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We study an inverse problem of determining a time-dependent potential appearing in the wave equation on conformally transversally anisotropic manifolds of dimension three or higher. These are compact Riemannian manifolds with boundary that are conformally embedded in a product of the real line and a transversal manifold. Under the assumption of the attenuated geodesic ray transform being injective on the transversal manifold, we prove the unique determination of time-dependent potentials from the knowledge of a certain partial Cauchy data set.
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This content will become publicly available on March 28, 2026
Blow-up sets of Ricci curvatures of complete conformal metrics
Abstract A version of the singular Yamabe problem in smooth domains in a closed manifoldyields 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.
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- Award ID(s):
- 2305038
- PAR ID:
- 10628420
- Publisher / Repository:
- Walter de Gruyter
- Date Published:
- Journal Name:
- Journal für die reine und angewandte Mathematik (Crelles Journal)
- ISSN:
- 0075-4102
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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