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Zuyi Zhang

Publications and source records attributed to Zuyi Zhang.

10 recordsLinked to original sources
arXiv2026

The existence criterion of holomorphic discs for higher $A_\infty$ operations via minimal discs

The main theorem of the paper provides an existence criterion of holomorphic discs for higher $A_\infty$ operations. The key step is to show that if a minimal disc in a Kähler manifold with boundary in a sequence of Lagrangian submanifolds intersecting transversely such that its partial Maslov indices are either all no less than $1$ or all no larger than $-1$, then there is a holomorphic disc with the same image as this minimal disc. As a by-product, we show that all minimal discs in $\C\mathrm{P}^m$ with boundary on $\R\mathrm{P}^m$ are holomorphic.

math.SG↗
arXiv2026

On the $\mathcal{D}^+_J$ operator on higher-dimensional almost Kähler manifolds

In this paper, we introduce $\mathcal{D}^+_J$, a generalization of $\partial\bar{\partial}$ operator on higher dimensional almost Kähler manifolds. Using the $\mathcal{D}^+_J$ operator, we investigate the $\bar{\partial}$-problem in almost Kähler geometry and explore the generalized Monge-Ampère equation on almost Kähler manifolds. We establish a uniqueness up to the addition of a constant and local existence theorem for this equation. At last, we find an elliptical system for $\mathcal{D}^+_J$ operator. As an application, we reorganize the result of Tosatti-Weinkove-Yau in \cite{TWY}.

math.DG↗
arXiv2026

Energy functionals on almost Kähler manifolds: I

In this paper, we consider the Donaldson gauge functional and the twisted Aubin functionals on almost Kähler manifolds. As in Kähler geometry, we generalize the inequality between Aubin functionals.

math.DG↗
arXiv2025

On a generalized Monge-Ampère equation on closed almost Kähler surfaces

We show the existence and uniqueness of solutions to a generalized Monge-Ampère equation on closed almost Kähler surfaces, where the equation depends only on the underlying almost Kähler structure. As an application, we prove Donaldson's conjecture for tamed almost complex 4-manifolds.

math.DG↗
arXiv2025

Uniqueness of holomorphic quilts lifted from holomorphic bigons on surfaces

In the author's previous paper, the author constructed holomorphic quilts from the bigons of the Lagrangian Floer chain group after performing Lagrangian composition. This paper proves the uniqueness of such holomorphic quilts. As a consequence, it provides a combinatorial method for computing the boundary map of immersed Lagrangian Floer chain groups when the symplectic manifolds are closed surfaces. One outcome is the construction of many examples exhibiting figure eight bubbling, which also confirms a conjecture of Cazassus Herald Kirk Kotelskiy.

math.SG↗
arXiv2024

Construction of holomorphic quilts in Cartesian product of closed surfaces

In this article, we modify the proof of holomorphic quilts from Wehrheim and Woodward in \cite{wehrheim2009floer} to construct a specific type of immersed holomorphic quilt, where the symplectic manifolds are closed surfaces. The application is to compare Lagrangian Floer theory with quilted Lagrangian Floer theory, as they relate through Lagrangian correspondence. A potential example is provided to support Bottman and Wehrheim's conjecture \cite{bottman2018gromov} regarding the isomorphism between Lagrangian Floer homology and quilted Lagrangian Floer homology after twisting by bounding cochains.

math.SG↗
arXiv2024

Singularities of Lagrangian Immersions and its Applications in Lagrangian Floer Theory

In this article, we study the singularities of Lagrangian immersions into Cartesian product of surfaces. After applying a Hamiltonian isotopy in the Weinstein tubular neighbourhood of the Lagrangian immersion, the singular points of the Lagrangian immersion can be expressed locally as fold points with finitely many cusp points. This result has applications in comparing two Lagrangian Floer complexes associated to curves on surfaces related by a certain Lagrangian correspondence and the quilt Floer complex induced by these three Lagrangian immersions.

math.GT↗