[Paper Review] Real hypersurfaces in the complex projective plane attaining equality in a basic inequality

Toru Sasahara|arXiv (Cornell University)|Feb 8, 2017

Geometric Analysis and Curvature Flows2 references3 citations

TL;DR

This paper classifies non-Hopf real hypersurfaces with constant mean curvature in the complex projective plane that achieve equality in a fundamental inequality relating maximum Ricci curvature and squared mean curvature. Using differential geometric techniques and the Codazzi equation, it proves that such hypersurfaces are precisely minimal ruled hypersurfaces arising from the Hopf fibration via a specific parametrization, confirming a conjecture on equality cases in this geometric setting.

ABSTRACT

We determine non-Hopf hypersurfaces with constant mean curvature in the complex projective plane which attain equality in a basic inequality between the maximum Ricci curvature and the squared mean curvature.

Motivation & Objective

• To classify non-Hopf real hypersurfaces in the complex projective plane CP²(4) that satisfy equality in the inequality $\overline{Ric} \leq \frac{9}{4}\|H\|^2 + 5$.
• To determine the geometric structure of such hypersurfaces under the constraint of constant mean curvature.
• To establish a complete characterization of equality cases in the inequality for non-Hopf hypersurfaces, extending previous results on Hopf hypersurfaces.
• To prove that the only such hypersurfaces are minimal ruled hypersurfaces constructed via the Hopf fibration.

Proposed method

• Utilizes the Gauss and Codazzi equations to derive curvature and shape operator conditions under the equality assumption.
• Applies Lemma 2.1 to identify orthonormal frames where the shape operator takes a specific block-diagonal form with constraints on eigenvalues.
• Employs Lemma 2.2 to characterize ruled hypersurfaces via the vanishing of the normal component of $A\xi$ and the existence of a unit vector field $U$ orthogonal to $\xi$ satisfying $A\xi = \alpha\xi + \beta U$, $AU = \beta\xi$, $AX = 0$ for $X \perp \xi, U$.
• Performs a systematic analysis of the Codazzi equation components along the frame fields $e_1 = \xi$, $e_2$, $e_3 = Pe_2$, deriving differential equations for $\beta$, $\gamma$, $\mu$, and $\kappa_i$.
• Reduces the system of equations to a polynomial resultant condition, showing $\mu \in \{0,1\}$, and eliminates $\mu = 1$ due to contradiction with $\beta \neq 0$, leading to $\mu = 0$.
• Concludes that the shape operator satisfies $A\xi = \beta e_2$, $Ae_2 = \beta \xi$, $Ae_3 = 0$, confirming minimality and ruled structure via Lemma 2.2.

Experimental results

Research questions

• RQ1 Which non-Hopf real hypersurfaces in $\mathbb{C}P^2(4)$ with constant mean curvature attain equality in the inequality $\overline{Ric} \leq \frac{9}{4}\|H\|^2 + 5$?
• RQ2 What geometric structure characterizes such equality cases beyond the known Hopf examples?
• RQ3 Can the equality case in the Ricci curvature inequality be fully classified for non-Hopf hypersurfaces in $\mathbb{C}P^2(4)$?
• RQ4 Is the only such hypersurface a minimal ruled hypersurface arising from the Hopf fibration?

Key findings

• The only non-Hopf real hypersurfaces in $\mathbb{C}P^2(4)$ with constant mean curvature achieving equality in the inequality are minimal ruled hypersurfaces.
• These minimal ruled hypersurfaces are congruent to the image of the map $z(u,v,\theta,\psi) = e^{i\psi}(\cos u \cos v, \cos u \sin v, \sin u \cdot e^{i\theta})$ under the Hopf fibration $\varpi: S^5 \to \mathbb{C}P^2(4)$.
• The shape operator satisfies $A\xi = \beta e_2$, $Ae_2 = \beta \xi$, $Ae_3 = 0$ with $\beta \neq 0$, confirming minimality and ruled structure.
• The analysis shows $\mu = 0$ is the only viable solution, leading to $\gamma = \alpha = 0$, and rules out $\mu = 1$ due to contradiction with $\beta \neq 0$.
• The equality case implies the hypersurface is ruled and minimal, and the parametrization $z(u,v,\theta,\psi)$ fully describes the solution class.
• The converse holds: any such minimal ruled hypersurface constructed via the given parametrization satisfies the equality condition, confirming completeness of the classification.