arXiv subjects
Publications and source records attributed to Gao Chen.
Xiuxiong Chen proved the existence of $C^{1, \bar 1}$ geodesics in the space of K\"{a}hler potentials and the convexity of the $J$-functional along $C^{1, \bar 1}$ geodesics. Analogous results were proved by Collins and Yau for the hypercritical Leung--Yau--Zaslow equation. In this paper, we study a similar picture for generalised Monge--Amp\`{e}re equations associated with two right-Noetherian polynomials in proper position.
We prove a Nakai-Moishezon-type criterion for complex Hessian-type equations on projective manifolds whose associated degree-$n$ polynomials are strongly strictly right-Noetherian. For strictly right-Noetherian polynomials of arbitrary degree, we prove a uniform Nakai-Moishezon-type criterion. This class includes the complex Hessian and Hessian quotient equations.
In this paper, we study the ellipticity of the vector bundle versions of the Monge-Amp\`ere, $J$, dHYM and $\sigma_{k}$-equations at a point. These are nonlinear geometric partial differential equations defined on a holomorphic vector bundle over a compact K\"ahler manifold. We show that when both the dimension of the manifold and the rank of the bundle are greater than or equal to three, these equations do not preserve ellipticity along continuity paths in the connected component of the trivial solution. However, the $\sigma_{2}$-equation does preserve ellipticity along continuity paths.
In this paper, we compute the expansion of some harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.
This study proposes an innovative evaluation method based on large language models (LLMs) specifically designed to measure the digital transformation (DT) process of enterprises. By analyzing the annual reports of 4407 companies listed on the New York Stock Exchange and Nasdaq from 2005 to 2022, a comprehensive set of DT indicators was constructed. The findings revealed that DT significantly improves a company's financial performance, however, different digital technologies exhibit varying effects on financial performance. Specifically, blockchain technology has a relatively limited positive impact on financial performance. In addition, this study further discovered that DT can promote the growth of financial performance by enhancing operational efficiency and reducing costs. This study provides a novel DT evaluation tool for the academic community, while also expanding the application scope of generative artificial intelligence technology in economic research.
There are two known classes of gravitational instantons with quadratic volume growth at infinity, known as type ALG and ALG$^*$. Gravitational instantons of type ALG were previously classified by Chen-Chen. In this paper, we prove a classification theorem for ALG$^*$ gravitational instantons. We determine the topology and prove existence of "uniform" coordinates at infinity for both ALG and ALG$^*$ gravitational instantons. We also prove a result regarding the relationship between ALG gravitational instantons of order $\mathfrak{n}$ and those of order $2$.
We study the asymptotic behavior of Hitchin's hyperkähler metric on the moduli space of rank two irregular Higgs bundles over $\mathbb{C}P^1$. Along a generic curve, we prove that the Hitchin metric is asymptotic to the semiflat metric at an arbitrary polynomial order. When there are no weakly parabolic singularities, the rate is exponential. In the case of four-dimensional moduli spaces, we prove that the semiflat metric is asymptotic to an ALG/ALG$^\ast$ model metric.
We develop a Fredholm Theory for the Hodge Laplacian in weighted spaces on ALG$^*$ manifolds in dimension four. We then give several applications of this theory. First, we show the existence of harmonic functions with prescribed asymptotics at infinity. A corollary of this is a non-existence result for ALG$^*$ manifolds with non-negative Ricci curvature having group $Γ= \{e\}$ at infinity. Next, we prove a Hodge decomposition for the first de Rham cohomology group of an ALG$^*$ manifold. A corollary of this is vanishing of the first betti number for any ALG$^*$ manifold with non-negative Ricci curvature. Another application of our analysis is to determine the optimal order of ALG$^*$ gravitational instantons.
These are the lecture notes for the Morningside Center of Mathematics Geometry Summer School on August 15-20, 2022. These lectures sketch the results by Yau, Demailly-Paun, the author, and Datar-Pingali about generalized Monge-Ampère equations and subvarieties and aim to use these results to study the Hodge conjecture.
We prove Torelli-type uniqueness theorems for both ALG$^*$ gravitational instantons and ALG gravitational instantons which are of order $2$. That is, the periods uniquely characterize these types of gravitational instantons up to diffeomorphism. We define a period mapping $\mathscr{P}$, which we show is surjective in the ALG cases, and open in the ALG$^*$ cases. We also construct some new degenerations of hyperkähler metrics on the K3 surface which exhibit bubbling of ALG$^*$ gravitational instantons.
