| Research Abstract |
We dealt with the system of differential equations z_<xx>= bz_y pz and z_<yy> = cz_x qz that defines a projectively minimal surface. The integrability condition of the system is
p_y = bc_x + 1/2b_xc - 1/2b_<yy>, q_x = cb_y + 1/2bc_y - 1/2c_<xx>,
b_<yyy> - bc_<xy> - 2bq_y - 2b_yc_x - 4qb_y = c_<xxx> - cb_<xy> - 2cp_x -2b_xc_y - 4pc_x
The following six vectors
U = z∧z_x, V = z∧z_y, N_1 = U_y, N_2 = V_x,
N_3 = 2z_y∧z_<xy> + bcV, N_4 = 2z_x∧z_y + bcU
define a frame T = ^t(U, V, N_1, N_2, N_3, N_4) in P^5. It satisfies a Pfaffin equation dT = ωT with a certain 1-form ω. A remarkable property of this frame is that the vectors satisfy the orthogonality condition
(U, N_3) = -1, (V, N_4) = 1, (N_1, N_1) = 1, (N_2, N_2) = -1,
relative to a certain canonical paring on P5 with the remaining parings being zero. We characterized such a frame. Namely, given a nondegenerate bilinear form {h_<ij>} on P^5, consider a projective frame t = ^t(t_1, …, t_6) that satisfies the orthogonality condition (t_i, t_j) = h_<ij> and denote the Pfaffian equation by dt = ωt. We assume the conditions that dt_1 ≡ 0 (mod t_1, t_2, t_3), dt_2 ≡ 0 (mod t_1, t_2, t_4), and that ω_1^3 and ω_2^4 are linearly independent. Then, we can find a change of the frame: t → gt by a transformation g with gh^tg = h such that the new frame gt satisfies a Pfaffian equation which has the same form as that satisfied by T, provided that the signature of h is (3, 3). Furthermore, when the signature is assumed to be (3, 3), the frame characterizes frames associated with Lie-minimal surfaces.
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