Complex Hyperbolic Immersions of Cheng--Yau Metrics on Thullen Domains
Abstract
We classify the homotheties of the complete Cheng--Yau Kähler--Einstein metrics on the two-dimensional Thullen domains M(μ), μ>0, that admit Kähler immersions into ℂℍ∞. With Ric(gCY)=−3gCY, an immersion of c gCY exists exactly when 0<μ≤μ+ and 0<c≤3min{1,μ}/(μ+2). The critical parameter μ+>1 is characterized by an explicit equation, and its endpoint is included. Every admissible metric has a proper global Kähler embedding. The same parameter interval classifies immersibility into flat and projective Hilbert space, where every positive homothety admits a proper global embedding. Beyond the wall, all three targets are excluded at every positive scale. The proof combines an algebraic normalization of the Cheng--Yau equation, all-order coefficient estimates, and complex continuation of a scalar flow. For μ≠1, the resulting complete nonhomogeneous examples disprove proposed Kähler--Einstein rigidity conjectures in the complex-hyperbolic and flat Hilbert settings. We also prove a complementary rigidity theorem for Bergman--Hartogs domains and construct complete η-Einstein Sasakian manifolds with nonhomogeneous Riemannian metrics and proper global Sasakian embeddings into the infinite-dimensional Heisenberg space form.
Related
- Equivariant isometric immersions of surfaces in hyperbolic space
- Counterexamples to the ordinary Fried conjecture near hyperbolic metrics
- CSC Bergman metrics are Einstein on pseudoconvex domains
- Schwarz lemma on bounded symmetric domains endowed with holomorphic invariant Kähler--Berwald metrics
- Rigidity of Extremal and cscK Bergman Metrics on Pseudoconvex Domains
BibTeX
@misc{caibar2026complex,
title = {{Complex Hyperbolic Immersions of Cheng--Yau Metrics on Thullen Domains}},
author = {Mirel Caibăr and Andrea Loi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.21911},
eprint = {2608.21911},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.21911},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Complex Hyperbolic Immersions of Cheng--Yau Metrics on Thullen Domains
AU - Mirel Caibăr
AU - Andrea Loi
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2608.21911
UR - https://arxiv.org/abs/2608.21911
AB - We classify the homotheties of the complete Cheng--Yau Kähler--Einstein metrics on the two-dimensional Thullen domains M(μ), μ>0, that admit Kähler immersions into ℂℍ^∞. With Ric(g_CY)=−3g_CY, an immersion of c g_CY exists exactly when 0<μ≤μ_+ and 0<c≤3min{1,μ}/(μ+2). The critical parameter μ_+>1 is characterized by an explicit equation, and its endpoint is included. Every admissible metric has a proper global Kähler embedding. The same parameter interval classifies immersibility into flat and projective Hilbert space, where every positive homothety admits a proper global embedding. Beyond the wall, all three targets are excluded at every positive scale. The proof combines an algebraic normalization of the Cheng--Yau equation, all-order coefficient estimates, and complex continuation of a scalar flow. For μ≠1, the resulting complete nonhomogeneous examples disprove proposed Kähler--Einstein rigidity conjectures in the complex-hyperbolic and flat Hilbert settings. We also prove a complementary rigidity theorem for Bergman--Hartogs domains and construct complete η-Einstein Sasakian manifolds with nonhomogeneous Riemannian metrics and proper global Sasakian embeddings into the infinite-dimensional Heisenberg space form.
ER -