arXiv research

Study infinite symplectic forms on ruled surfaces.

Study embeds ruled surfaces into symplectic manifolds, finds Stein fillability results.

We study symplectic deformation types of minimal symplectic fillings of links of quotient surface singularities. In particular, there are only finitely many symplectic deformation types for each quotient surface singularity.

This paper studies symplectic structures on elliptic surfaces with positive Euler number.

This paper studies symplectic critical surfaces in Hermite surfaces.

In this paper we show that every degree 2 homology class of a 2n-dimensional symplectic manifold is represented by an immersed symplectic surface if it has positive symplectic area. Moreover, the symplectic surface can be chosen to be embedded if 2n is at least 6. We also analyze the additional conditions under which e…

A study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is show…

In this paper we derive the Euler-Lagrange equation of the functional $L_β=\int_Σ\frac{1}{\cos^βα}dμ, ~~β\neq -1$ in the class of symplectic surfaces. It is $\cos^3α{\bf{H}}=β(J(J\nabla\cosα)^\top)^\bot$, which is an elliptic equation when $β\geq 0$. We call such a surface a $β$-symplectic critical surface. We first st…

We give a new characterization of symplectic surfaces in CP^2 via bridge trisections. Specifically, a minimal genus surface in CP^2 is smoothly isotopic to a symplectic surface if and only if it is smoothly isotopic to a surface in transverse bridge position. We discuss several potential applications, including the cla…

Constructs symplectic surface bundles with positive signatures.

Symplectic Torelli groups of positive rational surfaces are trivial or sphere braid groups.

We show that hyperelliptic symplectic Lefschetz fibrations are symplectically birational to two-fold covers of rational ruled surfaces, branched in a symplectically embedded surface. This reduces the classification of genus 2 fibrations to the classification of certain symplectic submanifolds in rational ruled surfaces…

Paper uses index theorem to relate symplectic bundle signature to surface group representation in real symplectic group.

Symplectic structure found on projective structures on surfaces with boundary.

New symplectic 4-manifolds with non-negative signatures are constructed using complex surfaces and quotients.

We derive an obstruction to representing a homology class of a symplectic 4-manifold by an embedded, possibly disconnected, symplectic surface.

Anti-symplectic involutions connect a sphere in a symplectic surface.

Defines log Floer cohomology for symplectic surfaces with a degenerate part.

Symplectic forms match on circle pattern space.

We construct a family of pairs of non-isotopic symplectic surfaces in the standard symplectic $4$-disk such that they are bounded by the same transverse knot in the standard contact $3$-sphere and fundamental groups of their complements are isomorphic. In the appendix, we prove explicitly that one can obtain a symplect…

The paper establishes criteria for symplectic surfaces in 4-manifolds.

Study shows symplectic mapping groups of K3 surfaces are infinitely generated.

We show that every negative definite configuration of symplectic surfaces in a symplectic 4--manifold has a strongly symplectically convex neighborhood. We use this to show that, if a negative definite configuration satisfies an additional negativity condition at each surface in the configuration, and if the complex si…

Develops a finite construction for self-duality and related moduli spaces over Riemann surfaces.

The symplectic cone of a closed oriented 4-manifold is the set of cohomology classes represented by symplectic forms. A well-known conjecture describes this cone for every minimal Kaehler surface. We consider the case of the elliptic surfaces E(n) and focus on a slightly weaker conjecture for the closure of the symplec…

In this paper, we demonstrate a relation among Seiberg-Witten invariants which arises from embedded surfaces in four-manifolds whose self-intersection number is negative. These relations, together with Taubes' basic theorems on the Seiberg-Witten invariants of symplectic manifolds, are then used to prove the symplectic…

We construct an infinite family of homologous, non-isotopic, symplectic surfaces of any genus greater than one in a certain class of closed, simply connected, symplectic four-manifolds. Our construction is the first example of this phenomenon for surfaces of genus greater than one.

We prove symplectic hypersurfaces in Weinstein domains and give obstructions for manifold boundaries.

We study neighborhoods of configurations of symplectic surfaces in symplectic 4-manifolds. We show that suitably `positive' configurations have neighborhoods with concave boundaries and we explicitly describe open book decompositions of the boundaries supporting the associated negative contact structures. This is used …

For each member of an infinite family of homology classes in the K3-surface E(2), we construct infinitely many non-isotopic symplectic tori representing this homology class. This family has an infinite subset of primitive classes. We also explain how these tori can be non-isotopically embedded as homologous symplectic …

Study Dehn-Seidel twists on Lagrangian spheres in K3 surfaces.

We prove that all minimal symplectic four-manifolds are essentially irreducible. We also clarify the relationship between holomorphic and symplectic minimality of Kähler surfaces. This leads to a new proof of the deformation-invariance of holomorphic minimality for complex surfaces with even first Betti number which ar…

In this note we introduce the notion of the relative symplectic cone. As an application, we determine the symplectic cone of certain T^2-fibrations. In particular, for some elliptic surfaces we verify a conjecture on the symplectic cone of minimal Kaehler surfaces raised by the second author.

This paper constructs symplectic surfaces in 4-manifolds with transversal intersections.

We initiate a study of positive multisections of Lefschetz fibrations via positive factorizations in framed mapping class groups of surfaces. Using our methods, one can effectively capture various interesting symplectic surfaces in symplectic 4-manifolds as multisections, such as Seiberg-Witten basic classes and except…

Study transverse knots and symplectic surfaces using Seiberg-Witten monopole equations.

This paper constructs a Weinstein trisection for a surface bundle.

Existence of symplectic structure shown on Seiberg-Witten moduli space product of Riemann surfaces.

Modulo trivial exceptions, we show that smoothly nontrivial symplectic sums of symplectic 4-manifolds along surfaces of positive genus are never rational or ruled, and we enumerate each case in which they have Kodaira dimension zero (i.e., are blowups of symplectic 4-manifolds with torsion canonical class). In particul…

Study on smooth moduli spaces of branes in symplectic 4-manifolds.

We develop an approach to affine symplectic invariant geometry of Lagrangian surfaces by the method of moving frames. The fundamental invariants of elliptic Lagrangian immersions in affine symplectic four-space are derived together with their integrability equations. The invariant setup is applied to discuss the questi…

We establish various stability results for symplectic surfaces in symplectic $4-$manifolds with $b^+=1$. These results are then applied to prove the existence of representatives of Lagrangian ADE-configurations as well as to classify negative symplectic spheres in symplectic $4-$manifolds with $κ=-\infty$. This involve…

Study shows Hamiltonian diffeomorphisms form a connected component in $C^0$-topology for most symplectic rational surfaces.

Study symplectic and Hamiltonian actions on irrational ruled surfaces, proving existence and non-existence of extensions.

Geometrically interprets symplectic structure in 3-manifold triangulations.

Lectures on symplectic aspects of surface degenerations at KIAS.

Let E(1)_p denote the rational elliptic surface with a single multiple fiber f_p of multiplicity p. We construct an infinite family of homologous non-isotopic symplectic tori representing the primitive class [f_p] in E(1)_p when p>1. As a consequence, we get infinitely many non-isotopic symplectic tori in the fiber cla…

The study explores pinwheels in symplectic surfaces and non-squeezing of rational homology balls.