Dice Question Streamline Icon: https://streamlinehq.com

Distinct-point separation on different boundary components by first Steklov eigenfunction maps

Establish that for any free boundary branched minimal immersion u: (N^2, ∂N) → (B^n, S^{n−1}) such that ∂N has at least two components, there exist points p, q on distinct boundary components of ∂N such that for every map F: (N, ∂N) → (B^n, S^{n−1}) by first Steklov eigenfunctions one has F(p) ≠ F(q).

Information Square Streamline Icon: https://streamlinehq.com

Background

Analogous to the Laplace case, the existence scheme for Steklov maximizers hinges on disallowing identification of boundary points from different components by all first-eigenfunction maps.

Proving this would complete the global existence theory for Σ1-maximizing metrics on orientable surfaces with boundary.

References

If N2\subset\mathbb{B}n is a free boundary branched minimal immersion by first Steklov eigenfunctions such that ∂N has at least two components, there exists a pair of points p,q in distinct components of ∂N such that for every map F\colon(N,∂N)\to (\mathbb{B}n,S{n−1}) by first Steklov eigenfunctions, F(p)\neq F(q).

— Embedded minimal surfaces in $\mathbb{S}^3$ and $\mathbb{B}^3$ via equivariant eigenvalue optimization (2402.13121 - Karpukhin et al., 20 Feb 2024) in Section 1.6 Equivariant optimization beyond orientable basic reflection surfaces