Some new functionals related to free boundary minimal submanifolds

The metrics induced on free boundary minimal surfaces in geodesic balls in the upper unit hemisphere and hyperbolic space can be characterized as critical metrics for the functionals $Θ_{r,i}$ and $Ω_{r,i}$, introduced recently by Lima, Menezes and the second author. In this paper, we generalize this characterization to free boundary minimal submanifolds of higher dimension in the same spaces. We also introduce some functionals of the form different from $Θ_{r,i}$ and show that the critical metrics for them are the metrics induced by free boundary minimal immersions into a geodesic ball in the upper unit hemisphere. In the case of surfaces, these functionals are bounded from above and not bounded from below. Moreover, the canonical metric on a geodesic disk in a 3-ball in the upper unit hemisphere is maximal for this functional on the set of all Riemannian metric of the topological disk.

💡 Research Summary

The paper investigates the relationship between free‑boundary minimal submanifolds (FBMI) and free‑boundary harmonic maps (FBHM) in geodesic balls of the upper unit hemisphere (S^{m}_{+}) and the hyperbolic space ( \mathbb H^{m}). The authors show that the metrics induced by such immersions can be characterized as extremal (critical) metrics for a family of functionals that combine Steklov eigenvalues with volume and boundary area terms.

Two previously introduced families of functionals, (\Theta_{r,i}) and (\Omega_{r,i}), are recalled and generalized to arbitrary dimension (k). For a compact (k)-dimensional manifold (\Sigma) with boundary, and for a radius (r\in(0,\pi/2)) (spherical case) or (r>0) (hyperbolic case), they are defined by
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