The sharp $L^2$-Michael--Simon inequality

Let $n\ge 3$. We prove that if $j\text{\hspace{0.17em}⁣}:({\mathrm{\Sigma }}^n,g)\to ({\mathbb{R}}^N,d{x}^2)$ is an isometric immersion, then \begin{equation*} \int_Σ\left( \lvert \nabla u \rvert^2 + \frac{n(n-2)}{4}\lvert H \rvert^2 u^2 \right) \geq \frac{n(n-2)}{4}$\mathrm{V}\mathrm{o}\mathrm{l}$(S^n)^{2/n}\left( \int_Σ\lvert u \rvert^{\frac{2n}{n-2}} \right)^{\frac{n-2}{n}} \end{equation*} for all $u\in W^{1,2}(\mathrm{\Sigma })$, and characterize immersions for which equality is realized. Our proof exploits the conformal invariance of the inequality, and hence extends to sharp Michael–Simon inequalities for submanifolds of the sphere, of hyperbolic space, and of other conformally developable manifolds. As an application, we derive Chen–Fenchel–Willmore-type inequalities relating $L^p$-norms of the mean curvature vector to the volume of a compact submanifold of a simply-connected spaceform.