Fredholm property of the Jacobi operator for singular minimizing tangent maps

Determine whether the Jacobi operator associated with a singular minimizing tangent map into a compact target manifold is a Fredholm operator, particularly when the energy density weight is only given by the singular map’s gradient squared.

Background

For a singular 0-homogeneous minimizing tangent map, the paper introduces the quantity $\lambda_1(u)$ through a Rayleigh quotient involving the second variation on the link. In the smooth case, this quantity is the lowest eigenvalue of a Fredholm Jacobi operator, but the corresponding operator-theoretic interpretation is uncertain for singular links.

The authors explain that compactness of the relevant weighted Sobolev embedding depends on the integrability of the weight $|du|^2$. For a general singular map, this weight need not have the regularity required by the cited compactness criterion, leaving the Fredholm property unresolved.