Thursday

In the round 3-sphere, there exists an family of minimal 2-spheres (the equatorial spheres) and an family of minimal tori (the Clifford tori). Almgren (1966) and Calabi (1967) proved that the equatorial spheres are the only minimal 2-spheres, and Brendle (2013) confirmed that the Clifford tori are the only minimal tori, thereby resolving the Lawson conjecture. The topology of these moduli spaces motivated Yau (1982) and White (1989) to conjecture the existence of at least four minimal spheres and five minimal tori, respectively, in any closed Riemannian 3-sphere. For other topological types, Lawson (1970) constructed minimal surfaces of arbitrary genus in , now known as Lawson surfaces. Inspired by Yau’s and White’s conjectures, it is further expected that for any genus g, there exist multiple genus-g minimal surfaces in any closed Riemannian 3-sphere, with the number related to the topology of the space of Lawson surfaces. In this talk, I will present the resolution of Yau’s conjecture by Wang–Zhou (2023) and of White’s conjecture by joint work of Adrian and myself (2024), in the setting of positive Ricci curvature. I will then discuss how the techniques we developed can be applied to construct multiple minimal surfaces of higher genus. This is based on joint work with Adrian Chu (Cornell University) and Zhihan Wang (Cornell University).