Quantum phenomenon of the energy density of a harmonic map to a sphere (Q1871511)

The author investigates the energy density \(e(f):=\frac 12 \|df \|^2\) of a harmonic map \(f \colon M \to \mathbb S^d\) on a \(m\)-dimensional compact Riemannian manifold \(M\). He proves that \(2e(f)=\lambda_k\) or \(2e(f)=\lambda_{k+1}\), i.e., \(e(f)\) is constant, provided \(\lambda_k \leq 2e(f) \leq \lambda_{k+1}\) pointwise. Here, \(\lambda_0=0 < \lambda_1 \leq \lambda_2 \leq \dots \to \infty\) denote the eigenvalues of the Laplacian \(\Delta_M\). Furthermore, he shows that the components of the function \(x:= i \circ f\) with respect to the eigenfunction decomposition consists only of a linear combination of the \(k\)th and \((k+1)\)st eigenfunction. Here, \(i : \mathbb S^d \hookrightarrow \mathbb R^{d+1}\) denotes the inclusion map. As an application he shows that if the square length of the second fundamental form \(\sigma\) of a closed hypersurface \(M\) in \(\mathbb R^{m+1}\) satisfies \(\lambda_k \leq \sigma \leq \lambda_{k+1}\) then \(\sigma=\lambda_k\) or \(\sigma=\lambda_{k+1}\). Finally, he proves an upper estimate on the first (non-zero) eigenvalue \(\lambda_1\) of \(\Delta_M\).