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strictly convex boundary

Proceedings of the Edinburgh Mathematical Society (1)

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Rigidity of capillary surfaces in compact 3-manifolds with strictly convex boundary
Part of
- Classical differential geometry
- Global differential geometry
Paulo Alexandre Sousa, Rondinelle Marcolino Batista, Barnabé Pessoa Lima, Bruno Vasconcelos Mendes Vieira
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 66 / Issue 1 / February 2023

Published online by Cambridge University Press:

03 April 2023, pp. 231-240
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In this paper, we obtain one sharp estimate for the length $L(\partial\Sigma)$ of the boundary $\partial\Sigma$ of a capillary minimal surface Σ2 in M3, where M is a compact three-manifolds with strictly convex boundary, assuming Σ has index one. The estimate is in term of the genus of Σ, the number of connected components of $\partial\Sigma$ and the constant contact angle θ. Making an extra assumption on the geometry of M along $\partial M$, we characterize the global geometry of M, which is saturated only by the Euclidean three-balls. For capillary stable CMC surfaces, we also obtain similar results.