For spectrally extremal minimal surfaces, Lawson surfaces have least area at large genus, and generic surfaces converge to a double equator, with analogous results in the ball.
Sharp eigenvalue bounds and minimal surfaces in the ball
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Large topology asymptotics for spectrally extremal minimal surfaces in $\mathbb{B}^3$ and $\mathbb{S}^3$
For spectrally extremal minimal surfaces, Lawson surfaces have least area at large genus, and generic surfaces converge to a double equator, with analogous results in the ball.