Journal of Mathematical Study (JMS) is a comprehensive mathematical journal published jointly by Global Science Press and Xiamen University. It publishes original research and survey papers, in English, of high scientific value in all major fields of mathematics, including pure mathematics, applied mathematics, operational research, and computational mathematics. JMS is edited by a group of leading researchers and aims to become a premier mathematical journal.
Preface: Special Issues in Honor of Professor Gang Tian
Copyright (c) 2025 Journal of Mathematical Study
Intrinsic Enumerative Mirror Symmetry: Takahashi’s Log Mirror Symmetry for ($\mathbb{P}^2$,$E$) Revisited
Let $E$ be a smooth cubic in the projective plane $\mathbb{P}^2$. Nobuyoshi Takahashi formulated a conjecture that expresses counts of rational curves of varying degree in ${\mathbb{P}^2}$ \ $E$ as the Taylor coefficients of a particular period integral of a pencil of affine plane cubics after reparametrizing the pencil using the exponential of a second period integral. The intrinsic mirror construction introduced by Mark Gross and the third author associates to a degeneration of ($\mathbb{P}^2$,$E$) a canonical wall structure from which one constructs a family of projective plane cubics that is birational to Takahashi’s pencil in its reparametrized form. By computing the period integral of the positive real locus explicitly, we find that it equals the logarithm of the product of all asymptotic wall functions. The coefficients of these asymptotic wall functions are logarithmic Gromov-Witten counts of the central fiber of the degeneration that agree with the algebraic curve counts in ($\mathbb{P}^2$,$E$) in question. We conclude that Takahashi’s conjecture is a natural consequence of intrinsic mirror symmetry. Our method generalizes to give similar results for log Calabi-Yau varieties of arbitrary dimension.
Some Remarks on Fibrations in Complex Geometry
In this article, we discuss some properties of holomorphic fibrations in the complex analytic setting.
One of the significant motivations for studying the Kähler-Ricci flow is its relation to the Analytic Minimal Model Program as initiated by Gang Tian. Carrying out this classification program requires careful analysis of the flow metric, particularly when it encounters singularities. In this note, we survey some results pertaining to the limit for the Kähler-Ricci flow.
Fukaya-Yamaguchi Conjecture in Dimension Four
Fukaya and Yamaguchi conjectured that if $M^n$ is a manifold with nonnegative sectional curvature, then the fundamental group is uniformly virtually abelian. In this short note we observe that the conjecture holds in dimensions up to four.
Zero Set of the Solution to One-Dimensional Parabolic Equation
In this paper, we investigate the one-dimensional parabolic equation. Using blow-up analysis, we prove that the zero set at each time slice is discrete and that the number of zeros decreases with respect to time. Moreover, it strictly declines at the singularity. This provides a new and simplified proof of the main results in existing literature, where the original proof relies on spectral theory. Besides, our results represent a localized version of those existing results with weaker conditions.
Quantitative Uniqueness for Mean Curvature Flow
We show that by applying a set of existing analytical arguments, a more robust effective uniqueness result for blowups can be obtained, with multiple implications following therefrom.
On a Flow Reducing Volume Within Hamiltonian Isotopy Class
We survey basic properties of the geometric flow for immersions within a Hamiltonian isotopy class and propose a definition for Type I singularities.
Higher order flow within Hamiltonian isotopy class
$p$-Laplace Equations, $p$-Superharmonic Functions, and Applications in Conformal Geometry
In this paper, we give an exposition of our recent work on nonlinear potential theory in conformal geometry. We apply nonlinear potential theory to study $p$-Laplace equations arising from conformal geometry and, in particular, the problems related to the asymptotic behavior near and the size of singularities in conformal geometry.
$W^{3,p}$ Estimate of Degenerate Complex Monge-Ampère Equations
Degenerate complex Monge-Ampère equations arise naturally in the study of geometry of singular varieties. In this paper, we prove gradient estimate and $W^{3,p}$ estimate for a class of degenerate complex Monge-Ampère equations.
