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Communications in Analysis and Geometry
Volume 31 (2023)
Number 5
Scalar curvature and harmonic one-forms on three-manifolds with boundary
Pages: 1259 – 1274
DOI: https://dx.doi.org/10.4310/CAG.2023.v31.n5.a6
Authors
Abstract
For a homotopically energy-minimizing map $u: N^3\to S^1$ on a compact, oriented $3$-manifold $N$ with boundary, we establish an identity relating the average Euler characteristic of the level sets $u^{-1}\{\theta\}$ to the scalar curvature of $N$ and the mean curvature of the boundary $\partial N$. As an application, we obtain some natural geometric estimates for the Thurston norm on $3$-manifolds with boundary, generalizing results of Kronheimer-Mrowka and the second named author from the closed setting. By combining these techniques with results from minimal surface theory, we obtain moreover a characterization of the Thurston norm via scalar curvature and the harmonic norm for general closed, oriented three-manifolds, extending Kronheimer and Mrowka’s characterization for irreducible manifolds to arbitrary topologies.
Received 27 February 2020
Accepted 15 April 2021
Published 16 July 2024