Some results for stationary maps of a functional related to pullback metrics (Q631696)

Let \((M,g), (N,h)\) be Riemannian manifolds without boundary and let \(f\) be a smooth map from \(M\) to \(N\). Assume that \(M\) is compact and consider the functional \(\Phi (f)=\int_M\|f^*h\|^2dv_g\). The authors study some properties of the stationary maps of the functional \(\Phi(f)\). Theorem 1. Let \(f\) be a stable stationary map for the functional \(\Phi (f)\) from the \(m\)-dimensional standard sphere \(S^m\) into a Riemannian manifold \(N\). If \(m\geq 5\), then \(f\) is a constant map. Theorem 2. Let \(f\) be a stable stationary map for the functional \(\Phi (f)\) from a compact Riemannian manifold \(M\) into the \(n\)-dimensional standard sphere \(S^n\). If \(n\geq 5\), then \(f\) is a constant map. Theorem 3. Let \(f\) be a stationary map for the functional \(\Phi (f)\) from a compact Riemannian manifold \(M\) into a Riemannian manifold \(N\). If there exists a strictly convex function on \(N\), then \(f\) is a constant map.