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arXiv · 2509.07608

Level sets of harmonic functions on three-dimensional manifolds with nonnegative scalar curvature

Yukai Sun
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Abstract

We investigate the level sets of harmonic functions on $(\mathbb{R}^{3}\setminus \{0\},g)$. Drawing inspiration from Miao, we adopt the method developed by Munteanu-Wang to derive a monotonic quantity associated with the level sets of harmonic functions on $(\mathbb{R}^{3}\setminus \{0\},g)$ with nonnegative scalar curvature, under certain conditions. Furthermore, we establish a rigidity result for this quantity. Additionally, we find an extra scalar-flat metric on $\mathbb{R}^{3}\setminus \{0\}$.

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Yukai Sun. 2025-09-09. Level sets of harmonic functions on three-dimensional manifolds with nonnegative scalar curvature. https://arxiv.org/abs/2509.07608

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