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arXiv · 2610.32734

A complete Kähler metric on $\mathbb{C}^n$ 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 $\mathbb{C}^n$ , $n\ge2$, 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 $\mathbb{C}\mathbb{P}^{n-1}$. 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.

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Jiwei Dai, Wancheng Zhang. 2026-10-23. A complete Kähler metric on $\mathbb{C}^n$ of nonnegative Ricci curvature with Euclidean volume growth and nonunique asymptotic cone. https://arxiv.org/abs/2610.32734

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