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Some new functionals related to free boundary minimal submanifolds

Canadian Journal of Mathematics. 2026. Vol. 1. No. 1. P. 1–27.
Medvedev V., Ma T.

The metrics induced on free boundary minimal surfaces in geodesic

balls in the upper unit hemisphere and hyperbolic space can be characterized as crit-

ical metrics for the functionals Θr,i and Ωr,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 Θr,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 function-

als 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 target: Mathematics
Language: English
Full text
DOI
Text on another site
Keywords: free boundary minimal submanifoldsминимальные подмногообразия со свободной границейRobin problemзадача Робэнаeigenvalue optimizationоптимизация собственных чисел
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