[Paper Review] Parallel mean curvature tori in $CP^{2}$ and $CH^{2}$

Katsuei Kenmotsu|arXiv (Cornell University)|Mar 2, 2016

Geometric Analysis and Curvature Flows15 references3 citations

TL;DR

This paper classifies parallel mean curvature tori in non-flat complex space forms, specifically $\mathbb{C}P^2$ and $\mathbb{C}H^2$, by proving that any such torus must be flat and totally real. Using special isothermal coordinates and generalized Hopf differentials, the authors show that non-zero parallel mean curvature implies vanishing Gaussian curvature, leading to explicit classification via known results on totally real flat tori in these spaces.

ABSTRACT

We explicitly determine tori that have a parallel mean curvature vector, both in the complex projective plane and the complex hyperbolic plane

Motivation & Objective

• To classify isometric immersions of the 2-torus into non-flat complex space forms with non-zero parallel mean curvature vector.
• To determine the geometric and curvature properties of such tori in $\mathbb{C}P^2$ and $\mathbb{C}H^2$.
• To resolve the complexity of the non-flat case compared to the flat $\mathbb{C}^2$ setting by showing that only flat, totally real tori can admit such immersions.
• To apply generalized Hopf differentials and structure equations to derive global constraints on the second fundamental form and mean curvature vector.

Proposed method

• Introduce special isothermal coordinates adapted to the geometry of the parallel mean curvature vector, enabling local analysis of the immersion.
• Use the unitary coframe formalism and the Gauss–Codazzi–Ricci equations to express curvature and second fundamental form in terms of the Kaehler angle $\alpha$ and complex functions $a, c$.
• Apply generalized Hopf differentials $\Phi_1 = (8ba - 3\rho\sin^2\alpha)\phi^2$ and $\Phi_2 = \bar{c}\phi^2$, proven holomorphic on $M_0$.
• Use the Gauss–Bonnet theorem to rule out non-flat solutions by contradiction, showing that non-vanishing curvature leads to inconsistency with compactness.
• Leverage known classification results for totally real submanifolds in symmetric spaces (Ohnita, Urbano) to complete the global classification.
• Establish that $a = \bar{a}$ identically on the torus, implying flatness and total reality via curvature bounds and topological constraints.

Experimental results

Research questions

• RQ1 What are the necessary geometric and curvature conditions for a torus to admit an isometric immersion with non-zero parallel mean curvature vector in $\mathbb{C}P^2$ or $\mathbb{C}H^2$?
• RQ2 How does the non-flatness of $\mathbb{C}P^2$ and $\mathbb{C}H^2$ affect the existence and classification of such tori compared to the flat $\mathbb{C}^2$ case?
• RQ3 Can the parallel mean curvature vector condition force the Gaussian curvature of the torus to vanish?
• RQ4 What role do holomorphic quadratic forms $\Phi_1$ and $\Phi_2$ play in constraining the second fundamental form and global structure?

Key findings

• Any isometric immersion of a 2-torus into $\mathbb{C}P^2$ or $\mathbb{C}H^2$ with non-zero parallel mean curvature vector must have identically zero Gaussian curvature.
• Such tori are necessarily totally real, meaning the complex structure of the ambient space maps the tangent space to a complex subspace orthogonal to the image.
• The condition $a = \bar{a}$ holds identically on the torus, implying symmetry in the second fundamental form and leading to flatness.
• The classification reduces to known results on flat totally real tori in $\mathbb{C}P^2$ and $\mathbb{C}H^2$, as established by Ohnita and Urbano.
• The non-flat case is significantly simpler than the flat $\mathbb{C}^2$ case, as only flat, totally real tori satisfy the parallel mean curvature condition.
• The Gauss–Bonnet theorem is used to rule out non-flat solutions, as negative curvature bounds would contradict the integral of curvature over a compact surface.