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3-dimensional null surfaces that are Killing horizons to the second order (in short: stationary horizons) are considered. They are embedded in 4- dimensional spacetimes that satisfy the vacuum Einstein equations with arbitrary cosmological constant. Intrinsic geometry of each stationary horizon consists of induced degenerate metric tensor and induced twist free connection. In the case of non-degenerate (non-extremal) stationary horizon the intrinsic geometry determines its spacetime Weyl tensor. The assumption that the Weyl tensor is of the Petrov type D amounts to a complex equation on invariants of the intrinsic geometry. The equation is shown to be an integrability condition for the so called Near Horizon Geometry equation. The emergence of the Near Horizon Geometry equation in this context is equivalent to the hyper-suface orthogonality of the transversal principal null direction of the Weyl tensor. In the case of bifurcated stationary horizon the type D equation implies the axial symmetry. In this way the axial symmetry is ensured without the rigidity theorem. All topologies of horizon cross-sections are considered. For cross-sections of genus > 0 the Petrov type D equation is solved completely due to methods of the algebraic topology. For topologically spherical cross-sections the equation can be exactly solved in the axisymmetric case. The result is the subject of an accompanying talk.

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Although the physical Hamiltonian operator can be constructed in the deparametrized model of loop quantum gravity coupled to a scalar field, its property is still unknown. This open issue is attacked in this paper by considering an operator Hˆv representing the square of the physical Hamiltonian op- erator acting nontrivially on a two-valent vertex of spin networks. The Hilbert space Hv preserved by the graphing changing operator Hˆv is consist of spin networks with a single two-valent non-degenerate vertex. The matrix element of Hˆv are explicitly worked out in a suitable basis. It turns out that the operator Hˆv is essentially self-adjoint, which implies a well-defined physical Hamiltonian operator in Hv for the deparametrized model.

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