Study of transformations of Lie-minimal surfaces

我们处理了定义射影极小曲面的微分方程组z_<xx>= bz_y pz和z_<yy> = cz_x qz。系统的可积性条件isp_y = bc_x + 1/2b_xc - 1/2b_<yy>, q_x = cb_y + 1/2bc_y - 1/2c_<xx>,b_<yyy> - bc_<xy> - 2b_q_y - 2b_yc_x - 4qb_y = c_<xxx> - cb_<xy> - 2cp_x - 2cp_x -2b_xc_y - 4pc_x。以下6个向量su = z∧z_x, V = z∧z_x, N_1 = U_y, N_2 = 2z_x∧z_ <xy> + bcV, N_4 = 2z_x∧z_y + bcu在P^5中定义了一个坐标系T = ^ T （U， V, N_1, N_2, N_3, N_4）。它满足Pfaffin方程dT = ω t具有一定的1-形式ω。该坐标系的一个显著性质是，相对于P5上的某个正则配对，向量满足正交性条件(U, N_3) = -1, (V, N_4) = 1, (N_1, N_1) = 1, (N_2, N_2) = -1，其余配对为零。我们描述了这样一个框架。即，给定P^5上的非退化双线性形式{h_<ij>}，考虑满足正交性条件(t_i, t_j) = h_<ij>的射影坐标系t = ^t(t_1，…，t_6)，用dt = ωt表示Pfaffian方程。我们假设dt_1≡0 (mod t_1, t_2, t_3), dt_2≡0 (mod t_1, t_2, t_4)，并且ω_1^3和ω_2^4是线性无关的。然后，我们可以通过g对gh^tg = h进行变换，求出坐标系t→gt的变化，使新坐标系gt满足与t满足的形式相同的Pfaffian方程，条件是h的特征为（3,3）。此外，当签名被假设为（3,3）时，框架表征了与Lie-minimal曲面相关的框架。

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 isp_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_xThe following six vectorsU = 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 + bcUdefine 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.