Aerosol effects on micro-/macro-physical properties of marine stratocumulus clouds over the Western North Atlantic Ocean (WNAO) are investigated using in-situ measurements and large-eddy simulations (LES) for two cold air outbreak (CAO) cases (February 28 and March 1, 2020) during the Aerosol Cloud meTeorology Interactions oVer the western ATlantic Experiment (ACTIVATE). The LES is able to reproduce the vertical profiles of liquid water content (LWC), effective radius r_eff and the cloud droplet number concentration Nc from fast cloud droplet probe (FCDP) in-situ measurements for both cases. Furthermore, we show that aerosols affect cloud properties (Nc, r_eff, and LWC) via the prescribed bulk hygroscopicity of aerosols and aerosol size distributions characteristics. Nc, r_eff, and liquid water path (LWP) are positively correlated to the bulk hygroscopicity of aerosols and aerosol number concentration (Na) while cloud fractional cover (CFC) is insensitive to the bulk hygroscopicity of aerosols and aerosol size distributions for the two cases. The changes to aerosol size distribution (number concentration, width, and the geometrical diameter) allow us to disentangle aerosol effects on cloud properties from the meteorological effects. We also use the LES results to evaluate cloud properties from two reanalysis products, ERA5 and MERRA-2. Comparing to LES, the ERA5 reanalysis is able to capture the time evolution of LWP and total cloud coverage within the study domain during both CAO cases while MERRA-2 underestimates them.
Large-eddy simulation (LES) is able to capture key boundary-layer (BL) turbulence and cloud processes. Yet, large-scale forcing and surface turbulent fluxes of sensible and latent heat are often poorly prescribed for LES simulations. We derive these quantities from measurements and reanalysis obtained for two cold air outbreak (CAO) events during Phase I of the Aerosol Cloud meTeorology Interactions oVer the western ATlantic Experiment (ACTIVATE) in February-March 2020. We study the two contrasting CAO cases by performing LES and test the sensitivity of BL structure and clouds to large-scale forcings and turbulent heat fluxes. Profiles of atmospheric state and large-scale divergence and surface turbulent heat fluxes obtained from the reanalysis data ERA5 agree reasonably well with those derived from ACTIVATE field measurements for both cases at the sampling time and location. Therefore, we adopt the time evolving heat fluxes, wind and advective tendencies profiles from ERA5 reanalysis data to drive the LES. We find that large-scale thermodynamic advective tendencies and wind relaxations are important for the LES to capture the evolving observed BL meteorological states characterized by the hourly ERA5 reanalysis data and validated by the observations. We show that the divergence (or vertical velocity) is important in regulating the BL growth driven by surface heat fluxes in LES simulations. The evolution of liquid water path is largely affected by the evolution of surface heat fluxes. The liquid water path simulated in LES agrees reasonably well with the ACTIVATE measurements.This study paves the path to investigate aerosol-cloud-meteorology interactions using LES informed and evaluated by ACTIVATE field measurements.
For any elliptic K3 surface $\mathfrak{F}: \mathcal{K} \rightarrow \mathbb{P}^1$, we construct a family of collapsing Ricci-flat Kähler metrics such that curvatures are uniformly bounded away from singular fibers, and which Gromov-Hausdorff limit to $\mathbb{P}^1$ equipped with the McLean metric. There are well-known examples of this type of collapsing, but the key point of our construction is that we can additionally give a precise description of the metric degeneration near each type of singular fiber, without any restriction on the types of singular fibers.
In this paper, we prove that for any Kähler metrics $ω_0$ and $χ$ on $M$, there exists $ω_φ=ω_0+\sqrt{-1}\partial\bar\partialφ>0$ satisfying the J-equation $\mathrm{tr}_{ω_φ}χ=c$ if and only if $(M,[ω_0],[χ])$ is uniformly J-stable. As a corollary, we can find many constant scalar curvature Kähler metrics with $c_1<0$. Using the same method, we also prove a similar result for the deformed Hermitian-Yang-Mills equation when the angle is in $(\frac{nπ}{2}-\fracπ{4},\frac{nπ}{2})$.
The goal of this paper is the construction of a compact manifold with G$_2$ holonomy and nodal singularities along circles using twisted connected sum method. This paper finds matching building blocks by solving the Calabi conjecture on certain asymptotically cylindrical manifolds with nodal singularities. However, by comparison to the untwisted connected sum case, it turns out that the obstruction space for the singular twisted connected sum construction is infinite dimensional. By analyzing the obstruction term, there are strong evidences that the obstruction may be resolved if a further gluing is performed in order to get a compact manifold with G$_2$ holonomy and isolated conical singularities with link $\mathbb{S}^3\times\mathbb{S}^3$.
In this paper, we study gravitational instantons (i.e., complete hyperkäler 4-manifolds with faster than quadratic curvature decay). We prove three main theorems: 1.Any gravitational instanton must have known end----ALE, ALF, ALG or ALH. 2.In ALG and ALH-non-splitting cases, it must be biholomorphic to a compact complex elliptic surface minus a divisor. Thus, we confirm a long-standing question of Yau in ALG and ALH cases. 3.In ALF-D_k case, it must have an O(4)-multiplet.
In this paper, we extend Lotay-Wei's Shi-type estimate from Laplacian flow to more general flows of G$_2$ structures including the modified Laplacian co-flow. Then we prove a version of $κ$-non-collapsing theorem. We will use both of them to study finite time singularities of general flows of G$_2$ structures.
In this paper, a metric with G$_2$ holonomy and slow rate of convergence to the cone metric is constructed on a ball inside the cone over the flag manifold.