Mass and volume of four-dimensional Einstein metrics
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917132451708928 |
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| author | Gursky, Matthew Malchiodi, Andrea |
| author_facet | Gursky, Matthew Malchiodi, Andrea |
| contents | Let $(M^4,\bar{g})$ be an Einstein manifold, where $M^4$ is a smooth, closed, oriented four-manifold $M^4$ and $\bar{g}$ has positive Einstein constant. Given a point $0 \in M^4$, let $G$ denote the (positive) Green's function $G$ of the conformal laplacian $L_{\bar{g}}$; then $g = G^2 \bar{g}$ is a complete, scalar-flat, asymptotically flat metric on $\widehat{M} = M \setminus \{ 0 \}$. We first show that the ADM mass of $g$ can be expressed as an integral over $\widehat{M}$, then use this identity to prove a lower bound for the mass of $g$ in terms of the volume of $\bar{g}$. As corollaries, we prove a 'mass times volume' inequality, plus various mass gap theorems characterizing the round metric on $S^4$ and the Fubini-Study metric on $\mathbb{CP}^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_07257 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mass and volume of four-dimensional Einstein metrics Gursky, Matthew Malchiodi, Andrea Differential Geometry 53C24, 53C25 Let $(M^4,\bar{g})$ be an Einstein manifold, where $M^4$ is a smooth, closed, oriented four-manifold $M^4$ and $\bar{g}$ has positive Einstein constant. Given a point $0 \in M^4$, let $G$ denote the (positive) Green's function $G$ of the conformal laplacian $L_{\bar{g}}$; then $g = G^2 \bar{g}$ is a complete, scalar-flat, asymptotically flat metric on $\widehat{M} = M \setminus \{ 0 \}$. We first show that the ADM mass of $g$ can be expressed as an integral over $\widehat{M}$, then use this identity to prove a lower bound for the mass of $g$ in terms of the volume of $\bar{g}$. As corollaries, we prove a 'mass times volume' inequality, plus various mass gap theorems characterizing the round metric on $S^4$ and the Fubini-Study metric on $\mathbb{CP}^2$. |
| title | Mass and volume of four-dimensional Einstein metrics |
| topic | Differential Geometry 53C24, 53C25 |
| url | https://arxiv.org/abs/2512.07257 |