Study of transformations of Lie-minimal surfaces

我们讨论了定义射影极小曲面的微分方程组z_&lt；xx&gt；=Bz_ypz和z_&lt；yy&gt；=Cz_xQz。系统isp_y=bc_x+1/2b_xc-1/2b_xc-1/2b_&lt；y&gt；，q_x=cb_y+1/2bc_y-1/2c_&lt；xx&gt；，b_&lt；yyy&gt；-bc_&lt；xy&gt；-2bq_y-2b_yc_x-4qb_y=c_&lt；xxx&gt；-cb_&lt；xy&gt；-2cp_x-2b_xc_y-4pc_x下列六个向量U=z∧z_x，V=z∧z_y，N_1=U_y，N_2=V_x，N_3=2z_y∧z_&lt；xy&gt；+bcv，N_4=2z_x∧z_y+bcU在P^5中定义一个框架T=^t(U，V，N_1，N_2，N_3，N_4)，它满足Pfaffin方程DT=ω具有一定的1-形式ω。该框架的一个显著性质是向量满足(U，N_3)=-1，(V，N_4)=1，(N_1，N_1)=1，(N_2，N_2)=-1的正交性条件。我们描述了这样一个框架。也就是说，给定P^5上的一个非退化双线性形式，考虑一个射影框架t=^t(t_1，…，t_6)满足正交性条件(t_i，t_j)=h_&lt；ij&gt，记为dt=ωt，假设dt_1≡0(mod t_1，t_2，t_3)，dt_2≡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来找到框架T→GT的变化，使得新的框架GT满足与T所满足的形式相同的Pfaffian方程，只要h的签名是(3，3)。此外，当假设签名为(3，3)时，框架刻画了与李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.