Blow-up sets of Ricci curvatures of complete conformal metrics
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Abstract
A version of the singular Yamabe problem in smooth domains in a closed manifold yields complete conformal metrics with negative constant scalar curvatures. In this paper, we study the blow-up phenomena of Ricci curvatures of these metrics on domains whose boundary is close to a certain limit set of a lower dimension. We will characterize the blow-up set according to the Yamabe invariant of the underlying manifold. In particular, we will prove that all points in the lower dimension part of the limit set belong to the blow-up set on manifolds not conformally equivalent to the standard sphere and that all but one point in the lower dimension part of the limit set belong to the blow-up set on manifolds conformally equivalent to the standard sphere. In certain cases, the blow-up set can be the entire manifold. We will demonstrate by examples that these results are optimal.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-2305038
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 12371208
Award Identifier / Grant number: 11901405
Award Identifier / Grant number: 12371236
Award Identifier / Grant number: 12001383
Funding statement: The first author acknowledges the support of NSF Grant DMS-2305038. The second author acknowledges the support of NSFC Grant 12371208 and NSFC Grant 11901405. The third author acknowledges the support of NSFC Grant 12371236 and NSFC Grant 12001383.
Acknowledgements
We would like to thank Matthew Gursky, Marcus Khuri, and Yuguang Shi for helpful discussions.
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Artikel in diesem Heft
- Frontmatter
- The closure ordering conjecture on local Arthur packets of classical groups
- Massey products in the étale cohomology of number fields
- Regular immersions directed by algebraically elliptic cones
- K-stable valuations and Calabi–Yau metrics on affine spherical varieties
- On Fontaine’s conjecture for torsion crystalline local systems
- Wavefront sets of unipotent representations of reductive 𝑝-adic groups II
- On canonical metrics of complex surfaces with split tangent and related geometric PDEs
- Blow-up sets of Ricci curvatures of complete conformal metrics
Artikel in diesem Heft
- Frontmatter
- The closure ordering conjecture on local Arthur packets of classical groups
- Massey products in the étale cohomology of number fields
- Regular immersions directed by algebraically elliptic cones
- K-stable valuations and Calabi–Yau metrics on affine spherical varieties
- On Fontaine’s conjecture for torsion crystalline local systems
- Wavefront sets of unipotent representations of reductive 𝑝-adic groups II
- On canonical metrics of complex surfaces with split tangent and related geometric PDEs
- Blow-up sets of Ricci curvatures of complete conformal metrics
