In this paper, we investigate the branched deformations of immersed compact special Lagrangian submanifolds. If there exists a nondegenerate \(\mathbb {Z}_2\) harmonic 1-form over a special Lagrangian submanifold L, we construct a family of immersed special Lagrangian submanifolds \(\tilde{L}_t\), that are diffeomorphic to a branched covering of L and \(\tilde{L}_t\) converge to 2L as current. This answers a question suggested by Donaldson (Deformations of multivalued harmonic functions, 2019. arXiv:1912.08274). As a corollary, we discover examples of special Lagrangian submanifolds that are rigid in the classical sense but exhibit branched deformations. In conjunction with the work of Abouzaid and Imagi in Nearby special lagrangians, 2021. arXiv:2112.10385, we derive constraints on the existence of nondegenerate \(\mathbb {Z}_2\) harmonic 1-forms.

在本文中，我们研究了浸入致密特殊拉格朗日子流形的分支变形。如果在特殊拉格朗日子流形L上存在非简并\(\mathbb {Z}_2\)调和 1-形式，我们构造一族沉浸式特殊拉格朗日子流形\(\tilde{L}_t\)，它们微分同胚于L和\(\tilde{L}_t\)的分支覆盖收敛到当前的2 L。这回答了唐纳森提出的问题（多值谐波函数的变形，2019。arXiv：1912.08274）。作为推论，我们发现了特殊拉格朗日子流形的例子，它们在经典意义上是刚性的，但表现出分支变形。结合 Abouzaid 和 Imagi 在 Nearby Special lagrangians, 2021. arXiv:2112.10385 中的工作，我们得出了对非简并\(\mathbb {Z}_2\)调和 1-形式存在的约束。