TY - JOUR

T1 - Extremal Kähler–Einstein Metric for Two-Dimensional Convex Bodies

AU - Klartag, Bo’az

AU - Kolesnikov, Alexander V.

N1 - Publisher Copyright:
© 2018, Mathematica Josephina, Inc.

PY - 2019/7/15

Y1 - 2019/7/15

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

UR - http://www.scopus.com/inward/record.url?scp=85052690544&partnerID=8YFLogxK

U2 - 10.1007/s12220-018-0077-4

DO - 10.1007/s12220-018-0077-4

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AN - SCOPUS:85052690544

SN - 1050-6926

VL - 29

SP - 2347

EP - 2373

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

IS - 3

ER -