• Corpus ID: 208077114

Scalar curvature and harmonic one-forms on three-manifolds with boundary

@article{Bray2019ScalarCA,
  title={Scalar curvature and harmonic one-forms on three-manifolds with boundary},
  author={Hubert L. Bray and Daniel L. Stern},
  journal={arXiv: Differential Geometry},
  year={2019}
}
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… 

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