Kiselman Minimum Principle and Rooftop Envelopes in Complex Hessian Equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914817522008064 |
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| author | Åhag, Per Czyż, Rafał Lu, Chinh H. Rashkovskii, Alexander |
| author_facet | Åhag, Per Czyż, Rafał Lu, Chinh H. Rashkovskii, Alexander |
| contents | We initiate the study of $m$-subharmonic functions with respect to a semipositive $(1,1)$-form in Euclidean domains, providing a significant element in understanding geodesics within the context of complex Hessian equations. Based on the foundational Perron envelope construction, we prove a decomposition of $m$-subharmonic solutions, and a general comparison principle that effectively manages singular Hessian measures. Additionally, we establish a rooftop equality and an analogue of the Kiselman minimum principle, which are crucial ingredients in establishing a criterion for geodesic connectivity among $m$-subharmonic functions, expressed in terms of their asymptotic envelopes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_04948 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Kiselman Minimum Principle and Rooftop Envelopes in Complex Hessian Equations Åhag, Per Czyż, Rafał Lu, Chinh H. Rashkovskii, Alexander Complex Variables Analysis of PDEs Differential Geometry Primary. 32U05, 32U35, 32W20, Secondary. 32F17, 53C22 We initiate the study of $m$-subharmonic functions with respect to a semipositive $(1,1)$-form in Euclidean domains, providing a significant element in understanding geodesics within the context of complex Hessian equations. Based on the foundational Perron envelope construction, we prove a decomposition of $m$-subharmonic solutions, and a general comparison principle that effectively manages singular Hessian measures. Additionally, we establish a rooftop equality and an analogue of the Kiselman minimum principle, which are crucial ingredients in establishing a criterion for geodesic connectivity among $m$-subharmonic functions, expressed in terms of their asymptotic envelopes. |
| title | Kiselman Minimum Principle and Rooftop Envelopes in Complex Hessian Equations |
| topic | Complex Variables Analysis of PDEs Differential Geometry Primary. 32U05, 32U35, 32W20, Secondary. 32F17, 53C22 |
| url | https://arxiv.org/abs/2405.04948 |