A new variational characterization of compact conformally flat 4-manifolds

Wu, Faen, and Zhao, Xinnuan. "A new variational characterization of compact conformally flat 4-manifolds." Communications in Mathematics 20.2 (2012): 71-77. <http://eudml.org/doc/251377>.

@article{Wu2012,
abstract = {In this paper, we give a new variational characterization of certain 4-manifolds. More precisely, let $R$ and $Ric$ denote the scalar curvature and Ricci curvature respectively of a Riemannian metric, we prove that if $(M^\{4\},g)$ is compact and locally conformally flat and $g$ is the critical point of the functional \[ F(g)=\int \_\{M^\{4\}\}(aR^\{2\}+b|Ric|^\{2\})\,\mathrm \{d\}v\_\{g\}\,,\] where \[(a,b)\in \mathbb \{R\}^\{2\}\setminus L\_\{1\}\cup L\_\{2\}\]\[L\_\{1\}\colon 3a+b=0\,;\quad L\_\{2\}\colon 6a-b+1=0\,,\] then $(M^\{4\},g)$ is either scalar flat or a space form.},
author = {Wu, Faen, Zhao, Xinnuan},
journal = {Communications in Mathematics},
keywords = {conformally flat; 4-manifold; variational characterization; conformally flat; 4-manifold; variational characterization},
language = {eng},
number = {2},
pages = {71-77},
publisher = {University of Ostrava},
title = {A new variational characterization of compact conformally flat 4-manifolds},
url = {http://eudml.org/doc/251377},
volume = {20},
year = {2012},
}

TY - JOUR
AU - Wu, Faen
AU - Zhao, Xinnuan
TI - A new variational characterization of compact conformally flat 4-manifolds
JO - Communications in Mathematics
PY - 2012
PB - University of Ostrava
VL - 20
IS - 2
SP - 71
EP - 77
AB - In this paper, we give a new variational characterization of certain 4-manifolds. More precisely, let $R$ and $Ric$ denote the scalar curvature and Ricci curvature respectively of a Riemannian metric, we prove that if $(M^{4},g)$ is compact and locally conformally flat and $g$ is the critical point of the functional \[ F(g)=\int _{M^{4}}(aR^{2}+b|Ric|^{2})\,\mathrm {d}v_{g}\,,\] where \[(a,b)\in \mathbb {R}^{2}\setminus L_{1}\cup L_{2}\]\[L_{1}\colon 3a+b=0\,;\quad L_{2}\colon 6a-b+1=0\,,\] then $(M^{4},g)$ is either scalar flat or a space form.
LA - eng
KW - conformally flat; 4-manifold; variational characterization; conformally flat; 4-manifold; variational characterization
UR - http://eudml.org/doc/251377
ER -