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In this paper we define the equivariant double-slice genus and equivariant super-slice genus of a strongly invertible knot. We prove lower bounds for both the equivariant double-slice genus and the equivariant super-slice genus. Using these bounds we find a family of knots which are double-slice and equivariantly slice but have arbitrarily large equivariant double-slice genus. From this, we construct equivariantly knotted symmetric 3-balls as well as unknotted symmetric 2-spheres which do not bound equivariant 3-balls. Additionally, using double-slice and super-slice genus we construct properly embedded surfaces with large 1-handle stabilization distance distance rel boundary., Comment: 18 pages, 13 figures. All comments are welcome!

We study the discretization of a Dirichlet form on the Koch snowflake domain and its boundary with the property that both the interior and the boundary can support positive energy. We compute eigenvalues and eigenfunctions, and demonstrate the localization of high energy eigenfunctions on the boundary via a modification of an argument of Filoche and Mayboroda. H\"older continuity and uniform approximation of eigenfunctions are also discussed.