Sharp focal radius estimate and rigidity of hypersurfaces in manifolds with positive curvature

Abstract

We prove a sharp Clifford-threshold focal-radius estimate and rigidity for immersed hypersurfaces. Under a $p$-form curvature condition, formulated by the Weitzenböck curvature term together with ${\mathrm{Ric}}_p\ge p$, any closed two-sided immersion $F:{\Sigma }^m\to M^{m+1}$ with $b_p(\Sigma ;\mathbb{R})\ne 0$ and $1\le p\le m/2$ satisfies \[ r_f(F,M)\le\fracπ{4}. \] The equality case is rigid: if the ambient manifold is complete, equality forces the hypersurface to be locally the Clifford hypersurface $S^p(1/\sqrt{2})\times S^{m-p}(1/\sqrt{2})\subset S^{m+1}(1)$; if the ambient manifold is compact and connected, it is a spherical space form. The curvature condition follows from $\sec \ge 1$ for $p=1$, from normalized $PIC1\ge 1$ for $p=2$, and from curvature operator bounded below by one in all degrees. By quotient lifting and the Hopf fibrations, we also obtain focal-radius estimates in ${\mathbb{CP}}^n$ and ${\mathbb{HP}}^n$, with projective Clifford rigidity, without any Betti-number assumption.