Singular metrics with nonnegative scalar curvature and RCD

Xianzhe Dai; Changliang Wang; Lihe Wang; Guofang Wei

doi:10.48550/arxiv.2412.09185

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Singular metrics with nonnegative scalar curvature and RCD

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Singular metrics with nonnegative scalar curvature and RCD

Xianzhe Dai, Changliang Wang, Lihe Wang and Guofang Wei

ArXiV.org

Cornell University

12/12/2024

DOI: 10.48550/arxiv.2412.09185

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https://doi.org/10.48550/arxiv.2412.09185View

Preprint (Author's original)This preprint has not been evaluated by subject experts through peer review. Preprints may undergo extensive changes and/or become peer-reviewed journal articles. Open Access

Abstract

We show that a uniformly Euclidean metric with isolated singularity on $M^n = T^n \# M_0$, $n=6, 7$ or $n\geq 6$, $M_0$ spin and nonnegative scalar curvature on the smooth part is Ricci flat and extends smoothly over the singularity. This confirms Schoen's Conjecture in these cases. The key to the proof is to show that the space has nonnegative synthetic Ricci curvature, i.e., an $RCD(0, n)$ space. Our result also holds when the singular set consists of a finite union of submanifolds (of possibly different dimensions) intersecting transversally under additional assumption on the co-dimension and the location of the singular set.

Mathematics - Differential Geometry

Details

Title: Subtitle: Singular metrics with nonnegative scalar curvature and RCD
Creators: Xianzhe Dai
Changliang Wang
Lihe Wang
Guofang Wei
Resource Type: Preprint
Publication Details: ArXiV.org
DOI: 10.48550/arxiv.2412.09185
ISSN: 2331-8422
Publisher: Cornell University; Ithaca, New York
Language: English
Date posted: 12/12/2024
Academic Unit: Mathematics
Record Identifier: 9984757743802771

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