Loi–Zedda optimal $L^2$ rigidity conjecture for projective manifolds

Let $M$ be an $n$-dimensional projective manifold in $\mathbb{P}^{n+r}$ with the induced metric. Let $\sigma$ be its second fundamental form, let $\ast1$ be the volume form, and define

$$|\sigma|^2_{L^2}:=\int_M|\sigma|^2\ast1.$$

Let $\operatorname{Vol}(\mathbb{P}^n)$ denote the volume of the standard $n$-dimensional projective linear subspace in $\mathbb{P}^{n+r}$. Loi–Zedda conjecture. If

$$|\sigma|^2_{L^2}<2n\operatorname{Vol}(\mathbb{P}^n),$$

then $M$ is isomorphic to the $n$-dimensional hyperplane $\mathbb{P}^n$. Equality holds if and only if $M$ is isomorphic to the complex quadric

$$Q^n:=\left\{[z^0:\cdots:z^{n+1}:\underbrace{0:\cdots:0}_{r-1}]\in\mathbb{P}^{n+r}\mathrel{\big|}(z^0)^2+\cdots+(z^{n+1})^2=0\right\}.$$

This is the proposed optimal $L^2$ rigidity threshold for projective manifolds: below the threshold only the totally geodesic projective space occurs, while equality is characterized by the quadric. The supplied status is unknown, so it is recorded as open.