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| Main Authors: | , |
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| Format: | Preprint |
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2018
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| Online Access: | https://arxiv.org/abs/1811.03756 |
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| _version_ | 1866915435103911936 |
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| author | Jin, Alvin Lee, Andrew S. |
| author_facet | Jin, Alvin Lee, Andrew S. |
| contents | We consider the embedding function $c_b(a)$ describing the problem of symplectically embedding an ellipsoid $E(1,a)$ into the smallest scaling of the polydisc $P(1,b)$. Previous work suggests that determining the entirety of $c_b(a)$ for all $b$ is difficult, as infinite staircases can appear for many sequences of irrational $b$. In contrast, we show that for every polydisc $P(1,b)$ with $b>2$, there is an explicit formula for the minimum $a$ such that the embedding problem is determined only by volume. That is, when the ellipsoid is sufficiently stretched, there is a symplectic embedding of $E(1,a)$ fully filling an appropriately scaled polydisc $P(λ,λb)$. Denoted $RF(b)$, this rigid-flexible ($RF$) value is piecewise smooth with a discrete set of discontinuities for $b>2$. At the same time, by exhibiting a sequence of obstructive classes for $b_n = \frac{n+1}{n}$ at $a=8$, we show % that $c_{b_n}(8)$ is above the volume constraint. So, in combination with the Frenkel-Müller result, it follows that $RF$ is also discontinuous at $b=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1811_03756 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | The rigid-flexible value for symplectic embeddings of four-dimensional ellipsoids into polydiscs Jin, Alvin Lee, Andrew S. Symplectic Geometry We consider the embedding function $c_b(a)$ describing the problem of symplectically embedding an ellipsoid $E(1,a)$ into the smallest scaling of the polydisc $P(1,b)$. Previous work suggests that determining the entirety of $c_b(a)$ for all $b$ is difficult, as infinite staircases can appear for many sequences of irrational $b$. In contrast, we show that for every polydisc $P(1,b)$ with $b>2$, there is an explicit formula for the minimum $a$ such that the embedding problem is determined only by volume. That is, when the ellipsoid is sufficiently stretched, there is a symplectic embedding of $E(1,a)$ fully filling an appropriately scaled polydisc $P(λ,λb)$. Denoted $RF(b)$, this rigid-flexible ($RF$) value is piecewise smooth with a discrete set of discontinuities for $b>2$. At the same time, by exhibiting a sequence of obstructive classes for $b_n = \frac{n+1}{n}$ at $a=8$, we show % that $c_{b_n}(8)$ is above the volume constraint. So, in combination with the Frenkel-Müller result, it follows that $RF$ is also discontinuous at $b=1$. |
| title | The rigid-flexible value for symplectic embeddings of four-dimensional ellipsoids into polydiscs |
| topic | Symplectic Geometry |
| url | https://arxiv.org/abs/1811.03756 |