Extremal Kähler–Einstein Metric for Two-Dimensional Convex Bodies

Bo’az Klartag; Alexander V. Kolesnikov

doi:10.1007/s12220-018-0077-4

Extremal Kähler–Einstein Metric for Two-Dimensional Convex Bodies

Bo’az Klartag, Alexander V. Kolesnikov^*

^*Corresponding author for this work

School of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

Abstract

Given a convex body K⊂ Rⁿ with the barycenter at the origin, we consider the corresponding Kähler–Einstein equation e^{- Φ}= det D²Φ. If K is a simplex, then the Ricci tensor of the Hessian metric D²Φ is constant and equals n-14(n+1). We conjecture that the Ricci tensor of D²Φ for an arbitrary convex body K⊆ Rⁿ is uniformly bounded from above by n-14(n+1) and we verify this conjecture in the two-dimensional case. The general case remains open.

Original language	English
Pages (from-to)	2347-2373
Number of pages	27
Journal	Journal of Geometric Analysis
Volume	29
Issue number	3
DOIs	https://doi.org/10.1007/s12220-018-0077-4
State	Published - 15 Jul 2019

Keywords

Kähler–Einstein equation
Monge–Ampère equation
Ricci tensors

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10.1007/s12220-018-0077-4

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abstract = "Given a convex body K⊂ Rn with the barycenter at the origin, we consider the corresponding K{\"a}hler–Einstein equation e- Φ= det D2Φ. If K is a simplex, then the Ricci tensor of the Hessian metric D2Φ is constant and equals n-14(n+1). We conjecture that the Ricci tensor of D2Φ for an arbitrary convex body K⊆ Rn is uniformly bounded from above by n-14(n+1) and we verify this conjecture in the two-dimensional case. The general case remains open.",

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AU - Kolesnikov, Alexander V.

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N2 - Given a convex body K⊂ Rn with the barycenter at the origin, we consider the corresponding Kähler–Einstein equation e- Φ= det D2Φ. If K is a simplex, then the Ricci tensor of the Hessian metric D2Φ is constant and equals n-14(n+1). We conjecture that the Ricci tensor of D2Φ for an arbitrary convex body K⊆ Rn is uniformly bounded from above by n-14(n+1) and we verify this conjecture in the two-dimensional case. The general case remains open.

AB - Given a convex body K⊂ Rn with the barycenter at the origin, we consider the corresponding Kähler–Einstein equation e- Φ= det D2Φ. If K is a simplex, then the Ricci tensor of the Hessian metric D2Φ is constant and equals n-14(n+1). We conjecture that the Ricci tensor of D2Φ for an arbitrary convex body K⊆ Rn is uniformly bounded from above by n-14(n+1) and we verify this conjecture in the two-dimensional case. The general case remains open.

KW - Kähler–Einstein equation

KW - Monge–Ampère equation

KW - Ricci tensors

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ER -

Extremal Kähler–Einstein Metric for Two-Dimensional Convex Bodies

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