Area gaps for minimal surfaces in spheres

In this paper, we prove that a linearly full minimal $2$-torus in ${\mathbb{S}}^N$ with $N\ge 5$ must have area $>2{\pi }^2$, and that a linearly full closed minimal surface in ${\mathbb{S}}^N$ with $N\ge 9$ must have area $>8\pi$. These results verify the Willmore conjecture and the generalized Willmore conjecture in higher codimension for minimal surfaces. The area gaps are obtained by applying the Cheng–Li–Yau comparison theorem and deriving a lower bound of the heat trace, where the monotonicity of the shifted normalized Riesz mean is used. We also establish a volume gap above the balanced Clifford product for linearly full closed minimal $n$-submanifolds in ${\mathbb{S}}^N$ when $n=3$, $N\ge 9$, or $n\ge 4$, $N\ge 3n-1$.