Solution of the Mumford-Shah conjecture

Abstract: We present a compact-translation argument for the classification of planar Mumford--Shah generalized global minimizers. Two perpendicular translations of a compact isolated portion of the crack, independently relaxed in the displacement, saturate a full-plane Hodge identity. The resulting normal conditions imply a holomorphic rigidity statement which excludes that portion. Established classification theorems then leave only the constant, pure-jump, triple-junction, and crack-tip models. The planar interior regularity conjecture follows from the known equivalence between this classification and local regularity. No homogeneity assumption is imposed on the generalized global minimizer.