A rigidity theorem for the pair \(2, \mathbb{C} P^n\) (complex hyperquadric, complex projective space) (Q1293184)

Given a compact Kähler manifold \(M\) of real dimension \(2n,\) let \(P\) be either a compact complex hypersurface of \(M\) or a compact totally real submanifold of dimension \(n.\) Let \(L\) (resp. \(\mathbb{R} P^n\)) be the complex hyperquadric (resp. the totally geodesic real projective space) in the complex projective space \(\mathbb{C} P^n\) of constant holomorphic sectional curvature \( 4\lambda.\) Theorem 1.1. Let \(\xi \) be the unit vector field defined as the gradient (in \( M\)) of the distance to \(P,\) and let \(\Pi \) be any totally real \((n-1)\)-plane tangent to \(M\) and orthogonal to \(\xi \) and \(J\xi .\) If (a) \(\rho (\xi ,\xi)\geq (2n+2)\lambda ,K(\xi ,\Pi)\geq (n-1)\lambda \) for all such \(\Pi ,\) (b) \(\overline{k}\geq \sqrt{\lambda },\) and (c) \(\frac{\text{vol}(P)}{\text{vol}(M)}= \frac{\text{vol}(L)}{\text{vol}(\mathbb{C} P^n)}\) or \(\mu _1(M-P_r)=\mu _1(\mathbb{C} P^n-L_r)\) for some \(r\) less than the distance from \(P\) to its cut locus, then there is a holomorphic isometry \(i:M\rightarrow \mathbb{C} P^n\) such that \(i|_P:P\rightarrow L\) is an holomorphic isometry. Let \(P\) be a totally real and totally geodesic submanifold of \(M\) of dimension \(n.\) Here, \(P\) totally real means that \(J(TP)\) is orthogonal to \( TP.\) Theorem 1.2. Let \(\xi \) be the unit vector field defined as the gradient (in \(M\)) of the distance to \(P,\) and let \(\Pi \) be any totally real \((n-1)\)-plane tangent to \(M\) and orthogonal to \(\xi \) and \(J\xi .\) If (a) \(K(\xi ,\Pi)\geq (n-1)\lambda \) for all such \(\Pi ,K_H(\xi)\geq 4\lambda ,\) (b) \(\frac{\text{vol}(P)}{\text{vol}(M)}= \frac{\text{vol}(\mathbb{R} P^n)}{\text{vol}(\mathbb{C} P^n)}\) or \(\mu _1(M-P_r)=\mu _1(\mathbb{C} P^n-\mathbb{R} P_r^n)\) for some \(r\) as above, then there is a holomorphic isometry \(i:M\rightarrow \mathbb{C} P^n\) such that \(i|_P:P\rightarrow \mathbb{R} P^n\) is an isometry.