Sharp uniform bound for the quaternionic Monge-Ampere equation on hyperhermitian manifolds
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866913331722321920 |
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| author | Sroka, Marcin |
| author_facet | Sroka, Marcin |
| contents | We provide the sharp $C^0$ estimate for the quaternionic Monge-Ampere equation on any hyperhermitian manifold. This improves previously known results concerning this estimate in two directions. Namely, it turns out that the estimate depends only on $L^p$ norm of the right hand side for any $p>2$ (as suggested by the local case studied in [Sr20a]). Moreover, the estimate still holds true for any hyperhermitian initial metric - regardless of it being HKT as in the original conjecture of Alesker-Verbitsky [AV10] - as speculated by the author in [Sr21]. For completeness, we actually provide a sharp uniform estimate for many quaternionic PDEs, in particular those given by the operator dominating the quaternionic Monge-Ampere operator, by applying the recent method of Guo and Phong [GP22a]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_00959 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Sharp uniform bound for the quaternionic Monge-Ampere equation on hyperhermitian manifolds Sroka, Marcin Analysis of PDEs Complex Variables Differential Geometry We provide the sharp $C^0$ estimate for the quaternionic Monge-Ampere equation on any hyperhermitian manifold. This improves previously known results concerning this estimate in two directions. Namely, it turns out that the estimate depends only on $L^p$ norm of the right hand side for any $p>2$ (as suggested by the local case studied in [Sr20a]). Moreover, the estimate still holds true for any hyperhermitian initial metric - regardless of it being HKT as in the original conjecture of Alesker-Verbitsky [AV10] - as speculated by the author in [Sr21]. For completeness, we actually provide a sharp uniform estimate for many quaternionic PDEs, in particular those given by the operator dominating the quaternionic Monge-Ampere operator, by applying the recent method of Guo and Phong [GP22a]. |
| title | Sharp uniform bound for the quaternionic Monge-Ampere equation on hyperhermitian manifolds |
| topic | Analysis of PDEs Complex Variables Differential Geometry |
| url | https://arxiv.org/abs/2211.00959 |