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A complete Kähler metric on ℂn of nonnegative Ricci curvature with Euclidean volume growth and nonunique asymptotic cone

  • Jiwei Dai
  • Wancheng Zhang
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Abstract

We construct complete smooth Kähler metrics on ℂn , n≥2, with nonnegative Ricci curvature, Euclidean volume growth, and nonunique tangent cones at infinity. More generally, we prescribe the entire tangent cone set within a class of smooth Hopf cones, using a compact connected family of positive transverse Ricci forms on ℂℙn−1. We identify all metric blow-downs and their Ricci currents, compute the exact asymptotic volume ratio, and classify isometries between the model cones. In particular, the Kähler condition does not ensure uniqueness at infinity under nonnegative Ricci curvature and Euclidean volume growth, even in complex dimension two.

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Cite

BibTeXRIS
BibTeX
@misc{dai2026complete,
  title = {{A complete Kähler metric on $\mathbbC^n$ of nonnegative Ricci curvature with Euclidean volume growth and nonunique asymptotic cone}},
  author = {Jiwei Dai and Wancheng Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32734},
  eprint = {2610.32734},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32734},
}
RIS
TY  - UNPB
TI  - A complete Kähler metric on ℂ^n of nonnegative Ricci curvature with Euclidean volume growth and nonunique asymptotic cone
AU  - Jiwei Dai
AU  - Wancheng Zhang
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.32734
UR  - https://arxiv.org/abs/2610.32734
AB  - We construct complete smooth Kähler metrics on ℂ^n , n≥2, with nonnegative Ricci curvature, Euclidean volume growth, and nonunique tangent cones at infinity. More generally, we prescribe the entire tangent cone set within a class of smooth Hopf cones, using a compact connected family of positive transverse Ricci forms on ℂℙ^n−1. We identify all metric blow-downs and their Ricci currents, compute the exact asymptotic volume ratio, and classify isometries between the model cones. In particular, the Kähler condition does not ensure uniqueness at infinity under nonnegative Ricci curvature and Euclidean volume growth, even in complex dimension two.
ER  -