Urs Frauenfelder
The Stark problem is a completely integrable system which describes the motion of an electron in a constant electric field and subject to the attraction of a proton. In this paper we show that in the planar case after Levi-Civita regularization the bounded component of the energy hypersurfaces of the Stark problem for energies below the critical value can be interpreted as boundaries of concave toric domains.
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Concave symplectic toric fillings
本文研究了接触toric3维流形的凹型辛填充结构，证明了其存在无限多个互不equivariant辛同构且不通过吹胀操作相关的填充。
Embedded contact homology gives a sequence of obstructions to four-dimensional symplectic embeddings, called ECH capacities. In "Symplectic embeddings into four-dimensional concave toric domains", the author, Choi, Frenkel, Hutchings and Ramos computed the ECH capacities of all "concave toric domains", and showed that these give sharp obstructions in several interesting cases. We show that these obstructions are sharp for all symplectic embeddings of concave toric domains into "convex" ones. In an appendix with Choi, we prove a new formula for the ECH capacities of convex toric domains, which …
ECH capacities give obstructions to symplectically embedding one symplectic four-manifold with boundary into another. We compute the ECH capacities of a large family of symplectic four-manifolds with boundary, called "concave toric domains". Examples include the (nondisjoint) union of two ellipsoids in $\mathbb{R}^4$. We use these calculations to find sharp obstructions to certain symplectic embeddings involving concave toric domains. For example: (1) we calculate the Gromov width of every concave toric domain; (2) we show that many inclusions of an ellipsoid into the union of an ellipsoid and…
ECH capacities of concave singular toric domains
本文推广四维环面域至边界为透镜空间 L(n,1) 的奇异环面域，并计算一类凹奇异环面域的 ECH 容量，结果可用于研究其去奇异化空间中的嵌入问题。
On the Lagrangian capacity of convex or concave toric domains
该研究针对拉格朗日容量展开计算研究，证明了四维凸环面域的拉格朗日容量等于其对角线，在特定虚拟扰动方案假设下将结果推广至任意凸或凹环面域，肯定了相关椭球拉格朗日容量猜想。
Obstruction to Symplectic Embeddings between Toric Domains
该研究利用嵌入接触同调，扩展Hutchings准则得到三类凸域辛嵌入的障碍，经应用验证其部分障碍强于ECH容量导出的障碍。
Symplectic embeddings of toric domains with boundary a lens space
本研究给出凹透镜空间的嵌入切触复形组合描述，定义带奇点的广义凹环面域，利用ECC计算其ECH容量，并给出可恢复该容量的辛流形填充方式。
Properties of contact toric structures and concave boundaries of linear plumbings
研究具有凹接触边界的球面上辛圆盘束的线性拼接，用复曲面几何工具确定接触结构紧或过度扭曲情况，开发工具包计算线性拼接凹边界的ECH代数挠率和接触不变量。
Barcode growth for toric-integrable Hamiltonian systems
本文研究完全可积哈密顿系统中Floer或辛同调持续性模的条形码增长，证明在凸/凹或实解析环面域及闭环面流形上条形码呈多项式增长且次数不超过维数一半，与许多非平凡动力学系统的指数增长形成对比，并将条形码增长函数用作区分开域的不变量。
We use positive S^1-equivariant symplectic homology to define a sequence of symplectic capacities c_k for star-shaped domains in R^{2n}. These capacities are conjecturally equal to the Ekeland-Hofer capacities, but they satisfy axioms which allow them to be computed in many more examples. In particular, we give combinatorial formulas for the capacities c_k of any "convex toric domain" or "concave toric domain". As an application, we determine optimal symplectic embeddings of a cube into any convex or concave toric domain. We also extend the capacities c_k to functions of Liouville domains whic…
The special concave toric domain for the rotating Kepler problem
针对能量≤-3/2的旋转开普勒问题（RKP），通过两种辛正则化识别相空间有界分量，构造特殊凹环面域（SCTD）及组合树结构，推进辛嵌入理解并提供哈密顿动力学研究新工具。
The Hamiltonian shape invariant of a domain $X \subset \mathbb R^4$, as a subset of $\mathbb R^2$, describes the product Lagrangian tori which may be embedded in $X$. We provide necessary and sufficient conditions to determine whether or not a path in the shape invariant can lift, that is, be realized as a smooth family of embedded Lagrangian tori, when $X$ is a basic $4$-dimensional toric domain such as a ball $B^4(R)$, an ellipsoid $E(a,b)$ with $\frac{b}{a} \in {\mathbb N}_{\geq 2}$, or a polydisk $P(c,d)$. As applications, via the path lifting, we can detect knotted embeddings of product L…
We establish computational results concerning the Lagrangian capacity from "Cieliebak and Mohnke - Punctured holomorphic curves and Lagrangian embeddings". More precisely, we show that the Lagrangian capacity of a 4-dimensional convex toric domain is equal to its diagonal. The proof involves comparisons between the Lagrangian capacity, the McDuff-Siegel capacities from "McDuff and Siegel - Symplectic capacities, unperturbed curves, and convex toric domains", and the Gutt-Hutchings capacities from "Gutt and Hutchings - Symplectic capacities from positive S1-equivariant symplectic homology". Wor…
Curvy points, the perimeter, and the complexity of convex toric domains
研究环面边界的曲率和周长对四维辛填充问题的影响，证明ECH容量次主导渐近恢复周长，周长阻碍完全填充，并分类具有无限阶梯的光滑凸环面域。
ECH capacities are rich obstructions to symplectic embeddings in 4-dimensions that have also been seen to arise in the context of algebraic positivity for (possibly singular) projective surfaces. We extend this connection to relate general convex toric domains on the symplectic side with towers of polarised toric surfaces on the algebraic side, and then use this perspective to show that the sub-leading asymptotics of ECH capacities for all convex and concave toric domains are $O(1)$. We obtain sufficient criteria for when the sub-leading asymptotics converge in this context, generalising resul…
The Lagrange Problem from the Viewpoint of Toric Geometry
本文从环面几何角度研究拉格朗日问题，证明在特定能量和参数条件下，正则化能量超曲面的有界分量为严格单调或凸环面区域的边界，并给出动态凸性及凹凸性结论。
Given a strictly concave rational PL function $ϕ$ on a complete $n$-dimensional fan $Σ$, we construct an exact symplectic structure of finite volume on $(\mathbb{C}^\times)^n$ and a family of functions $H_{ϕ,ε}$ called polyhedral Hamiltonians. We prove that for each $ε$ the one-periodic orbits of $H_{ϕ,ε}$ come in families corresponding to finitely many primitive lattice points of $Σ$ and determine their topology. When $ϕ$ is negative on the rays of $Σ$, we show that the level sets of polyhedral Hamiltonians are hypersurfaces of contact type. As a byproduct, this construction provides a dynami…
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