$L^\infty$-metrics on tori and Schoen's conjecture

Jian Wang; Jinmin Wang; Zhizhang Xie

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arXiv · 2606.21325

$L^\infty$-metrics on tori and Schoen's conjecture

Jian Wang·Jinmin Wang·Zhizhang Xie

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Abstract

We prove Schoen's conjecture on $L^\infty$-metrics for tori. More precisely, we show that any $L^\infty$-metric on a torus that is smooth and has non-negative scalar curvature away from an embedded submanifold with codimension at least three extends to a smooth flat metric. We also prove a LLarull-type theorem for $L^\infty$-metric. Our proof uses weighted scalar curvature and the relative index theorem.

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Jian Wang, Jinmin Wang, Zhizhang Xie. 2026-06-19. $L^\infty$-metrics on tori and Schoen's conjecture. https://arxiv.org/abs/2606.21325

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