A new sphere theorem via eigenmaps

In this talk we show that if a closed $n$-dimensional Riemannian manifold $(M^n, g)$ has an eigenmap $\Phi: M^n \to \mathbb{R}^{n+1}$ whose pull-back $\Phi^*g_{\mathbb{R}^{n+1}}$ is close to the original Riemannian metric $g$ quantitatively in the $L^1$-average sense, then the manifold $M^n$ is diffeomorphic to the standard $n$-sphere. The proof is based on regularity results on metric measure spaces with Ricci curvature bounded from below, so-called $\mathrm{RCD}$ spaces.