Sharp Obstructions Research Articles

Symplectic embeddings of four-dimensional polydisks into half integer ellipsoids

We obtain new sharp obstructions to symplectic embeddings of four-dimensional polydisks P(a, 1) into four-dimensional ellipsoids E(bc, c) when $$1\le a< 2$$ and b is a half-integer. When $$1 \le a < 2-O(b^{-1})$$ we demonstrate that P(a, 1) symplectically embeds into E(bc, c) if and only if $$a+b\le bc$$ . Our results show that inclusion is optimal and extend the result by Hutchings (Geom Topol 20(2):1085–1126, 2016) when b is an integer. Our proof is based on a combinatorial criterion developed by Hutchings [14] to obstruct symplectic embeddings.

Symplectic embeddings from concave toric domains into convex ones

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 shows that they are determined by the ECH capacities of a corresponding collection of balls.

A remark on the stabilized symplectic embedding problem for ellipsoids

This note constructs sharp obstructions for stabilized symplectic embeddings of an ellipsoid into a ball, in the case when the initial four-dimensional ellipsoid has ‘eccentricity’ of the form $$3\ell -1$$ for some integer $$\ell $$ .

Symplectic embeddings and the Lagrangian bidisk

In this paper we obtain sharp obstructions to the symplectic embedding of the lagrangian bidisk into four-dimensional balls, ellipsoids and symplectic polydisks. We prove, in fact, that the interior of the lagrangian bidisk is symplectomorphic to a concave toric domain using ideas that come from billiards on a round disk. In particular, we answer a question of Ostrover. We also obtain sharp obstructions to some embeddings of ellipsoids into the lagrangian bidisk.

Beyond ECH capacities

ECH (embedded contact homology) capacities give obstructions to symplectically embedding one four-dimensional symplectic manifold with boundary into another. These obstructions are known to be sharp when the domain and target are ellipsoids (proved by McDuff), and more generally when the domain is a "concave toric domain" and the target is a "convex toric domain" (proved by Cristofaro-Gardiner). However ECH capacities often do not give sharp obstructions, for example in many cases when the domain is a polydisk. This paper uses more refined information from ECH to give stronger symplectic embedding obstructions when the domain is a polydisk, or more generally a convex toric domain. We use these new obstructions to reprove a result of Hind-Lisi on symplectic embeddings of a polydisk into a ball, and generalize this to obstruct some symplectic embeddings of a polydisk into an ellipsoid. We also obtain a new obstruction to symplectically embedding one polydisk into another, in particular proving the four-dimensional case of a conjecture of Schlenk.

On symplectic capacities of toric domains

A toric domain is a subset of $(\mathbb{C}^n,\omega_{\text{std}})$ which is invariant under the standard rotation action of $\mathbb{T}^n$ on $\mathbb{C}^n$. For a toric domain $U$ from a certain large class for which this action is not free, we find a corresponding toric domain $V$ where the standard action is free, and for which $c(U)=c(V)$ for any symplectic capacity $c$. Michael Hutchings gives a combinatorial formula for calculating his embedded contact homology symplectic capacities for certain toric four-manifolds on which the $\mathbb{T}^2$-action is free. Our theorem allows one to extend this formula to a class of toric domains where the action is not free. We apply our theorem to compute ECH capacities for certain intersections of ellipsoids, and find that these capacities give sharp obstructions to symplectically embedding these ellipsoid intersections into balls.

Symplectic embeddings into four-dimensional concave toric domains

ECH (embedded contact homology) 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 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 a cylinder are ‘optimal’; and (3) we find a sharp obstruction to ball packings into certain unions of an ellipsoid and a cylinder.

Symplectic embeddings of polydisks

We show that the polydisk $$P(1,2)$$ , the product of disks of areas $$1$$ and $$2$$ , can be symplectically embedded in a ball $$B(R)$$ of capacity $$R$$ if and only if $$R \ge 3$$ . Hence, the inclusion map gives the optimal embedding and neither the Embedded Contact Homology nor Ekeland–Hofer capacities give sharp obstructions in this situation. Our proof applies the theory of pseudoholomorphic curves in manifolds with cylindrical ends, and in particular finite energy foliations.

