If M = { ( x, f ( x ) ) ∣ ( x 1, …, x n ) ∈ Ω } is an affine maximal hypersurface, and if M is complete with respect to the Calabi metric G = ∑ ∂ 2 f ∂ x i ∂ x j d x i d x j, then, in the case where n = 2 or n = 3, M must be an elliptic paraboloid.
In fact, we study more general surfaces satisfying a fourth order partial differential equation (PDE), which include affine maximal hypersurface equations (see [8]), an α-relative extremal hypersurface equation (see [12]) and the Abreu equation (see [13]), Δ ρ = − β ∥ ∇ ρ ∥ 2 ρ, (1.1).
Let ((M, g, N )) be a closed normalized null hypersurface with rigged vector field ξ.
A null hypersurface with a specific screen distribution is given by (( M,g,mathscr{S}(N))).
Let ((M,g,N)) be a normalized null hypersurface with rigged vector field ξ.
The conducting sheet is modeled as an idealized hypersurface with an effective electric conductivity.
Let Mn be a compact hypersurface with constant mean curvature H in Sn+1.
The equality holds for all p ∈ M if and only if either M is a screen homothetic lightlike hypersurface with φ = − 1 or M is a totally geodesic lightlike hypersurface.
Finally, the complexity of the trained SVM is further reduced by approximating the separation hypersurface with a subset of the support vectors.
Let ((M, g, N )) be a closed normalized null hypersurface with rigged vector field ξ and (tau^{N} (xi) = 0 ) in a pseudo-Riemannian manifold.
