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Distinct-point separation by first Laplace eigenfunction maps

Establish that for any branched minimal immersion u: M^2 → S^n of a closed surface by first Laplace eigenfunctions, there exist points p, q ∈ M such that for every map F: M → S^n by first Laplace eigenfunctions one has F(p) ≠ F(q).

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Background

The authors’ existence scheme reduces global maximizer existence to ruling out the possibility that all first-eigenfunction maps identify prescribed pairs of points.

Proving this separation property would complete the existence theory for Λ1-maximizing metrics on all orientable surfaces.

References

If M2\subset Sn is a branched minimal immersion by first eigenfunctions for the Laplacian, there exists some pair of points p,q\in M such that for every map F\colon M\to Sn by first 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