Minimal surfaces in the three-sphere by desingularizing intersecting Clifford tori

For each integer k >= 2 we apply PDE gluing methods to desingularize certain collections of intersecting Clifford tori, thus producing sequences of minimal surfaces embedded in the round three-sphere. The collections of the Clifford tori we use consist of either k Clifford tori intersecting with maximal symmetry along two orthogonal great circles (lying on orthogonally complementary two-planes) or of the same k Clifford tori supplemented by an additional Clifford torus equidistant from the original two circles of intersection so that the latter torus orthogonally intersects each of the former k tori along a pair of disjoint orthogonal circles. The former two circles get desingularized by using singly periodic Karcher-Scherk towers of order k as models, so that after rescaling the sequences of minimal surfaces converge smoothly on compact subsets to the Karcher-Scherk tower of order k. Near the other 2k circles (in the latter case) the corresponding rescaled sequences converge to a singly periodic Scherk surface. The simpler examples of the first type, where the number of handles desingularizing each circle is the same, resemble surfaces constructed by Choe and Soret (Math Ann 364(3-4):763-776, 2016) by different methods. There are many new examples which are more complicated and on which the numbers of handles for the two circles differ. All examples of the latter type are new as well.