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Jost Schultze
Jost Schultze
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There is a lot of important recent work on the construction of Calabi-Yau metrics on non-compact complex manifolds, such as $\mathbb{C}^n$. For example:

[1] SzékelyhidiLi, GY. Degenerations of $\mathbb{C}^n$ andA new complete Calabi-Yau metricsmetric on $\mathbb{C}^3$. Duke MathInvent. JMath. 168217 (2019), no. 141, 2651–27001–34.

[2] LiSzékelyhidi, YG. A new completeDegenerations of $\mathbb{C}^n$ and Calabi-Yau metric on $\mathbb{C}^3$metrics. Invent.Duke Math. 217J. 168 (2019), no. 114, 1–342651–2700.

Being a complete outsider, I have a very hard time understanding what exactly they mean by a Calabi-Yau metric on a non-compact manifoldsmanifold such as $\mathbb{C}^n$ (they don't seem to define it in their papers). For compact manifolds, one usually definedefines a Calabi-Yau metric to be a metric with holonomy in $\mathrm{SU}(n)$ (see, e.g., Joyce's book Riemannian holonomy groups and calibrated geometries on page 54). For simply connected manifolds, this is equivalent to a Ricci-flat Kähler metric. So is a Calabi-Yau metric on $\mathbb{C}^n$ the same as a Ricci-flat Kähler metric (probably not)?

What is the definition of a Calabi-Yau metric on $\mathbb{C}^n$? Are there several equivalent definitions?

There is a lot of important recent work on the construction of Calabi-Yau metrics on non-compact complex manifolds, such as $\mathbb{C}^n$. For example:

[1] Székelyhidi, G. Degenerations of $\mathbb{C}^n$ and Calabi-Yau metrics. Duke Math. J. 168 (2019), no. 14, 2651–2700.

[2] Li, Y. A new complete Calabi-Yau metric on $\mathbb{C}^3$. Invent. Math. 217 (2019), no. 1, 1–34.

Being a complete outsider, I have a very hard time understanding what exactly they mean by a Calabi-Yau metric on non-compact manifolds such as $\mathbb{C}^n$ (they don't seem to define it in their papers). For compact manifolds, one usually define a Calabi-Yau metric to be a metric with holonomy in $\mathrm{SU}(n)$ (see, e.g., Joyce's book Riemannian holonomy groups and calibrated geometries on page 54). For simply connected manifolds, this is equivalent to a Ricci-flat Kähler metric. So is a Calabi-Yau metric on $\mathbb{C}^n$ the same as a Ricci-flat Kähler metric (probably not)?

What is the definition of a Calabi-Yau metric on $\mathbb{C}^n$? Are there several equivalent definitions?

There is a lot of important recent work on the construction of Calabi-Yau metrics on non-compact complex manifolds, such as $\mathbb{C}^n$. For example:

[1] Li, Y. A new complete Calabi-Yau metric on $\mathbb{C}^3$. Invent. Math. 217 (2019), no. 1, 1–34.

[2] Székelyhidi, G. Degenerations of $\mathbb{C}^n$ and Calabi-Yau metrics. Duke Math. J. 168 (2019), no. 14, 2651–2700.

Being a complete outsider, I have a very hard time understanding what exactly they mean by a Calabi-Yau metric on a non-compact manifold such as $\mathbb{C}^n$ (they don't seem to define it in their papers). For compact manifolds, one usually defines a Calabi-Yau metric to be a metric with holonomy in $\mathrm{SU}(n)$ (see, e.g., Joyce's book Riemannian holonomy groups and calibrated geometries on page 54). For simply connected manifolds, this is equivalent to a Ricci-flat Kähler metric. So is a Calabi-Yau metric on $\mathbb{C}^n$ the same as a Ricci-flat Kähler metric?

What is the definition of a Calabi-Yau metric on $\mathbb{C}^n$? Are there several equivalent definitions?

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Source Link
Jost Schultze
Jost Schultze
  • 473
  • 2
  • 7

What is the definition of a Calabi-Yau metric on a non-compact manifold?

There is a lot of important recent work on the construction of Calabi-Yau metrics on non-compact complex manifolds, such as $\mathbb{C}^n$. For example:

[1] Székelyhidi, G. Degenerations of $\mathbb{C}^n$ and Calabi-Yau metrics. Duke Math. J. 168 (2019), no. 14, 2651–2700.

[2] Li, Y. A new complete Calabi-Yau metric on $\mathbb{C}^3$. Invent. Math. 217 (2019), no. 1, 1–34.

Being a complete outsider, I have a very hard time understanding what exactly they mean by a Calabi-Yau metric on non-compact manifolds such as $\mathbb{C}^n$ (they don't seem to define it in their papers). For compact manifolds, one usually define a Calabi-Yau metric to be a metric with holonomy in $\mathrm{SU}(n)$ (see, e.g., Joyce's book Riemannian holonomy groups and calibrated geometries on page 54). For simply connected manifolds, this is equivalent to a Ricci-flat Kähler metric. So is a Calabi-Yau metric on $\mathbb{C}^n$ the same as a Ricci-flat Kähler metric (probably not)?

What is the definition of a Calabi-Yau metric on $\mathbb{C}^n$? Are there several equivalent definitions?