arXiv subjects

Jianyu Ou

Publications and source records attributed to Jianyu Ou.

9 recordsLinked to original sources

arXiv2026

The dimension of polynomial growth holomorphic functions and forms on gradient Kähler Ricci shrinkers

We study polynomial growth holomorphic functions and forms on complete gradient shrinking Ricci solitons. By relating to the spectral data of the $f$-Laplacian, we show that the dimension of the space of polynomial growth holomorphic functions or holomorphic $(p,0)$-forms are finite. In particular, a sharp dimension estimate for the space of linear growth holomorphic functions was obtained. Under some additional curvature assumption, we prove an almost sharp estimate for the frequency of polynomial growth holomorphic functions, which was used to obtain dimension upper bound as a power function of the polynomial order.

math.DG↗

arXiv2025

Dimension estimate and existence of holomorphic sections with polynomial growth on gradient Kähler Ricci shrinkers

We prove an upper bound for the dimension of the linear space of holomorphic functions with polynomial growth on gradient Kähler Ricci shrinkers with bounded curvature. The upper bound is given as a power function of the growth rate. Similar results hold for holomorphic $(p, 0)-$forms, and holomorphic sections of the pluri-anticanonical line bundle $K_M^{-q}$. We also prove the existence of holomorphic sections of $K_M^{-q}$ with polynomial growth when the Kähler Ricci shrinker is asymptotically conical, provided $q$ is sufficiently large; as an application, we show that the Kodaira map constructed using such sections is a holomorphic embbedding into a complex projective space.

math.DG↗

arXiv2025

Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

Let $(M, g, f)$ be a $5$-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+\nabla^2f= λg$, where $\text{Ric}$ is the Ricci tensor and $\nabla^2f$ is the Hessian of the potential function $f$. We prove that it is a finite quotient of $\mathbb{R}^2\times \mathbb{S}^3$ if $M$ has constant scalar curvature $R=3 λ$.

math.DG↗

arXiv2025

Rigidity of Five-Dimensional shrinking gradient Ricci solitons

Suppose $(M, g, f)$ is a 5-dimensional complete shrinking gradient Ricci soliton with $R=1$. If it has bounded curvature, we prove that it is a finite quotient of $\mathbb{R}^3\times \mathbb{S}^2$.

math.DG↗

arXiv2024

Some rigidity results on shrinking gradient Ricci soliton

Suppose $(M^n, g, f)$ is a complete shrinking gradient Ricci soliton. We give several rigidity results under some natural conditions, generalizing the results in \cite{Petersen-Wylie,Guan-Lu-Xu}. Using maximum principle, we prove that shrinking gradient Ricci soliton with constant scalar curvature $R=1$ is isometric to a finite quotient of $\mathbb{R}^2\times \mathbb{S}^2$, giving a new proof of the main results of Cheng-Zhou \cite{Cheng-Zhou}.

math.DG↗

arXiv2023

Three Balls Theorem for Eigenfunctions of Dirac Operator in Clifford Analysis

In this paper we establish the three balls theorem for functions $u$ satisfying $Du=λu$ in Clifford analysis, where $D$ is the Dirac operator. As an application, we generalize Hadamard's three circles theorem to monogenic function in $\mathbb R^{n+1}.$

math.CV↗

arXiv2022

Liouville Theorem on Ricci shrinkers with constant scalar curvature and its application

In this paper we consider harmonic functions on gradient shrinking Ricci solitons with constant scalar curvature. A Liouville theorem is proved without using gradient estimate : any bounded harmonic function is constant on gradient shrinking Ricci solitons with constant scalar curvature. As an application, we show that the space of harmonic functions with polynomial growth has finite dimension.

math.DG↗

arXiv2019

Uncertainty Principle and its rigidity on complete gradient shrinking Ricci solitons

We prove rigidity theorems for shrinking gradient Ricci solitons supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in $\mathbb{R}^n$. In addtion, we partially give analogous rigidity results of the Caffarelli-Kohn-Nirenberg inequalities on shrinking Ricci solitons.

math.DG↗

arXiv2017

Symplectic aspects of polar actions

An isometric compact group action $G \times (M,g) \rightarrow (M,g)$ is called polar if there exists a closed embedded submanifold $Σ\subseteq M$ which meets all orbits orthogonally. Let $Π$ be the associated generalized Weyl group. We study the properties of the lifting action $G$ on the cotangent bundle $T^*M$. In particular, we show that the restriction map $(C^{\infty}(T^*M))^G \rightarrow (C^{\infty}(T^* Σ))^Π$ is a surjective homomorphism of Poisson algebras. As a corollary, the singular symplectic reductions $T^*M // G $ and $T^* Σ// Π$ are isomorphic as stratified symplectic spaces, which gives a partial answer to a conjecture of Lerman, Montgomery and Sjamaar.

math.DG↗