Study of transformations of Lie-minimal surfaces

讨论了定义射影极小曲面的微分方程组z_<xx>= bz_ypz和z_<yy>= cz_xqz。系统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_x以下六个向量U = z &lt;$z_x，V = z &lt;$z_y，N_1 = U_y，N_2 = V_x，N_3 = 2z_y &lt;$z_<xy>+ bcV，N_4 = 2z_x &lt;$z_y + bcU定义了P^5中的帧T = ^t（U，V，N_1，N_2，N_3，N_4）。它满足一个Pfaffin方程dT = ωT，具有一定的1-形式ω。该框架的一个显著性质是向量满足正交条件（U，N_3）=-1，（V，N_4）= 1，（N_1，N_1）= 1，（N_2，N_2）=-1，相对于P5上的某个标准对，其余对为零.我们描述了这样一个框架。也就是说，给定P^5上的非退化双线性形式{h_<ij>}，考虑满足正交条件（t_i，t_j）= h_的投射框架t = ^t（t_1，...，t_6），<ij>并将Pfiran方程记为dt = ωt。我们假设dt_1 &lt;$0（mod t_1，t_2，t_3），dt_2 &lt;$0（mod t_1，t_2，t_4），ω_1^3和ω_2^4线性无关。然后，我们可以通过具有gh^tg = h的变换g找到框架的变化：t → gt，使得新的框架gt满足与T满足的形式相同的Pfiran方程，条件是h的签名是（3，3）。此外，当签名被假定为（3，3）时，框架表征与Lie极小曲面相关联的框架。

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.