Construction des applications harmoniques non rigides d'un tore dans la sphère
A harmonic map f: \(M\to S^ n\) of a compact Riemannian manifold M into the Euclidean n-sphere \(S^ n\) is said to be infinitesimally rigid if every projectable divergence-free Jacobi field v along f is of the form \(v=X{\mathbb{O}}f\), where \(X\in so(n+1)\) is an infinitesimal isometry on \(S^ n\). Here we show that the vector function \^v: \(M\to {\mathbb{R}}^{n+1}\) induced by such v satisfies the equation \(\Delta^ M \hat v=2 e(f) \hat v\), where e(f) is the energy density of f. Using this we prove that every harmonic embedding of \(S^ 1\times S^ 1\) into \(S^ n\) with \(e(f)=1/2\) is infinitesimally non-rigid.
scientific article; zbMATH DE number 3844568
scientific article