A Cartan-the Hartogs Domain Extremum Problem

One of the key issues in the theory of several complex variables is the classification of the domain in the biholomorphic reflection. In the theory of one complex variable, Riemann mapping theorem has resolved the problem about simply connected domains with at least one boundary points in the complex plane D will be mapped onto the unit disk by biholomorphic functions f. But in the theory of several complex variable, many of the simply connected domains are not holomorphic equivalent. We need to prove their equivalence with some suitable methods. Extremal problem is the promotion of Schwarz Lemma in a high-dimensional. Through the study of extremal problem, we can get the extremal mapping which maps a domain onto the unit disk, and we can get extremal distanceμusing extremal distanceμwe can measure whether the two domains are biholomorphic equivalent.In this paper, we study the extremal problem between the Cartan-Hartogs domain of the second type and the unit hyperball. We also study the extremal problem between the Cartan-Hartogs domain of the third type and the unit hyperball. We get the maximal inscribed Hermitian ellipsoids of the Cartan-Hartogs domain of the second type and the third type. Then we obtain the Caratheodory extremal mappings, the Caratheodory extremal, the extremal distances. These results show that we can study extremal problems in the Cartan-Hartogs domain of the second type and the third type, thereby they improve the development of Cartan-Hartogs domain research.In 1998, Yin Weiping structured four types of Cartan-Hartogs domains. Here the Cartan-Hartogs domain of the second and the third type are following: Here R_Ⅱ(p) ,R_Ⅲ(q) denotes respectively the Cartan domain of the second type in the sense of L.K.Hua with Z a symmetrical matrix of order p and Cartan domain of the third type in the sense of L.K.Hua with Z a skew-symmetrical matrix of order q, Z^t denotes the transposed of Z, Z denotes the conjugate of Z, det denotes the determinant of a square matrix, N is a positive integer, and K is a positive real number. We note respectively the Cartan-Hartogs domain of the second and the third type Y_Ⅱ(N,p, K) and Y_Ⅲ(N, q, K).In order to find the specific forms of the maximal inscribed Hermitian ellipsoid of Y_Ⅱ(N,p, K) and Y_Ⅲ(N,q, K), first we find the necessary and sufficient conditionfor the Hermitian ellipsoid S(a, b) being the maximal inscribed Hermitian ellipsoid of Y_Ⅱ(N,p,K) and Y_Ⅲ(N,q, K), we can get a function h(μ) by general form of Y_Ⅱ(N,p,K) and Y_Ⅲ(N,q,K), and then in accordance with the maximal inscribed Hermitian ellipsoid definition, we find when S(a,b) is the maximal inscribedHermitian ellipsoid of Y_Ⅱ(N,p,K) and Y_Ⅲ(N,q,K), this function h(μ) required to meet some conditions.Combining with the sufficient and necessary conditionsfor S(a, b) being the maximal inscribed Hermitian ellipsoid of Y_Ⅱ(N,p,K) and Y_Ⅲ(N, q, K), we obtain the specific forms of the maximal inscribed Hermitian ellipsoid in the form of three situations after some discussion and calculation. Then according to the specific form of the maximal inscribed Hermitian ellipsoid, we obtainthe Caratheodory extremal mapping, the Caratheodory extremal value and the Caratheodory extremal distance. We receive the Caratheodory extremal mapping, the Caratheodory extremal value and the Caratheodory extremal distance between B^N+Mand Y_Ⅱ(N,p,K) if 0 < K < 1 and the corresponding conclutions between B^N+M and Y_Ⅲ(N, q, K) if 0 < K < 2:(1)The Caratheodory extremal mappings of Y_Ⅱ(N,p, K) and Y_Ⅲ(N,q, K) are the following:If 00 =(?)If 0 < K < 2, A =Ⅲ, t=p, then: a₀ =(?)(2) The Caratheodory extremal values of Y_Ⅱ(N,p,K) and Y_Ⅲ(N,q,K) are(?) (02).(3) The Caratheodory extremal distances of Y_Ⅱ(N,p, K) and Y_Ⅲ(N, q, K) are1/2 log(?)(0