Discrete Four-Order Schrödinger Equation on the Hexagonal Triangulation
In this paper, we establish a decay estimate for the discrete four-order Schrödinger equation on the hexagonal triangulation with $γ$=0. The proof is based on the uniform estimates of oscillatory integrals, as developed by Karpushkin, along with a key result by Varchenko. Our result is to show the $l^1$→$l^∞$ dispersive decay rate is $⟨t⟩^{−σ}$ for any $0<\sigma<\frac{1}{2}.$ Additionally, we provide estimates for the inhomogeneous discrete fourth-order Schrödinger equation with $γ$=0.
A Blow up Solution of the Navier-Stokes Equations with a Super Critical Forcing Term
A forced solution $v$ of the axially symmetric Navier-Stokes equation in a finite cylinder $D$ with suitable boundary condition is constructed. The forcing term, whose order of scaling is slightly worse than the critical order $−2,$ is in the mildly super critical space $L^q_tL^1_x$ for all $q>1.$ The velocity, which is smooth until its final blow up moment, is in the energy space throughout.
blow up example with supercritical forcing term
On the Fundamental Group of Non-Collapsed Ancient Ricci Flows
We show that any manifold admitting a non-collapsed, ancient Ricci flow must have finite fundamental group. This generalizes what was known for $κ$-solutions in dimensions 2, 3. We furthermore show that this fundamental group must be a quotient of the fundamental group of the regular part of any tangent flow at infinity.
A Note on the Global Existence for the Strong Solution of the Stochastic Three-Dimensional Axially Symmetric Euler Equations
This paper establishes the global existence of the pathwise solutions for the stochastic 3-D incompressible Euler equations with the axially symmetric initial data and axially symmetric additive noise.
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On a Class of Degenerate Nonlinear Elliptic Equations in Weighted Sobolev Spaces
In this paper we are interested in the existence of solutions for Dirichlet problem associated to the degenerate nonlinear elliptic equations in the setting of the weighted Sobolev spaces $W^{1,p}_ 0 (Ω,ω).$
A Regularity Result for the Incompressible Elastodynamics with a Free Interface
We consider the incompressible inviscid elastodynamics equations with a free surface and establish the regularity of solutions for elastic system. Compared with the previous result on this free boundary problem [ Gu X and Wang F, Well-posedness of the free boundary problem in incompressible elastodynamics under the mixed type stability condition, J. Math. Anal. Appl., 2020, 482(1): 123529] in space $H^3,$ we are able to establish the regularity in space $H^{2.5+δ}.$ It is achieved by reformulating the system into the Lagrangian formulation, presenting the uniform estimates for the pressure, the tangential estimates for the system, as well as the divergence and curl estimates.
Generalized Estimation of Numerical Radius for the off-Diagonal 2×2 Operator Matrices and Its Application
We give some generalized upper bounds for the numerical radius of off-diagonal 2×2 operator matrices. These inequalities are mainly based on the extension Buzano inequality and the generalized Young inequality. And our bounds refine and generalize the existing related upper bounds. Moreover, the conclusion is applied to the non-monic operator polynomials and gives a new bound for the eigenvalues of these operator polynomials.
Some $p$-Adic Hardy Operators and Their Commutators
In this paper, we will study the sharp estimates of $p$-adic weighted Hardy operator on central and noncentral weighted $p$-adic Morrey spaces. Moreover, we can obtain the sharp bound of generalized $p$-adic Hardy operator. In addition, the commutator that is generated by the generalized $p$-adic Hardy operator and the central BMO function is also bounded on $p$-adic Morrey spaces.
Common Fixed Point Results for a Pair of Generalized Nonlinear Mappings in Complete Metric Spaces
In this paper, under the asymptotically regular condition, we investigate the existence of a unique common fixed point for a pair of generalized nonlinear mappings in the framework of complete metric spaces. Additionally, we extend our analysis to the case involving two control functions. Our work generalizes some results in recent papers.
