Examples around the strong Viterbo conjecture
Examples around the strong Viterbo conjecture
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DOI：
10.1007/s11784-022-00949-6
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发表时间：
2022-06-01
影响因子：
1.8
通讯作者：
Ramos, Vinicius G. B.
中科院分区：
文献类型：
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作者：
Gutt, Jean;Hutchings, Michael;Ramos, Vinicius G. B.
维泰博猜想的一个强版本断言，所有正规化辛能力同意凸域。我们回顾已知的结果表明，某些特定的归一化辛能力同意凸域。我们还回顾了为什么所有的归一化辛容量同意S-1不变凸域。我们引入一类新的例子称为“单调环面域”，这不一定是凸的，其中包括所有的动态凸环面域在四维。我们证明了，在四维单调环面域，所有正规化辛容量同意。对于任意维的单调环面域，证明了Gromov宽度与第一等变容度一致。我们还研究了一个家庭的例子，非单调环面域，并确定强维泰博猜想的结论时，这些例子持有。沿着这条路，我们计算了一大类“弱凸复曲面域”在四维空间中的柱面容量。
A strong version of a conjecture of Viterbo asserts that all normalized symplectic capacities agree on convex domains. We review known results showing that certain specific normalized symplectic capacities agree on convex domains. We also review why all normalized symplectic capacities agree on S-1-invariant convex domains. We introduce a new class of examples called "monotone toric domains", which are not necessarily convex, and which include all dynamically convex toric domains in four dimensions. We prove that for monotone toric domains in four dimensions, all normalized symplectic capacities agree. For monotone toric domains in arbitrary dimension, we prove that the Gromov width agrees with the first equivariant capacity. We also study a family of examples of non-monotone toric domains and determine when the conclusion of the strong Viterbo conjecture holds for these examples. Along the way, we compute the cylindrical capacity of a large class of "weakly convex toric domains" in four dimensions.