Journal article
Singular Metrics with Nonnegative Scalar Curvature and RCD
Communications in contemporary mathematics
02/20/2026
DOI: 10.1142/S0219199726500343
Abstract
We show that a uniformly Euclidean metric with isolated singularity on closed [Formula: see text], where [Formula: see text] or [Formula: see text], [Formula: see text] spin, and nonnegative scalar curvature on the smooth part is flat and extends smoothly over the singularity. This confirms Schoen’s Conjecture in these cases. The novel approach here, which is the key to the proof, is to show that the space has nonnegative synthetic Ricci curvature, i.e., an [Formula: see text] 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.
Details
- Title: Subtitle
- Singular Metrics with Nonnegative Scalar Curvature and RCD
- Creators
- Xianzhe Dai - University of California, Santa BarbaraChangliang Wang - Tongji UniversityLihe Wang - University of IowaGuofang Wei - University of California, Santa Barbara
- Resource Type
- Journal article
- Publication Details
- Communications in contemporary mathematics
- DOI
- 10.1142/S0219199726500343
- ISSN
- 0219-1997
- eISSN
- 1793-6683
- Publisher
- World Scientific
- Grant note
- National Science Foundation: DMS-1928930 Simons FoundationNSFS: 25ZR1401357 Fundamental Research Funds for the Central UniversitiesNSF DMS grant: 2403557
This material is based upon work supported by the National Science Foundation under Grant No. DMS-1928930, while XD and GW were in residence at the Simons Laufer Mathematical Sciences Institute (formerly MSRI) in Berkeley, California, during the Fall semester of 2024. XD and GW thank SLMath for providing a stimulating environment and acknowledge useful discussions with Rick Schoen, Vincent Guedj, Xinyu Zhu, Conghan Dong. XD and CW would like to thank Jian Wang and Simone Cecchini for valuable conversations. Finally we would like to thank Gabor Szekelyhidi for very fruitful communication. XD was partially supported by Simons Foundation. CW was partially supported by NSFS (25ZR1401357) and the Fundamental Research Funds for the Central Universities. GW was partially supported by NSF DMS grant 2403557.
- Language
- English
- Electronic publication date
- 02/20/2026
- Academic Unit
- Mathematics
- Record Identifier
- 9985141900802771
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