A low-regularity Riemannian positive mass theorem for non-spin manifolds with distributional curvature
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866910010158612480 |
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| author | Hafemann, Eduardo |
| author_facet | Hafemann, Eduardo |
| contents | This article establishes a low-regularity Riemannian positive mass theorem for non-spin manifolds whose metrics are only $C^0 \cap W_{\mathrm{loc}}^{1,n}$ and smooth outside a compact set. The main theorem asserts that asymptotically flat manifolds with nonnegative distributional scalar curvature have nonnegative ADM mass. The proof uses smooth approximations of the metric together with a Sobolev version of Friedrichs' Lemma, which yields improved convergence for commutators between differentiation and convolution operators. Rigidity is obtained for $C^0 \cap W_{\mathrm{loc}}^{1,p}$ metrics with $p>n$ via the comparison theory of $\sf{RCD}$-spaces and a rigidity theorem for compact manifolds with metrics of nonnegative distributional curvature by Jiang-Sheng-Zhang. The argument relies on either elementary techniques or generalisations of the standard argument. In essence, a version of the main theorem of Lee-LeFloch is presented in which the spin condition is removed under the assumption that the metric is smooth outside a compact set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03451 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A low-regularity Riemannian positive mass theorem for non-spin manifolds with distributional curvature Hafemann, Eduardo Differential Geometry General Relativity and Quantum Cosmology This article establishes a low-regularity Riemannian positive mass theorem for non-spin manifolds whose metrics are only $C^0 \cap W_{\mathrm{loc}}^{1,n}$ and smooth outside a compact set. The main theorem asserts that asymptotically flat manifolds with nonnegative distributional scalar curvature have nonnegative ADM mass. The proof uses smooth approximations of the metric together with a Sobolev version of Friedrichs' Lemma, which yields improved convergence for commutators between differentiation and convolution operators. Rigidity is obtained for $C^0 \cap W_{\mathrm{loc}}^{1,p}$ metrics with $p>n$ via the comparison theory of $\sf{RCD}$-spaces and a rigidity theorem for compact manifolds with metrics of nonnegative distributional curvature by Jiang-Sheng-Zhang. The argument relies on either elementary techniques or generalisations of the standard argument. In essence, a version of the main theorem of Lee-LeFloch is presented in which the spin condition is removed under the assumption that the metric is smooth outside a compact set. |
| title | A low-regularity Riemannian positive mass theorem for non-spin manifolds with distributional curvature |
| topic | Differential Geometry General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2602.03451 |