辛流形T^2 x M的谱柔度
Spectral Flexibility of Symplectic Manifolds T^2 x M
摘要：
我们考虑了与T^2 × M上的自然辛结构相容的黎曼度量，其中T^2是辛2-环面，M是闭辛流形。对于每个这样的度量，我们附加相应的拉普拉斯算子并考虑其第一个正特征值λ_1。我们证明了在保持辛结构不变的情况下，通过对度量结构进行变形，λ_1可以变得任意大。猜想是，对于维数>= 4的任何辛流形，这一点同样成立。我们减少一般的猜想，一个纯粹的辛问题。
英文摘要：
We consider Riemannian metrics compatible with the natural symplectic structure on T^2 x M, where T^2 is a symplectic 2-Torus and M is a closed symplectic manifold. To each such metric we attach the corresponding Laplacian and consider its first positive eigenvalue λ_1. We show that λ_1 can be made arbitrarily large by deforming the metric structure, keeping the symplectic structure fixed. The conjecture is that the same is true for any symplectic manifold of dimension >= 4. We reduce the general conjecture to a purely symplectic question.
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