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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.14667 |
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Table of Contents:
- Let $γ$ be a Riemannian metric on $Σ= S^1 \times T^{n-2}$, where $3 \leq n \leq 7$. Consider $Ω= B^2 \times T^{n-2}$ with boundary $\partial Ω= Σ$, and let $g$ be a Riemannian metric on $Ω$ such that the scalar curvature $R_g \geq -n(n - 1)$ and $g|_{\partial Ω} = γ$. Assuming the mean curvature of $\partial Ω$ with respect to the outward normal is positive, we establish that the total mean curvature of $\partial Ω$ is bounded from above by a constant depending only on $n$ and $γ$. Furthermore, we compute the sharp constant for this estimate when $γ$ is a flat metric. This result resolves a special case of a conjecture by Gromov concerning total mean curvature of fill-in with scalar curvature bounded from below. The proof combines techniques developed by Shi-Tam, Shi-Wang-Wei, as well as recent work by Brendle-Hung on the systolic inequality.