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

Autor:	Bray, Hubert L., Stern, Daniel L.
Rok vydání:	2019
Předmět:	Mathematics - Differential Geometry Mathematics - Geometric Topology
Druh dokumentu:	Working Paper
Popis:	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. Comment: 13 pages; all comments welcome
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/1911.06803 Zobrazit plný text záznamu View this record from Arxiv