The curvature of a minimal disc $S^2 \subset \mathbf{R}^3$ can be bounded in terms of the curvature of its boundary via the Gauss–Bonnet formula: \begin{equation} \frac{1}{2}\int_S \lvert A \rvert^2 \leq -2\pi + \int_{\partial S} k_g, \end{equation} where $k_g$ is the geodesic curvature of $\partial S$. (In fact this is an identity rather than an inequality.)

Is there an analog of this valid for minimal hypersurfaces $S^n \subset \mathbf{R}^{n+1}$?

For example, is there an inequality like $\int_S \lvert A \rvert^2 \leq C + \int_{\partial S} \lvert A_{\partial S} \rvert^2$, with $C$ depending on the topology of $S$?

The curvature of a minimal disc $S^2 \subset \mathbf{R}^3$ can be bounded in terms of the curvature of its boundary via the Gauss–Bonnet formula: \begin{equation} \frac{1}{2}\int_S \lvert A \rvert^2 \leq -2\pi + \int_{\partial S} k_g, \end{equation} where $k_g$ is the geodesic curvature of $\partial S$. (In fact this is an identity rather than an inequality.)

Is there an analog of this valid for minimal hypersurfaces $S^n \subset \mathbf{R}^{n+1}$?

For example, is there an inequality like $\int_S \lvert A \rvert^2 \leq C + \int_{\partial S} \lvert A_{\partial S} \rvert^2$, with $C$ depending on the topology of $S$? (This might be missing some normalization, but I'm just trying to give an idea of the 'spirit' of the inequality that I am interested in.)