Seiberg-Witten Equations and Einstein Metrics on Finite Volume 4-Manifolds with Asymptotically Hyperbolic Ends

2024/02/15 by Alex Xu, Xu, Alex
Mathematics · #53C21 #57K41 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2402.10366

openalex publication_date 2024/02/15 · openalex created_date 2024/02/20 · openalex updated_date 2026/07/28

Abstract

We construct infinitely many examples of finite volume 4-manifolds with T³ ends that do not admit any cusped asymptotically hyperbolic Einstein metrics yet satisfy a strict logarithmic version of the Hitchin-Thorpe inequality due to Dai-Wei. This is done by using estimates from Seiberg-Witten theory due to LeBrun as well as a method for constructing solutions to the Seiberg-Witten equations on noncompact manifolds due to Biquard. We also use constructions coming from the Pin^-(2) monopole equations to obtain a larger class of manifolds where these techniques apply.

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