Non-Traveling Wave Solutions of (3+1)-Dimensional Variable Coefficients BLMP Equation with a Compound Technology
We investigate non-traveling wave solutions of the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation with time-dependent coefficients, which describes the propagation of waves in incompressible fluids. We creatively combine the extended three-wave method with the generalized variable separation method and successfully obtain sixty exact non-traveling solutions including kink-like solutions, singular solitary wave-like solutions, periodic solitary wave-like solutions, periodic kink-like solutions, periodic cross-kink-like waves, homoclinic breather wave-like solutions and so on. The variable coefficients and arbitrary functions in the obtained solutions are easy to exhibit abundant soliton structures, which may be of great significance for explaining some practical physical phenomena. By contour plots, 2D plots, and 3D plots, we analyze the dynamic characteristics of periodic cross-kink-like solution, singular solitary wave-like solution, homoclinic breather wave-like solution. Additionally, we show changes of solutions under different tails to illustrate the influence of tails on solutions.
A Note on the Determinant of a Special Class of $Q$-Walk Matrices
For a graph $G$ of order $n,$ its $Q$-walk matrix is defined by $W_Q(G) = [e,Qe,···,Q^{n−1}e],$ where $Q$ is the signless Laplacian matrix of $G$ and $e$ denotes the all-one column vector. Let $G \circ P_k$ represent the rooted product graph of $G$ and a path $P_k.$ In this note, we establish the relationship between determinants of $W_Q(G)$ and $W_Q(G \circ P_k )$ for $k=2,3.$
Characterizations of Umbilical Hypersurfaces by Partially Overdetermined Problems in Space Forms
In this paper, we characterize the rigidity of umbilical hypersurfaces by a Serrin-type partially overdetermined problem in space forms, which generalizes the similar results in Euclidean half-space and Euclidean half-ball. Guo-Xia first obtained these rigidity results when the Robin boundary condition on the support hypersurface is homogeneous, at this time the target umbilical hypersurface has orthogonal contact angle with the support. However, in this paper we can obtain any contact angle $θ ∈ (0,π)$ by changing the Robin boundary condition to be inhomogeneous.
On Conical Asymptotically Flat Manifolds
We prove a conjecture of Petrunin and Tuschmann on the non-existence of asymptotically flat 4-manifolds asymptotic to the half plane. We also survey recent progress and questions concerning gravitational instantons, which serve as our motivation for studying this question.
An Alternative Proof for the Upper Bound of Curvature Integral on Manifolds with Lower Sectional Curvature Bound
Petrunin proved that the integral of scalar curvature in a unit ball is bounded from above in terms of the dimension of the manifold and the lower bound of the sectional curvature. In this paper, we give an alternative proof for this result. The main difference between this proof and Petrunin’s original proof is that we construct a stratified finite covering and apply it directly to the given manifold, rather than arguing by contradiction for a sequence of manifolds, which satisfy some technical lifting properties.
Moduli Spaces of Metrics with Lower or Pinched Curvature Bounds and Their Compactifications
We present a metric approach to the study of moduli spaces of metrics with certain curvature bounds and suitable other geometric constraints, and their compactifications. This is accompanied by a deeper discussion of related results, questions and problems in the realm of positive and positively pinched sectional as well as Ricci curvature. Regarding the latter two topics, we place special emphasis on corresponding works and contributions of Rong X and his collaborators.
Rigidity for Einstein Manifolds under Bounded Covering Geometry
In this note, we prove three rigidity results for Einstein manifolds with bounded covering geometry. (1) An almost flat manifold $(M,g)$ must be flat if it is Einstein, i.e. ${\rm Ric}_g =λg$ for some real number $λ.$ (2) A compact Einstein manifold with a non-vanishing and almost maximal volume entropy is hyperbolic. (3) A compact Einstein manifold admitting a uniform local rewinding almost maximal volume is isometric to a space form.
Pair-of-Pants Decompositions of 4-Manifolds Diffeomorphic to General Type Hypersurfaces
In this paper, we show that a smooth 4-manifold diffeomorphic to a complex hypersurface in $\mathbb{CP}^3$ of degree $d≥5$ can be decomposed as the union of $d(d−4)^2$ copies of 4-dimensional pair-of-pants and certain subsets of ${\rm K3}$ surfaces.
Volume Gap Theorems for Ricci Nonnegative Metrics and Einstein Metrics
In this note, we prove some gap theorems of asymptotic volume ratio for Ricci nonnegative metrics, and gap theorems of volume for Einstein metrics.
In this survey paper, we discuss various examples of Ricci solitons and their constructions. Some open questions related to the rigidity and classification of Ricci solitons will be also discussed through those examples.