New York Journal of Mathematics
Volume 28 (2022), 659-671


H. Contreras Peruyero and P. Suárez-Serrato

Collapsing and group growth as obstructions to Einstein metrics on some smooth 4-manifolds

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Published: March 30, 2022.
Keywords: Einstein metrics, 4-manifolds, graph manifolds, Thurston geometries.
Subject: Primary 53C25, Secondary 53C23, 57R57.

We show that a combination of collapsing and excessive growth from the fundamental group impedes the existence of Einstein metrics on several families of smooth 4-manifolds. These include infrasolvmanifolds whose fundamental group is not virtually nilpotent, most elliptic surfaces of zero Euler characteristic, geometrizable manifolds with hyperbolic factor geometries in their geometric decomposition, and higher graph 4-manifolds without purely negatively curved pieces.


PSS thankfully acknowledges partial support by DGAPA-UNAM PAPIIT grant IN104819.

Author information

H. Contreras Peruyero:
Centro de Ciencias Matemáticas
Universidad Nacional Autónoma de México UNAM
Antigua Carretera a Pátzcuaro # 8701
Col. Ex Hacienda San José de la Huerta, 58089, Morelia, Michoacán, Mexico


P. Suárez-Serrato:
Instituto de Matemáticas
Universidad Nacional Autónoma de México UNAM
Area de la Investigacion Cientifica, Circuito exterior
Ciudad Universitaria, 04510, Coyoacán, Mexico City, Mexico