2019/08/26 by Daniel Stern, Stern, Daniel · 9 citations
Mathematics · #Analytic and geometric function theory #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #math.DG #math.GT
paper · pdf · doi:10.48550/arxiv.1908.09754
v2: minor edits--corrected statements of rigidity/splitting results; comments welcome
arxiv created 2019/09/10 · arxiv updated 2019/09/11
For a harmonic map u:M3→ S1 on a closed, oriented 3--manifold, we establish the identity 2π∫θ∈ S1χ(Σθ)≥ (1)/(2)∫θ∈ S1∫Σθ(|du|-2|Hess(u)|2+RM) relating the scalar curvature RM of M to the average Euler characteristic of the level sets Σθ=u-1\θ\. As our primary application, we extend the Kronheimer--Mrowka characterization of the Thurston norm on H2(M;ℤ) in terms of ‖RM-‖L2 and the harmonic norm to any closed 3--manifold containing no nonseparating spheres. Additional corollaries include the Bray--Brendle--Neves rigidity theorem for the systolic inequality (min RM)sys2(M)≤ 8π, and the well--known result of Schoen and Yau that T3 admits no metric of positive scalar curvature.