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Discs area-minimizing in mean convex Riemannian n-manifolds
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 152 / Issue 6 / December 2022
- Published online by Cambridge University Press:
- 06 September 2021, pp. 1361-1382
- Print publication:
- December 2022
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- Article
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In this work, we consider oriented compact manifolds which possess convex mean curvature boundary, positive scalar curvature and admit a map to
$\mathbb {D}^{2}\times T^{n}$ with non-zero degree, where 
$\mathbb {D}^{2}$ is a disc and 
$T^{n}$ is an 
$n$-dimensional torus. We prove the validity of an inequality involving a mean of the area and the length of the boundary of immersed discs whose boundaries are homotopically non-trivial curves. We also prove a rigidity result for the equality case when the boundary is strongly totally geodesic. This can be viewed as a partial generalization of a result due to Lucas Ambrózio in (2015, J. Geom. Anal., 25, 1001–1017) to higher dimensions.
