有界域间的全纯Bergman等距映射与Lu定理的推广
Holomorphic Bergman isometries between bounded domains and the extension of Lu's theorem
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中文总结 AI 辅助
该研究探讨复流形间保持Bergman度量的全纯映射，得出等维情形下全纯Bergman等距映射的性质，推广Lu定理并给出相关应用。
中文摘要 AI 辅助
我们研究复流形间保持Bergman度量（相差一个正常数）的全纯映射。在维数相等的情形下，若源为Stein流形、目标为复欧氏空间中的有界域，或源为有界拟凸域、目标为复流形，则全纯Bergman等距映射是到相对闭局部pluripolar集补集上的双全纯映射，且该常数必为1。复欧氏空间中的有界域为此结论的特殊情形。我们得到了其在保曲率映射上的若干应用，还进一步证明了映入强拟凸域笛卡尔积的局部全纯Bergman等距映射的刚性结果，并将其应用于有限解析对应与模对应。
英文摘要
We study holomorphic maps between complex manifolds that preserve the Bergman metric up to a positive constant. In the equal dimensional case, either the source is a Stein manifold and the target is a bounded domain in the complex Euclidean space, or the source is a bounded pseudoconvex domain and the target is a complex manifold, holomorphic Bergman isometry is a biholomorphism onto the complement of a relatively closed locally pluripolar set, and the constant is necessarily one. These include the bounded domains in the complex Euclidean space as special cases. Some applications to curvature-preserving maps are obtained. We further prove the rigidity result for local holomorphic Bergman isometries into Cartesian product of strongly pseudoconvex domains, with applications to finite analytic and modular correspondences.
