The one-small-trace case of Bowditch's conjecture

Bowditch's conjecture characterizes quasifuchsian punctured-torus characters by two conditions on simple-curve traces. We prove the conjecture for characters in the Bowditch domain having exactly one essential nonperipheral simple curve with trace modulus at most two. More precisely, each such character admits an explicit path to a Fuchsian character within that domain, and every other simple-curve trace remains outside the closed disk of radius two throughout the path. Bowditch's component theorem then gives quasifuchsianity. The construction combines a real-length deformation with simultaneous contraction of finitely many relevant phases in the standard fan coordinates. A quantitative phase estimate controls the central fan indices, while the fork lemma protects all complementary branches. We also obtain a finite fan criterion and the restriction Re(x^2)>2 for the unique small trace x.