We prove a normal form for contact forms close to a Zoll one and deduce that Zoll contact forms on any closed manifold are local maximizers of the systolic ratio. Corollaries of this result are: (1) sharp local systolic inequalities for Riemannian and Finsler metrics close to Zoll ones, (2) the perturbative case of a conjecture of Viterbo on the symplectic capacity of convex bodies, (3) a generalization of Gromov’s non-squeezing theorem in the intermediate dimensions for symplectomorphisms that are close to linear ones.

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关于 Zoll 接触形式的局部收缩最优性

我们证明了接近 Zoll 的接触形式的正常形式，并推断出任何封闭流形上的 Zoll 接触形式都是收缩比的局部最大化。这个结果的推论是：(1) 黎曼和芬斯勒度量接近 Zoll 度量的尖锐局部收缩不等式，(2) Viterbo 关于凸体辛容量的猜想的扰动情况，(3) Gromov 非-在接近线性的辛同胚的中间维度中压缩定理。