We prove that the punctured generalized conifolds and punctured orbifolded conifolds are mirror symmetric under the SYZ program with quantum corrections. This mathematically confirms the gauge-theoretic prediction by Aganagic–Karch–L¨ust–Miemiec, and also provides a supportive evidence to Morrison’s conjecture that geometric transitions are reversed under mirror symmetry.

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New spaces of Lipschitz type on metric-measure spaces are introduced and they are shown to be just the well-known Besov spaces or Triebel–Lizorkin spaces when the smooth index is less than 1. These theorems also hold in the setting of spaces of homogeneous type, which include Euclidean spaces, Riemannian manifolds and some self-similar fractals. Moreover, the relationships amongst these Lipschitz-type spaces, Hajłasz–Sobolev spaces, Korevaar–Schoen–Sobolev spaces, Newtonian Sobolev space and Cheeger–Sobolev spaces on metric-measure spaces are clarified, showing that they are the same space with equivalence of norms. Furthermore, a Sobolev embedding theorem, namely that the Lipschitz-type spaces with large orders of smoothness can be embedded in Lipschitz spaces, is proved. For metric-measure spaces with heat kernels, a Hardy–Littlewood–Sobolev theorem is establish, and hence it is proved that Lipschitz-type spaces with small orders of smoothness can be embedded in Lebesgue spaces.

An n dimensional minimal submanifold $\sum$  of $R^{n+m}$ is called nonparametric if $\sum$  can be represented as the graph of a vector-valued function $f : D \subset \mathbb{ R}^n \to \mathbb{R}^n$. This note provides a sufficient condition for the stability of such $\sum$ in terms of the norm of the differential $df$.