Hodge theory on ALG$^*$ manifolds
| Autor: | Chen, Gao, Viaclovsky, Jeff, Zhang, Ruobing |
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| Rok vydání: | 2021 |
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| Druh dokumentu: | Working Paper |
| Popis: | We develop a Fredholm Theory for the Hodge Laplacian in weighted spaces on ALG$^*$ manifolds in dimension four. We then give several applications of this theory. First, we show the existence of harmonic functions with prescribed asymptotics at infinity. A corollary of this is a non-existence result for ALG$^*$ manifolds with non-negative Ricci curvature having group $\Gamma = \{e\}$ at infinity. Next, we prove a Hodge decomposition for the first de Rham cohomology group of an ALG$^*$ manifold. A corollary of this is vanishing of the first betti number for any ALG$^*$ manifold with non-negative Ricci curvature. Another application of our analysis is to determine the optimal order of ALG$^*$ gravitational instantons. Comment: 35 pages; final version; to appear in J. Reine Angew. Math. (Crelle's Journal) |
| Databáze: | arXiv |
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