Variational problems of normal curvature tensor and concircular scalar fields. (Q1411696)

It is well known that, for a submanifold \(M^m\) in a space form \(\tilde M(c)\), the normal curvature tensor \(R^\perp\) is invariant under conformal transformations of the ambient space. Therefore, the functional \({\mathcal R}^\perp_q[\phi]=\int_M \| R^\perp\| ^qdv\), defined on the space of immersions \(\phi\colon M^m\to \tilde M(c)\), is a conformal invariant if \(q=m/2\). If \(q=2\), then the functional \({\mathcal R}^\perp_2[\phi]\) is the Yang-Mills integral of the normal bundle. In this article the author gives a detailed investigation of the functional \({\mathcal R}^\perp_q[\phi]\), in particular \({\mathcal R}^\perp_2[\phi]\), for immersions of manifolds into space forms. He derives the first variational formula of \({\mathcal R}^\perp_q[\phi]\), using in an elegant way the second Bianchi identity, and, when the submanifold is two-dimensional, its Euler-Lagrange equation is expressed in terms of isothermal coordinates. The first variation is used to prove that if \(\phi\) is an immersion of a 4-dimensional compact oriented manifold \(M^4\) into an \(n\)-dimensional space form and the normal connection is self-dual or anti-self-dual, then \(\phi\) is a critical immersion of \({\mathcal R}^\perp_2[\phi]\). The author also studies the relation between critical surfaces of the Willmore functional and critical surfaces of \({\mathcal R}^\perp_2[\phi]\) giving formulas relating the sum of residues of logarithmic singularities of \(S\)-Willmore points in a compact oriented Willmore surface with conformal invariants. For minimal immersions \(\phi\colon M^2\to S^{n}(c)\) from a compact surface \(M^{2}\) the author proves that if \(\phi\) is a critical surface for \({\mathcal R}^\perp_2[\phi]\) and the curvature ellipses are circles, then the Gauss curvature is constant and the immersion is a standard minimal immersion of a sphere or a constant isotropic minimal immersion of a flat torus. In the last part of the paper, in order to terminate the proof of the latter result, the author studies two-dimensional Riemannian manifolds admitting concircular scalar fields whose characteristic functions are polynomials of degree \(2\) or \(3\).