First eigenvalue of symmetric minimal surfaces in $\mathbb{S}^3$

Abstract

Let λ 1 be the first nontrivial eigenvalue of the Laplacian on a compact surface without boundary. We show that λ 1 = 2 on compact embedded minimal surfaces in S 3 which are invariant under a finite group of reflections and whose fundamental piece is simply connected and has less than six edges. In particular λ 1 =2 on compact embedded minimal surfaces in S 3 that are constructed by Lawson and by Karcher-Pinkall-Sterling.

Metrics on a closed surface of genus two which maximize the first eigenvalue of the Laplacian

“…It is also remarked in [7] that the conjecture implies that the inequality Λ(ds 2 ) ≤ 16π is sharp in the class of smooth metrics, although the equality may not be attained. It is worth mentioning that the Lawson minimal surface of genus two in S 3 has λ 1 = 2 [2] and Area ≈ 21.91 [5], and therefore Λ ≈ 43.82 < 16π.…”

A characterization of the critical catenoid

“…Similar reflection techniques were developed in [5] in the setting of closed embedded minimal surfaces in the round three-sphere S 3 . These arguments together with ideas from the proof of Theorem 1 above yield a new proof of a result of Ros (Theorem 2 above).…”

“…Therefore, the inverse image of Σ ∩ O under stereographic projection is a simply connected fundamental domain for M . We may then use Theorem 2 from [5] to conclude the first eigenvalue λ 1 of the laplacian on M is 2. By a result of Montiel and Ros ([11], Theorem 4) this implies M is congruent to the Clifford torus.…”

“…In light of this result, to prove Theorem 1, it suffices to prove that the lowest Steklov eigenvalue σ 1 (Σ) on a free boundary minimal annulus Σ with the assumed symmetries is 1. For this, we adapt a symmetrization technique of Choe and Soret [5], who proved that the lowest eigenvalue of the Laplacian on the Lawson and Karcher-Pinkall-Sterling minimal surfaces in S 3 is 2. A similar technique was used to estimate the first Steklov eigenvalue for domains in the plane with reflectional symmetries [10].…”

Minimal surfaces in $$S^3$$ : a survey of recent results

“…Choe and Soret were recently able to verify Yau's conjecture for the Lawson surfaces and the Karcher-Pinkall-Sterling examples (cf. [12], [13]). The following result is a consequence of Courant's nodal theorem and plays an important role in the argument: If ψ| D + and ψ| D − are of the same sign, then Σ ψ = 0, which is impossible.…”