Sharp Sobolev inequalities on noncompact Riemannian manifolds with bounded Ricci curvature
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917254543704064 |
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| author | Morpurgo, Carlo Nardulli, Stefano Qin, Liuyu |
| author_facet | Morpurgo, Carlo Nardulli, Stefano Qin, Liuyu |
| contents | Given a smooth, complete Riemannian manifold $M$ with bounded Ricci curvature and positive injectivity radius, we derive a sharp Sobolev inequality for the embedding of $W^{1,p}(M)$ into $L^{\frac{np}{n-p}}(M)$, when $1\le p< n$. We will first reduce the inequality to functions having support with small enough volume. In turn, we will show that the inequality for small volumes is implied by a first order uniform asymptotic expansion of the isoperimetric profile for $M$, for small volumes. We will then show that such an expansion follows from a local, uniform Sobolev inequality for functions in $W^{1,1}$, having support with small enough diameter. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06648 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp Sobolev inequalities on noncompact Riemannian manifolds with bounded Ricci curvature Morpurgo, Carlo Nardulli, Stefano Qin, Liuyu Analysis of PDEs Differential Geometry 46E35, 58J60 Given a smooth, complete Riemannian manifold $M$ with bounded Ricci curvature and positive injectivity radius, we derive a sharp Sobolev inequality for the embedding of $W^{1,p}(M)$ into $L^{\frac{np}{n-p}}(M)$, when $1\le p< n$. We will first reduce the inequality to functions having support with small enough volume. In turn, we will show that the inequality for small volumes is implied by a first order uniform asymptotic expansion of the isoperimetric profile for $M$, for small volumes. We will then show that such an expansion follows from a local, uniform Sobolev inequality for functions in $W^{1,1}$, having support with small enough diameter. |
| title | Sharp Sobolev inequalities on noncompact Riemannian manifolds with bounded Ricci curvature |
| topic | Analysis of PDEs Differential Geometry 46E35, 58J60 |
| url | https://arxiv.org/abs/2602.06648 |