Weighted Sobolev inequalities under lower Ricci curvature bounds
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We obtain sharp weighted Poincaré and Sobolev inequalities over complete, noncompact Riemannian manifolds with polynomial volume growth and a quadratically decaying lower bound on Ricci. This improves and extends earlier work of Tian-Yau and Minerbe. We deduce a sharp existence result for bounded solutions of the Poisson equation on such manifolds, highlighting the well-known distinction between spaces of volume growth ≤ 2 and > 2 in terms of their Green’s functions. We also show that if the manifold is nonparabolic and carries a smooth function which behaves like the radius function of a cone, then these solutions almost decay at the rates expected from a cone.
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If f ∈ C^{2,α}(M) satisfies |f| ≤ C r^{-μ} for some μ > 2, and if β ≤ 2 then ∫(e^f - 1)ω_0^m = 0, then there exists ᾱ ∈ (0,α] and u ∈ C^{4,ᾱ}(M) such that (ω_0 + i∂∂u)^m = e^f ω_0^m; if β ≤ 2, then ∫|∇u|^2 ω_0^m < ∞.
“Let f ∈ C 2,α (M ) satisfy |f | ≤ Cr -μ for some μ > 2. If β ≤ 2, then assume in addition that (e f -1)ω m 0 = 0. Then there exists an ᾱ ∈ (0, α] and u ∈ C 4, ᾱ(M ) such that (ω 0 + i∂ ∂u) m = e f ω m 0 .If β ≤ 2, then |∇u| 2 ω m 0 < ∞.”
From Gravitational instantons from rational elliptic surfaces (2011) · cites this paper as [23]
The Introduction (page unavailable) describes a complex Monge–Ampère analogue with the stated condition ∫(e^f−1)ω^m=0 when β≤2 and the regularity f∈C^{2,α} ⇒ u∈C^{4,ᾱ}. It makes that regularity conditional on bounded curvature and scalar-curvature derivatives, which the quoted claim omits. The paper also assumes SOB(β) throughout. (LLM, conf=0.85)
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Research works printed but not linked: 17
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Gravitational instantons from rational elliptic surfaces
Hein, H.-J. (2011) · Journal of the American Mathematical Society · doi:10.1090/s0894-0347-2011-00723-6 · Cited by 87 · see citing works
Self-citation
- Authors
- Hein, H.-J.
- Affiliations
- Imperial College London (GB)
- Corresponding author
- Hein, Hans-Joachim
- ORCID
- Hein, Hans-Joachim
- Venue
- Proceedings of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Published
- 2011-01-04
- Volume
- 139
- Issue
- 8
- Pages
- 2943-2955
- ISSN
- 0002-9939 (ISSN-L)
- 1088-6826
- OpenAlex
- W1999456950
- Cited by
- 30
- References
- 0
- Citations by year
- 2025: 1 · 2024: 4 · 2023: 2 · 2022: 2 · 2021: 3 · 2020: 4 · 2019: 4 · 2018: 3 · 2016: 1 · 2015: 1 · 2014: 1 · 2013: 2 · 2012: 1
- FWCI
- 2.50
- Type
- journal article
- Language
- en
- Open access
- hybrid
- doi.org
- Licence
- Locations
- Proceedings of the American Mathematical Society (publishedVersion, public-domain)
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Physical Sciences › Mathematics › Applied Mathematics › Nonlinear Partial Differential Equations
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Advanced Differential Geometry Research
- Keywords
- weighted Poincaré inequality, weighted Sobolev inequalities, Ricci curvature lower bound, complete noncompact Riemannian manifolds, polynomial volume growth, Poisson equation, boundedness of solutions, Green's function, sharp constants
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