On the structure of the moduli space of harmonic eigenmaps (Q1898881)

Let \(F: M\to S^N\) be a harmonic map of a Riemannian manifold \(M\) into the Euclidean \(N\)-sphere \(S^N\subset \mathbb{R}^{N+ 1}\). If \(A\) is a \((n+ 1)\times (N+ 1)\)-matrix, \(f= A\cdot F\) defines a harmonic map of \(M\) into \(S^n\) provided that \(A\) maps the image of \(F\) into \(S^n\). In this case one says that \(f\) is derived of \(F\) and defines a symmetric endomorphism \(\langle f\rangle= A^\tau A- I_{N+ 1}\in S^2(\mathbb{R}^{n+ 1})\). Two maps \(f'', f': M\to S^n\), derived from \(F\) are said to be equivalent if \(f''= U\cdot f'\) for some \(U\in O(n+ 1)\). Given a full harmonic map \(F: M\to S^N\) (the image spans the range), the equivalence classes of full harmonic maps \(f: M\to S^n\) that are derived from \(F\) can be parametrized, via \(f\to \langle f\rangle\), by the convex body \[ {\mathcal L}_F= \{C\in E_F:C+ I_{N+ 1}\text{ is positive semidefinite}\}, \] where \[ E_F= (\text{span}\{\text{proj } [F(x)]:x\in M\})^\perp\subset S^2(\mathbb{R}^{N+ 1}). \] The author constructs and studies a finite stratification on \({\mathcal L}_F\) with almost everywhere smooth strata.