Extremal Kähler–Einstein Metric for Two-Dimensional Convex Bodies

Bo’az Klartag, Alexander V. Kolesnikov*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Pages (from-to)2347-2373
Number of pages27
JournalJournal of Geometric Analysis
Issue number3
StatePublished - 15 Jul 2019


  • Kähler–Einstein equation
  • Monge–Ampère equation
  • Ricci tensors


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