A finiteness theorem for harmonic maps into Hilbert Grassmannians (Q2701678)

Let \(S\) be the complex algebra of Hilbert-Schmidt operators acting on an infinite-dimensional separable complex Hilbert space \(H\). Let \(\text{GL}_{\text{HS}}(H)\) be the Lie group of invertible operators of the form \(I+x\), \(x\in S\) and let \(U_{\text{HS}}(H)\) be its Lie subgroup of unitary operators.NEWLINENEWLINENEWLINEBy extending the results of [\textit{R. P. Gomez}, Manuscr. Math. 93, No. 3, 325-335 (1997; Zbl 0895.58014)], the main theorem here states that every harmonic map \(f: M\to U_{\text{HS}}(H)\) from a closed Riemannian manifold has its image contained inside a finite-dimensional subgroup of \(U_{\text{HS}}(H)\), provided \(f(p)= \text{Id}\) for some \(p\in M\).NEWLINENEWLINENEWLINEThe proof uses the space of solutions of a second-order linear elliptic differential equation of the sections of a Hilbert vector bundle. Therefore, any harmonic map from a closed Riemannian manifold into Hilbert Grassmannians arises from the composition of harmonic maps into finite-dimensional Grassmannians with the inclusion of finite-dimensional Grassmannians into the Hilbert Grassmannians.