The no invariant line fields conjecture

Let $f$ be a polynomial, an entire transcendental function with finitely many singular values, or a rational map that is not double covered by an integral torus endomorphism. An invariant line field on the Julia set is a measurable Beltrami differential that is invariant almost everywhere and has modulus $1$ on a positive-measure set and $0$ elsewhere. No invariant line fields conjecture. The map $f$ supports no invariant line field on its Julia set. This conjecture is related to density of hyperbolicity and remains open; positive-area Julia sets are known, but no invariant line fields supported on quadratic Julia sets have been constructed.