Rapid assessment of ship grounding over large contact surfaces

A simple method for the rapid assessment of ship bottom structures subjected to grounding over seabed obstructions with large contact surfaces is proposed in this paper. It has been recognised that the shape and size of the seabed obstruction is of crucial importance in relation to the characteristics of bottom damage. Most studies on ship grounding are concerned with ‘rock-type’ sharp obstructions, where plate tearing is the dominating failure mode. However, very few studies are found related to grounding over blunt obstructions with large contact surfaces such as ‘shoals’. Denting rather than tearing is more likely for the bottom plating as observed from actual grounding incidents. The sharp obstruction may cause earlier penetration and result in very unfavourable consequences such as compartment flooding. In contrast, the bottom plating may not fracture when moving over a blunt-type sea floor. But it may threaten the global hull girder resistance and give rise to even worse consequences such as hull collapse. The proposed simple method is established on the basis of a series of closed-form solutions for individual structural members developed from the plastic mechanism analysis. The primary deformation modes for the major bottom structural members are sliding deformation of longitudinal girders, denting and crushing of transverse members and indentation of bottom plating. The effect of friction is considered and estimated in a simple manner. The vertical resistance that governs the vertical ship motion is derived. It is found that the vertical resistance is free of friction. The proposed simple method for bottom strength is verified against large-scale non-linear finite-element analyses, where a good correlation is obtained.

Rapid assessment of ship grounding over large contact surfaces

A simple method for the rapid assessment of ship bottom structures subjected to grounding over seabed obstructions with large contact surfaces is proposed in this paper. It has been recognised that the shape and size of the seabed obstruction is of crucial importance in relation to the characteristics of bottom damage. Most studies on ship grounding are concerned with ‘rock-type’ sharp obstructions, where plate tearing is the dominating failure mode. However, very few studies are found related to grounding over blunt obstructions with large contact surfaces such as ‘shoals’. Denting rather than tearing is more likely for the bottom plating as observed from actual grounding incidents. The sharp obstruction may cause earlier penetration and result in very unfavourable consequences such as compartment flooding. In contrast, the bottom plating may not fracture when moving over a blunt-type sea floor. But it may threaten the global hull girder resistance and give rise to even worse consequences such as hull collapse. The proposed simple method is established on the basis of a series of closed-form solutions for individual structural members developed from the plastic mechanism analysis. The primary deformation modes for the major bottom structural members are sliding deformation of longitudinal girders, denting and crushing of transverse members and indentation of bottom plating. The effect of friction is considered and estimated in a simple manner. The vertical resistance that governs the vertical ship motion is derived. It is found that the vertical resistance is free of friction. The proposed simple method for bottom strength is verified against large-scale non-linear finite element analyses, where a good correlation is obtained.

Comparison of Roughness Parameters for Various Microchannel Surfaces in Single-Phase Flow Applications

Recently, a set of new roughness parameters was proposed by Kandlikar et al. and Taylor et al. for reporting surface roughness as related to fluid flow. The average roughness R a parameter is often used in microfluidic applications, but this parameter alone is insufficient for describing surface roughness; a specimen with deep grooves and sharp obstructions can share the same average roughness value as a relatively smooth surface with low uniform surface roughness. Because the average roughness parameter is broad, it is difficult to assess the surface topography features that result from different machining processes or etches. A profilometer and a digital microscope are used to examine the surface roughness profiles of various materials submitted to different machining techniques. The materials studied will be similar to those used for microchannels including aluminum, stainless steel, copper, and silicon. Depending on the material, these samples are submitted to several machining processes, including milling, grinding, fly cutting, and microfabrication techniques. These machining processes and microfabrication techniques are of practical interest in microfluidics applications. After studying the surface roughness patterns exhibited in these samples, the roughness parameters employed in some of the recent roughness models are evaluated. This study is expected to provide more understanding of assorted surface roughness.