On the Willmore functional of 2-tori in some product Riemannian manifolds (Q2902675)

The Willmore functional is an important conformal invariant in the conformal geometry of submanifolds. One of its expressions is NEWLINE\[NEWLINEW(M)= \int_M (|H|^2-R/(m(m-1))+1/(m(m-1))\sum_{ij}\widetilde R_{ijij})^{m/2} dM,NEWLINE\]NEWLINE where \(m\) is the dimension of the immersed manifold \(M\), \(H\) is its mean curvature vector, \(R\) is the scalar curvature and \(\widetilde R_{ijkl}\) are the components of the curvature tensor field of the ambient space, restricted to the manifold \(M\). After presenting the variational equations for Willmore submanifolds the author presents a small application. Next he discusses the minimum of the Willmore functional of a torus \(T^2\) in a Riemannian manifold \(N\), especially when \(N\) is a product manifold. He shows that when \(N=S^2\times S^1\) the minimum of \(W(T^2)\) is \(0\). If \(N=\mathbb{R}^2\times S^1\) there exists no torus having least Willmore functional. If \(N=H^2(-c)\times S^1\) and \(M = \gamma \times S^1\) the minimum of \(W(M)\) is \(2\pi^2 \sqrt c\). Here, \(\gamma \) is a an immersed curve with arc length as parameter.