Singular metrics with nonnegative scalar curvature and RCD

2024/12/12 by Xianzhe Dai, Changliang Wang, Dai, Xianzhe +5 · 1 citation
Physics and Astronomy · #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #Differential Geometry (math.DG) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2412.09185

openalex publication_date 2024/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that a uniformly Euclidean metric with isolated singularity on Mⁿ = Tⁿ # M₀, where 4≤ n≤ 7 or n≥ 4, M₀ 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.

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