arXiv subjects
Publications and source records attributed to Hirotaka Kiyohara.
Timelike minimal surfaces in the three-dimensional Lorentzian Heisenberg group are shown to be constructed from Lorentzian harmonic maps into the de-Sitter two-sphere, and they naturally admit singular points. In particular, we provide criteria for cuspidal edges, swallowtails, and cuspidal cross caps, and present several explicit examples.
We construct a class of Lorentzian harmonic maps into the de-Sitter $2$-space satisfying the eigenvalue equation $\Box N=2H^2N$ for the d'Alambert operator $\Box$ and a non-zero constant $H$ from framed null curves. We also investigate two classes of timelike surfaces associated with these Lorentzian harmonic maps: the first one is timelike surfaces with constant mean curvature $H$ in Lorentz-Minkowski $3$-space and the second one is timelike minimal surfaces in the three-dimensional Lorentzian Heisenberg group $\operatorname{Nil}_3(H)$. In particular, we characterize some properties of singularities on timelike minimal surfaces in $\operatorname{Nil}_3(H)$ via an invariant of framed null curves.
In the three-dimensional Heisenberg group equipped with a certain left invariant Lorentzian metric, timelike minimal surfaces which have the Abresch-Rosenberg differentials with vanishing multiplication of the coefficient function and its para-complex conjugate are characterized as the surfaces defined by the multiplication of null curves and affine null lines. Moreover, the constructions of these surfaces with prescribed curvatures of null curves or prescribed null lines are given.
Timelike surfaces in the three-dimensional Heisenberg group with left invariant semi-Riemannian metric are studied. In particular, non-vertical timelike minimal surfaces are characterized by the non-conformal Lorentz harmonic maps into the de Sitter two phere. On the basis of the characterization, the generalized Weierstrass type representation will be established through the loop group decompositions.