New minimal surface doublings of the Clifford torus and contributions to questions of Yau

TL;DR

This paper introduces new minimal surface doublings of the Clifford torus in the 3-sphere, establishes a quadratic lower bound on the number of such surfaces with fixed genus, and verifies Yau's conjecture for their first eigenvalues.

Contribution

It generalizes previous doubling constructions, provides a new lower bound on minimal surfaces in S^3, and confirms Yau's conjecture for specific minimal surface cases.

Findings

Constructed new minimal doublings of the Clifford torus with catenoidal bridges.

Proved a quadratic lower bound on the number of embedded minimal surfaces of fixed genus.

Verified Yau's conjecture for the first eigenvalue in specific minimal surface cases.

Abstract

The purpose of this article is three-fold. First, we apply a general theorem from our earlier work to produce many new minimal doublings of the Clifford Torus in the round three-sphere. This construction generalizes and unifies prior doubling constructions for the Clifford Torus, producing doublings with catenoidal bridges arranged along parallel copies of torus knots. Ketover has also constructed similar minimal surfaces by min-max methods as suggested by Pitts-Rubinstein, but his methods apply only to surfaces which are lifts of genus two surfaces in lens spaces, while ours are not constrained this way. Second, we use this family to prove a new, quadratic lower bound for the number of embedded minimal surfaces in $\mathbb{S}^3$ with prescribed genus. This improves upon bounds recently given by Ketover and Karpukhin-Kusner-McGrath-Stern, and contributes to a question of Yau about the structure of the space of minimal surfaces in $\mathbb{S}^3$ with fixed genus. Third, we verify Yau's conjecture for the first eigenvalue of minimal surfaces in $\mathbb{S}^3$ in the following cases. First, for all minimal surface doublings of the equatorial two-sphere constructible by our earlier general theorem. Second, for all the Clifford Torus doublings constructed in this article.