Normal Critical Surfaces in ${\mathbb {C}}P^{2}$

Abstract

Let $F: M\rightarrow {\mathbb {C}}P^{2}$ be an isometric immersion of a closed surface in the complex projective plane ${\mathbb {C}}P^{2}$. In this paper, we consider the functional $W_{N}(F)=\int _{M}({\overline{K}}^{\perp }-K^{\perp })\textrm{d}M$, which is a global conformal invariant. The critical surfaces of $W_{N}(F)$ are called normal critical surfaces. We compute the first variation of $W_{N}(F)$. Moreover, we build an index formula for the normal critical surfaces in the spirits of Webster (J Differ Geom 20:463–470, 1984.), Wolfson (J Diff Geom 29:281–294, 1989), Chen-Tian (Geom Funct Anal 7:873–916, 1997) and Han-Li (J Eur Math Soc 12:505–527, 2010). In particular, when the Kähler angle $\alpha$ satisfies $\cos \alpha \leqslant -\tfrac{\sqrt{5}}{3}$ or $-\tfrac{\sqrt{5}}{3}\leqslant \cos \alpha \leqslant \tfrac{\sqrt{5}}{3}$ or $\cos \alpha \geqslant \tfrac{\sqrt{5}}{3}$, M is the normal critical surface if and only if it is the minimal surface with constant Kähler angle $\alpha$.