Holomorphic Bergman isometries between bounded domains and the extension of Lu's theorem

Abstract: 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.

• The paper proves that holomorphic Bergman isometries between suitable bounded domains, Stein manifolds, and complex manifolds are biholomorphisms onto complements of relatively closed pluripolar sets, with the scaling constant necessarily equal to 1.
• The paper combines Calabi diastasis, Bergman–Bochner map continuation, monodromy, covering Riemann domains, and envelopes of holomorphy to establish global rigidity while identifying precisely how incompleteness creates exceptional sets.
• The paper shows that pluripolar defects are optimal, derives rigidity for local isometries, curvature-preserving maps, products, and finite analytic correspondences, and recovers genuine biholomorphisms when the source Bergman metric is complete.

Overview

This paper by Yuan Yuan studies holomorphic maps between complex manifolds that preserve the Bergman metric up to a positive constant $\lambda$, and establishes a mapping-theoretic extension of Lu's uniformization theorem (2608.15991). The central theme is that a holomorphic Bergman isometry between bounded domains — or more generally between Stein manifolds and bounded pseudoconvex domains — must be a biholomorphism onto the complement of a relatively closed pluripolar set, and that the scaling constant is necessarily $\lambda = 1$. The paper also proves a rigidity theorem for local Bergman isometries into Cartesian products of strongly pseudoconvex domains and derives applications to finite analytic correspondences and curvature-preserving maps.

The work is situated at the intersection of two research lines: Mok's theory of holomorphic isometric embeddings of bounded symmetric domains and their rigidity properties, and the recent program initiated by Huang–Li on removing the completeness hypothesis from Lu's theorem. Huang and Li showed that for a bounded pseudoconvex domain whose Bergman metric has constant holomorphic sectional curvature, the correct conclusion without completeness is biholomorphism to the unit ball with a relatively closed pluripolar set removed; they subsequently extended this to arbitrary Stein manifolds.

Main results

The paper's principal theorems share a common structure. Let $f : D_1 \to D_2$ be a holomorphic map satisfying $f^*\omega_{D_2} = \lambda \omega_{D_1}$ with $\lambda > 0$. The first result treats two bounded domains with $D_1$ pseudoconvex; the second replaces the source by an $n$-dimensional Stein manifold with well-defined Bergman metric; the third reverses the roles, taking $D$ bounded pseudoconvex as source and an arbitrary complex manifold $M$ (with the Bergman-Bochner map an immersion) as target. In each case the conclusion is that there exists a relatively closed (locally) pluripolar set $E$ such that $\lambda = 1$0 is a biholomorphism onto the complement of $\lambda = 1$1, and moreover $\lambda = 1$2.

Two local versions follow from Mok's germ-extension theorem and its Huang–Li analogue: if a local holomorphic Bergman isometry is defined only on a connected open subset, it extends to a global biholomorphism onto the complement of a pluripolar set, provided the relevant completeness assumptions hold. These local-to-global statements are significant because they show that agreement of Bergman metrics on an arbitrarily small open set forces the global biholomorphic structure up to a thin exceptional set.

A corollary worth emphasizing: when $\lambda = 1$3 is complete, the exceptional set must be empty, so any holomorphic Bergman isometry out of a Bergman-complete domain is a genuine biholomorphism. This recovers classical rigidity in the complete case while showing precisely how completeness enters.

Optimality and the isometric constant

Two features demonstrate that the conclusions are sharp. First, the paper proves that any holomorphic Bergman isometry between bounded domains satisfies $\lambda = 1$4. The argument compares vanishing orders of the kernels $\lambda = 1$5 and $\lambda = 1$6 along hypersurfaces of their common zero support, obtaining $\lambda = 1$7, then constructs a meromorphic quotient $\lambda = 1$8 that would violate a vanishing lemma (a bounded holomorphic function dominated by a negative power of the kernel must vanish identically) if $\lambda = 1$9.

Second, the conclusion "up to a pluripolar set" cannot be strengthened: the author constructs, in every dimension, a bounded pseudoconvex domain $f : D_1 \to D_2$0 and a bona fide holomorphic Bergman self-isometry that is not surjective. In dimension one, one removes a discrete orbit $f : D_1 \to D_2$1 of a disc automorphism from $f : D_1 \to D_2$2; since the removed set is pluripolar, the Bergman space and metric are unchanged, but the automorphism restricted to this domain maps it properly into itself, missing one point. Taking products with polydiscs gives examples in all dimensions. Thus even for self-maps, the pluripolar defect is unavoidable.

Methodology

The proofs combine several techniques. Injectivity follows from Calabi's diastasis: the identity $f : D_1 \to D_2$3 propagates off the zero set of a kernel product, and evaluating at a hypothetical second preimage forces equality in the Cauchy–Schwarz inequality, hence linear dependence of reproducing kernels, contradicting point separation of the Bergman space.

