Extremal Kähler–Einstein metric for two-dimensional convex bodies

Bo'az Klartag; Alexander V. Kolesnikov

doi:https://doi.org/10.1007/s12220-018-0077-4

Extremal Kähler–Einstein metric for two-dimensional convex bodies

Bo'az Klartag, Alexander V. Kolesnikov

Weizmann Institute of Science, Tel Aviv University

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=detD2Φ. 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
Early online date	28 Aug 2018
DOIs	https://doi.org/10.1007/s12220-018-0077-4
State	Published - 15 Jul 2019

Keywords

Kähler–Einstein equation
Monge–Ampère equation
Ricci tensors

All Science Journal Classification (ASJC) codes

Geometry and Topology

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https://doi.org/10.1007/s12220-018-0077-4

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title = "Extremal K{\"a}hler–Einstein metric for two-dimensional convex bodies",

abstract = "Given a convex body K⊂Rn with the barycenter at the origin, we consider the corresponding K{\"a}hler–Einstein equation e−Φ=detD2Φ. 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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T1 - Extremal Kähler–Einstein metric for two-dimensional convex bodies

AU - Klartag, Bo'az

AU - Kolesnikov, Alexander V.

PY - 2019/7/15

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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=detD2Φ. 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=detD2Φ. 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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DO - https://doi.org/10.1007/s12220-018-0077-4

M3 - مقالة

SN - 1050-6926

VL - 29

SP - 2347

EP - 2373

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

IS - 3

ER -

Extremal Kähler–Einstein metric for two-dimensional convex bodies

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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