Regularity and compactness of harmonic-Einstein equations
type: Item
instance of: scholarly article
title: Regularity and compactness of harmonic-Einstein equations
zbMATH Open document ID: 1390.53029
author: Huabin Ge
author: Wen Shuai Jiang
published in: Mathematische Annalen
publication date: 2018-04-06T00:00:00Z
review text: The authors consider a pointed Riemannian manifold \((M,x,g)\) with \(\mathrm{Vol}(B_1(x))\geq v>0\) that satisfies the harmonic-Einstein equation \(\operatorname{Ric}g-\nabla u\otimes \nabla u=\lambda g\) with \(|\lambda|\leq n-1\), where \(u:(M,g)\to (N,h)\) is a harmonic map to a fixed compact Riemannian manifold \((N,h)\). For any \(p<1\), they prove the \(L^ p\) curvature estimate \(\int_{B_1 (x)} |Rm|^p \leq C(p,N,v)\). As a consequence, if \((N,h)\) has nonpositive sectional curvature, then \(|\operatorname{Ric} |\leq C(n,v,N)\). This means that the harmonic-Einstein equation automatically implies bounded Ricci curvature provided nonpositivity of the sectional curvature of \((N,h)\).
reviewed by: Andreas Arvanitoyeorgos
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53C43
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 6856701
zbMATH Keywords: Einstein equation
zbMATH Keywords: harmonic Einstein equation
MaRDI profile type: Publication
full work available at URL: s00208-017-1523-5
OpenAlex ID: W2591440976
Wikidata QID: Q124841683
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DOI: 10.1007/S00208-017-1523-5
description: scientific article; zbMATH DE number 6856701