On static manifolds satisfying an overdetermined Robin type condition on the boundary
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- by Tiarlos Cruz and Ivaldo Nunes;
- Proc. Amer. Math. Soc. 151 (2023), 4971-4982
- DOI: https://doi.org/10.1090/proc/16497
- Published electronically: July 3, 2023
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Abstract:
In this work, we consider static manifolds $M$ with nonempty boundary $\partial M$. In this case, we suppose that the potential $V$ also satisfies an overdetermined Robin type condition on $\partial M$. We prove a rigidity theorem for the Euclidean closed unit ball $B^3$ in $\mathbb {R}^3$. More precisely, we give a sharp upper bound for the area of the zero set $\Sigma =V^{-1}(0)$ of the potential $V$, when $\Sigma$ is connected and intersects $\partial M$. We also consider the case where $\Sigma =V^{-1}(0)$ does not intersect $\partial M$.References
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Bibliographic Information
- Tiarlos Cruz
- Affiliation: Universidade Federal de Alagoas, Instituto de Matemática, Maceió, Alagoas 57072-970, Brazil
- MR Author ID: 1095786
- ORCID: 0000-0002-6682-9474
- Email: cicero.cruz@im.ufal.br
- Ivaldo Nunes
- Affiliation: Departamento de Matemática, Universidade Federal do Maranhão, São Luís, Maranhão 65080 - 805, Brazil
- MR Author ID: 1019640
- Email: ivaldo.nunes@ufma.br
- Received by editor(s): December 29, 2022
- Received by editor(s) in revised form: March 14, 2023
- Published electronically: July 3, 2023
- Additional Notes: The first author was supported in part by CNPq/Brazil grant 311803/2019-9.
The first and second authors were supported by the ICTP through the Associates Programme (2018-2023 and 2019-2024, respectively). The authors were partially supported by the Brazilian National Council for Scientific and Technological Development (CNPq Grant 405468/2021-0). - Communicated by: Jiaping Wang
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 4971-4982
- MSC (2020): Primary 53C21, 35Q75; Secondary 53C24
- DOI: https://doi.org/10.1090/proc/16497
- MathSciNet review: 4634898