We prove that if fg: (Σ, g) (S^2+p, ) is a smooth minimal isometric embedding of a Riemannian surface (Σ, g), and 0, 1 t gₜ is a path of area preserving conformal deformations of g on fg (Σ), then there exists a path of conformal diffeomorphism Fₜ: (S^2+p, Fₜ^*) (S^2+p, ) that starts at ₒ^₂+, set theoretically fixes fg (Σ) for all t, and it is such that F^*ₜ g₅₆ (₌) =gₜ with f₆䂻: (Σ, gₜ) (S^2+p, ) a path of minimal embedding deformations of the initial fg. We apply this result to the Lawson surface (Σ, g) = (ξ₊/₌, ₌, g⏝_₊/₌, ₌), m|k>1, to conclude that if a=μ₆_⏝_{₊/₌, ₌} (Σ), and 0, 1 t gₜ is a path of area a metrics conformal deformations of g₊/₌, ₌ to a metric gₐ of scalar curvature 4πχ (Σ) /a, then f₆_⏝_{₊/₌, ₌}: (ξ₊/₌, ₌, g⏝_₊/₌, ₌) (S³, ) has associated minimal isometric conformal deformations f₆䂻 to the isometric embedding f₆䂯 of gₐ, in sharp contrast with the situation of the standard sphere ξ₀, ₁ and Clifford torus (ξ₁, ₁, which are the only orientable Riemannian surfaces of genus 0 and 1 isometrically embedded into (S³, ) as minimal surfaces. If σ² (Σ): =₆ ₂ (⏢) (4πχ (Σ) ) ²/ (14₆ ₆W₅₆ (Σ) ), W₅₆ (Σ) the Willmore energy of fg and C (Σ) the space of classes, then (4πχ (Σ) ) ²/ (1 4W₅₆ (Σ) ) σ² (Σ) = (4πχ (Σ) ) ²/ (14W₅_₆_{⏝_{₊, ₁}} (Σ) ), and we describe the fgs for which the equality is achieved.
Santiago R. Simanca (Thu,) studied this question.