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Free, publicly-accessible full text available April 1, 2026
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Abstract We study holomorphic mapsFfrom a smooth Levi non-degenerate real hypersurface$$ M_{\ell }\subset {\mathbb {C}}^n $$ into a hyperquadric$$ {\mathbb {H}}_{\ell '}^N $$ with signatures$$ \ell \le (n-1)/2 $$ and$$ \ell '\le (N-1)/2,$$ respectively. Assuming that$$ N - n < n - 1,$$ we prove that if$$ \ell = \ell ',$$ thenFis either CR transversal to$$ {\mathbb {H}}_{\ell }^N $$ at every point of$$ M_{\ell },$$ or it maps a neighborhood of$$ M_{\ell } $$ in$$ {\mathbb {C}}^n $$ into$$ {\mathbb {H}}_{\ell }^N.$$ Furthermore, in the case where$$ \ell ' > \ell ,$$ we show that ifFis not CR transversal at$$0\in M_\ell ,$$ then it must be transversally flat. The latter is best possible.more » « less
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Abstract The paper studies complex manifolds whose Bergman metrics are incomplete but have constant holomorphic sectional curvature.We will construct a real analytic unbounded domain in \mathbb{C}^{2}whose Bergman metric is well-defined and has a positive constant holomorphic sectional curvature, which appears to be the first example of this kind.We will answer a long standing folklore conjecture that a Stein manifold has a negative constant holomorphic sectional curvature if and only if it is biholomorphic to a ball with a pluripolar set removed.Together with the uniqueness of a moment problem in the appendix of the paper provided by John Treuer, we will show that, under natural assumptions, there is no complex manifold whose Bergman metric is flat.more » « lessFree, publicly-accessible full text available March 22, 2026
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In this paper, we first study a mapping problem between indefinite hyperbolic spaces by employing the work established earlier by the authors. In particular, we generalize certain theorems proved by Baouendi-Ebenfelt-Huang [Amer. J. Math. 133 (2011), pp. 1633–1661] and Ng [Michigan Math. J. 62 (2013), pp. 769–777; Int. Math. Res. Not. IMRN 2 (2015), pp. 291–324]. Then we use these results to prove a rigidity result for proper holomorphic mappings between type I classical domains, which confirms a conjecture formulated by Chan [Int. Math. Res. Not., doi.org/10.1093/imrn/rnaa373] after the work of Zaitsev-Kim [Math. Ann. 362 (2015), pp. 639-677], Kim [ Proper holomorphic maps between bounded symmetric domains , Springer, Tokyo, 2015, pp. 207–219] and himself.more » « less
