A Willmore functional for compact surfaces of complex projective plane

TL;DR

This paper introduces a conformally invariant functional for surfaces in the complex projective plane, analyzing its critical points, minima, and bounds, with special focus on minimal, Lagrangian, and specific geometric surfaces.

Contribution

It defines a new Willmore-like functional for complex projective surfaces, explores its critical points, and characterizes surfaces attaining bounds, extending classical surface theory.

Findings

Minimal surfaces are critical points of the functional.

Constructed minima using twistor spaces.

Identified surfaces attaining lower bounds.

Abstract

We propose the study of a conformally invariant functional for surfaces of complex projective plane which is closely related to the classical Willmore functional. We show that minimal surfaces of complex projective plane are critical for this functional and construct some minima for it via the twistors spaces of complex projective plane. Also, we find lower bounds for this functional and for its restriction to the class of Lagrangian surfaces and characterize the complex lines and the Lagrangian totally geodesic surfaces and the Whitney spheres as the only attaining those bounds.