Variational problems of normal curvature tensor and concircular scalar fields

Abstract

We consider the integral of (the square of) the length of the normal curvature tensor for immersions of manifolds into real space forms, especially into spheres. The first variation formula is given and the Euler-Lagrange equation is expressed in terms of the isothermal coordinates when the submanifold is two-dimensional. The relations between the critical surfaces and Willmore surfaces are discussed. We also give formulas concerning the residue of logarithmic singularities of $S$-Willmore points or estimate it by a conformal invariant.

We show that if a compact critical surface satisfies certain conditions and the immersion is minimal, then the Gauss curvature is a non-negative constant and the immersion is a standard minimal immersion of a sphere or a constant isotropic minimal immersion of a flat torus. To prove this result, we study two-dimensional Riemannian manifolds admitting concircular scalar fields whose characteristic functions are polynomials of degree $2$. Moreover, the case that the characteristic functions are polynomials of degree $3$ is studied.