We rigorously establish the existence of many free boundary minimal annuli with boundary in a geodesic sphere of \(\mathbb {S}^3\) . These arise as compact subdomains of a one-parameter family of complete minimal immersions of \(\mathbb {R} \times \mathbb {S}^1\) into \(\mathbb {S}^3\) described by do Carmo and Dajczer [1]. While the immersed free boundary minimal annuli we exhibit may in general fail to be embedded or contained in a geodesic ball, we show that there is at least a one-parameter family of embedded examples that are contained in geodesic balls whose radius may be less than, equal to or greater than \(\frac{\pi }{2}\) . After explaining the connection to Otsuki tori [2], we establish lower bounds on the number of immersed free boundary minimal annuli contained in each Otsuki torus in terms of the corresponding rational number. Finally, we show that some of the recent work of Lee and Seo [3] on isoperimetric inequalities and of Lima and Menezes [4] on index bounds extends to geodesic balls equal to or larger than a hemisphere.
