# 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…

## 15 Citations

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### Harmonic Functions and the Mass of 3-Dimensional Asymptotically Flat Riemannian Manifolds

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An explicit lower bound for the mass of an asymptotically flat Riemannian 3-manifold is given in terms of linear growth harmonic functions and scalar curvature. As a consequence, a new proof of the…

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We study the mass of asymptotically flat $3$-manifolds with boundary using the method of Bray-Kazaras-Khuri-Stern. More precisely, we derive a mass formula on the union of an asymptotically flat…

### Spacetime Harmonic Functions and the Mass of 3-Dimensional Asymptotically Flat Initial Data for the Einstein Equations

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### A sharp systolic inequality for $3$-manifolds with boundary

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We prove two sharp systolic inequalities for compact $3$-manifolds with boundary. They relate the $1$-systole and $2$-systole of the manifold to its scalar curvature and mean curvature of the…

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## References

SHOWING 1-10 OF 16 REFERENCES

### Scalar curvature and harmonic maps to $S^1$

- Mathematics
- 2019

For a harmonic map $u:M^3\to S^1$ on a closed, oriented $3$--manifold, we establish the identity $$2\pi \int_{\theta\in S^1}\chi(\Sigma_{\theta})\geq \frac{1}{2}\int_{\theta\in…

### On the structure of manifolds with positive scalar curvature

- Mathematics
- 1979

Publisher Summary This chapter discusses some recent results by Richard Schoen and Shing-Tung Yau on the structure of manifolds with positive scalar curvature. The chapter presents theorems which are…

### Scalar curvature and the Thurston norm

- Mathematics
- 1997

the supremum being taken over all connected, oriented surfaces Σ embedded in Y whose genus g is at least 2 [8]. If Y contains spheres or non-separating tori, the definition is extended by declaring…

### SCALAR CURVATURE AND THE THURSTON

- Mathematics
- 2004

the supremum being taken over all connected, oriented surfaces Σ embedded in Y whose genus g is at least 2 [8]. If Y contains spheres or non-separating tori, the definition is extended by declaring…

### Scalar curvature and hammocks

- Mathematics
- 1999

Scalar curvature is the simplest generalization of Gaussian curvature to higher dimensions. However there are many questions open with regard to its relation to other geometric quantities and…

### ON HARMONIC SPINORS

- Mathematics
- 1998

We study the question to what extend classical Hodge–deRham theoryfor harmonic diﬀerential forms carries over to harmonic spinors. Despitesome special phenomena in very low dimensions and despite the…

### On the proof of the positive mass conjecture in general relativity

- Mathematics
- 1979

LetM be a space-time whose local mass density is non-negative everywhere. Then we prove that the total mass ofM as viewed from spatial infinity (the ADM mass) must be positive unlessM is the flat…

### Positive scalar curvature and the Dirac operator on complete riemannian manifolds

- Mathematics
- 1983

Operateurs de Dirac generalises sur une variete complete. Theoremes d'annulation. Estimations pour la dimension du noyau et du conoyau. Le theoreme d'indice relatif. Varietes hyperspheriques et…

### A norm for the homology of 3-manifolds

- Mathematics
- 1986

On construit une norme naturelle aisement calculable sur l'homologie des 3-varietes. Cette norme est une extension de la notion de genre d'un nœud

### A new proof of the positive energy theorem

- Mathematics
- 1981

A new proof is given of the positive energy theorem of classical general relativity. Also, a new proof is given that there are no asymptotically Euclidean gravitational instantons. (These theorems…