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.

在这项工作中，我们考虑具有凸平均曲率边界、正标量曲率的定向紧凑流形，并允许映射到$\mathbb {D}^{2}\times T^{n}$具有非零度，其中$ \mathbb {D}^{2}$是一个圆盘，$T^{n}$是一个$n$维环面。我们证明了一个不等式的有效性，该不等式涉及面积的均值和边界为同伦非平凡曲线的浸入圆盘的边界长度。当边界完全是测地线时，我们还证明了相等情况的刚性结果。这可以看作是将 Lucas Ambrózio 在 (2015, J. Geom. Anal ., 25 , 1001–1017) 中的结果部分推广到更高维度。