Commun. Math. Phys. 249, 529-548(2004) ] and Maerten [Ann. Henri Poincaré7, 975-1011(2006) ] independently generalized the positive energy the-orem to the case of nonzero second fundamental form respectively. Chapter 3 deals with a spacetime version of the positive energy theorem for the asymptotically AdS spacetime with arbitrary cosmological constant when the nonzero second fundamen-tal form appears. Joint with Zhang, we obtain the following theorems: Theorem 3．1 corresponds to Zhang's Riemannian version while Theorem 3．3 includes Maerten's inequality of the Lorentzian length of the total mass vector. Theorem 3．2 gives a positive energy theorem in the important case when M is maximal.In the last chapter, we discuss the Liouville-type theorems for wave maps over gravitational fields. Hu et al. [Chinese Ann. Math. B 5, 737-740(1984) ; Lett. Math. Phys. 14, 253-262(1987) ; Lett. Math. Phys. 14, 343-351(1987) ; et al.] gave a series of nonexistence theorems for the wave maps from the flat spacetime as well as Schwarzschild spacetime. Under the assumption of finite energy (or slowly divergent energy), the author obtains an absence theorem for static wave maps defined from the Schwarzschild-AdS spacetime into any Riemannian manifold (Theorem 4．3) .