Pidstryhach Institute for Applied Problems in Mechanics and Mathematics of NAS of Ukraine,
3B Naukova Str., Lviv, 79601, Ukraine

E-mail:  pelykh@lms.lviv.ua

Knot manifolds of double-covariant systems of elliptic equations and a theorem about positive definition of gravitation energy

Abstract:
Solving a problem about correlation between Witten's spinor method for proof the theorem about positive definition of energy and alternative tensor methods demands the investigations knot manifolds for Sen-Witten equation. We prove that sufficient condition of zeros absence for solution of Sen-Witten equation which satisfy Reula condition in asymptotically flat three-dimensional space is the nearness of this space to maximal in asympotically minkowskian manifold.
With the aim of obtaining more general conditions we prove the principle of maximum for the norm of solution of elliptic system of second-order equations, which is locally SU(2)-covariant and covariant relatively to arbitrary coordinate transformations in three-dimensional Riemannian space and obtain more general conditions by analyzing knot surfaces of separate equations for diagonalized system. On this basis we prove the theorem, which completely solves the known problem on comparison between Witten spinor method and Nester tensor method for proof the theorem of positive definition of gravitation energy