The key new ingredient is a continuation argument for the Bergman-Bochner map. Calabi's extension theorem, in the form due to Huang–Li, continues the germ of the projective isometry along every path in the ambient domain. Monodromy is handled by passing to a covering Riemann domain determined by the stabilizer subgroup of the initial germ; on this cover every $f : D_1 \to D_2$4-function extends single-valuedly. Irgens' $f : D_1 \to D_2$5-envelope of holomorphy then identifies the complement of the distinguished sheet as pluripolar, and Josefson's theorem upgrades local pluripolarity to global pluripolarity. Once the exceptional set $f : D_1 \to D_2$6 is known to be relatively closed and pluripolar, restriction induces a unitary isomorphism of Bergman spaces, forcing $f : D_1 \to D_2$7 and hence $f : D_1 \to D_2$8.

For the case where pseudoconvexity is dropped but $f : D_1 \to D_2$9 is imposed, the paper obtains a partial result via moment theory: the complement of the image is Bergman-negligible in the sense of Ebenfelt–Treuer–Xiao. The proof shows the kernel ratio extends to a nowhere-vanishing multiplier, builds a unitary multiplier operator between reproducing kernel Hilbert spaces, and concludes via Stone–Weierstrass that the measures agree, so the complement has measure zero and all $f^*\omega_{D_2} = \lambda \omega_{D_1}$0-holomorphic functions extend.

Applications to curvature-preserving maps

By the Nomizu–Yano theorem, a strongly curvature-preserving local biholomorphism between locally irreducible Kähler metrics is automatically a homothetic Bergman isometry. Combined with the main theorems, this yields: a strongly curvature-preserving local biholomorphism between locally irreducible bounded domains, with the source pseudoconvex, is a biholomorphism modulo a pluripolar defect; with both metrics complete, it extends to a genuine biholomorphism. Analogous corollaries hold for Kähler–Einstein domains with nonzero scalar curvature: preservation of Ricci forms forces the Einstein constants to agree and yields biholomorphism (up to the pluripolar defect in the incomplete case). Notably, neither Bergman metric needs to be locally symmetric — agreement of full curvature jets suffices.

Rigidity for products and correspondences

The second half of the paper addresses the equidimensional analogue of Mok–Ng rigidity outside the homogeneous setting. Let $f^*\omega_{D_2} = \lambda \omega_{D_1}$1 be simply connected, bounded, strongly pseudoconvex with real analytic boundary, $f^*\omega_{D_2} = \lambda \omega_{D_1}$2, and let $f^*\omega_{D_2} = \lambda \omega_{D_1}$3 be holomorphic maps, each of full rank somewhere, satisfying $f^*\omega_{D_2} = \lambda \omega_{D_1}$4 on a connected open $f^*\omega_{D_2} = \lambda \omega_{D_1}$5. Then each $f^*\omega_{D_2} = \lambda \omega_{D_1}$6 extends to an automorphism of $f^*\omega_{D_2} = \lambda \omega_{D_1}$7 and $f^*\omega_{D_2} = \lambda \omega_{D_1}$8.

The proof extends the tuple $f^*\omega_{D_2} = \lambda \omega_{D_1}$9 to a proper holomorphic map via Mok's extension theorem, uses Zorn's boundary extension of the Bergman kernel to realize the graph as an irreducible analytic subvariety meeting the boundary, applies Hopf lemma and Baire category to find a boundary point where some component is transversal, and invokes the Nemirovski–Shafikov uniformization theorem to conclude each component is an automorphism.

Following Clozel–Ullmo, the paper introduces finite analytic correspondences on analytic spaces associated to $\lambda > 0$0 (quotients $\lambda > 0$1 pulling back $\lambda > 0$2 to $\lambda > 0$3) and shows that a correspondence preserving $\lambda > 0$4 reduces, over a suitable open set, to the product situation above. Consequently, metric-preserving finite analytic correspondences are special (all branches automorphisms), and on regular quotients $\lambda > 0$5 they are modular, i.e., induced by elements of the commensurator of $\lambda > 0$6 in $\lambda > 0$7. This extends the modular correspondence framework from Shimura varieties to nonhomogeneous strongly pseudoconvex domains.

Limitations and open questions

Several restrictions are explicit. The product rigidity theorem requires strong pseudoconvexity, real analytic boundary, simple connectivity, and $\lambda > 0$8; the author conjectures the same conclusion holds for arbitrary smooth bounded domains of dimension at least 2, but notes that Mok's $\lambda > 0$9-th root map gives counterexamples in dimension one and analogous non-standard examples exist from polydiscs. The positive-codimensional problem is deferred to forthcoming work. For maps preserving Bergman $D_1$0-forms, the reduction to metric preservation is available only for $D_1$1 (via Chan–Yuan); no analogous result is known for $D_1$2, and it remains open whether the product rigidity theorem extends to preservation of $D_1$3-forms with $D_1$4. Finally, the general (non-pseudoconvex) case is handled only under the normalization $D_1$5 and yields only Bergman-negligibility rather than pluripolarity of the complement.

Conclusion

The paper establishes that holomorphic Bergman isometries in the equidimensional setting are rigid: they are biholomorphisms onto complements of pluripolar sets with scaling constant forced to equal one, extending Lu-type uniformization from intrinsic curvature conditions to mapping problems between arbitrary bounded domains and Stein manifolds. The combination of Calabi extension, monodromy control on covering Riemann domains, Irgens envelopes, and moment-theoretic arguments provides a flexible toolkit, and the resulting rigidity of correspondences brings the Clozel–Ullmo modularity picture to a substantially broader class of domains.