arXiv subjects

Jingrui Cheng

Publications and source records attributed to Jingrui Cheng.

14 recordsLinked to original sources

arXiv2026

An improvement of regularity result for pseudo Calabi flow

In this paper, we observe that if the initial data of pseudo Calabi flow has volume form $C^0$ close to a smooth one, then the flow is immediately smooth for $t>0$. As an application, we show that if the initial data has volume form $C^0$ close to that of a cscK metric, then the pseudo Calabi flow exists for $t\in (0,+\infty)$. We also prove similar improvement of regularity and long time existence result for pseudo Calabi flow on a Fano manifold when the volume form is bounded and the class is close to $c_1(M)$.

math.DG↗

arXiv2025

Viscosity solution to complex Hessian quotient equations

In this paper, we prove the existence of viscosity solutions to complex Hessian equations on compact Hermitian manifolds, assuming the existence of a strict subsolution in the viscosity sense. The results cover the complex Hessian quotient equations. This generalized our previous results where the equation needs to satisfy a determinant domination condition.

math.AP↗

arXiv2024

Viscosity solution to complex Hessian equations on compact Hermitian manifolds

We prove the existence of viscosity solutions to complex Hessian equations on a compact Hermitian manifold that satisfy a determinant domination condition. This viscosity solution is shown to be unique when the right hand is strictly monotone increasing in terms of the solution. When the right hand side does not depend on the solution, we reduces it to the strict monotonicity of the solvability constant.

math.AP↗

arXiv2023

Interior $W^{2,p}$ estimate for small perturbations to the complex Monge-Ampere equation

Let $w_0$ be a bounded, $C^3$, strictly plurisubharmonic function defined on $B_1\subset \mathbb{C}^n$. Then $w_0$ has a neighborhood in $L^{\infty}(B_1)$ with the following property: for any continuous, plurisubharmonic function $u$ in this neighborhood solving $1-\eps \le MA(u)\le 1+\eps$, one has $u\in W^{2,p}(B_{\frac{1}{2}})$, as long as $\eps>0$ is small enough depending only on $n$ and $p$. This partially generalizes Caffarelli's interior $W^{2,p}$ estimates for real Monge-Ampere to the complex version.

math.AP↗

arXiv2022

Regularization Of $m$-subharmonic Functions And HÖlder Continuity

We use sup-convolution to find upper approximations of a bounded $m$-subharmonic function on a compact Kähler manifold with nonnegative holomorphic bisectional curvature. As an application, we show the Hölder continuity of solutions to $σ_m$ equation when the right hand side is in $L^p$, $p>\frac{n}{m}$. All these results generalize to more general complex Hessian equations.

math.AP↗

arXiv2022

Gradient estimate for complex Monge-Ampere equation with continuous right hand side

In this note, we consider complex Monge-Ampere equation posed on a compact Kähler manifold. We show how to get $L^p$($p<\infty$) and $L^{\infty}$ estimate for the gradient of the solution in terms of the continuity of the right hand side.

math.AP↗

arXiv2022

The $L^{\infty}$ estimates for parabolic complex Monge-Ampere and Hessian equations

In this paper, we consider a version of parabolic complex Monge-Ampere equations, and use a PDE approach similar to Phong et al to establish $L^{\infty}$ and Hölder estimates. We also generalize the $L^{\infty}$ estimates to parabolic Hessian equations.

math.AP↗

arXiv2018

On the constant scalar curvature Kähler metrics, general automorphism group

In this paper, we derive estimates for scalar curvature type equations with more singular right hand side. As an application, we prove Donaldson's conjecture on the equivalence between geodesic stability and existence of cscK when $Aut_0(M,J)\neq0$. Moreover, we also show that when $Aut_0(M,J)\neq0$, the properness of $K$-energy with respect to a suitably defined distance implies the existence of cscK.

math.DG↗

arXiv2018

On the constant scalar curvature Kähler metrics, existence results

In this paper, we generalize our apriori estimates on cscK(constant scalar curvature Kähler) metric equation to more general scalar curvature type equations (e.g., twisted cscK metric equation). As applications, under the assumption that the automorphism group is discrete, we prove the celebrated Donaldson's conjecture that the non-existence of cscK metric is equivalent to the existence of a destabilized geodesic ray where the $K$-energy is non-increasing. Moreover, we prove that the properness of $K$-energy in terms of $L^1$ geodesic distance $d_1$ in the space of Kähler potentials implies the existence of cscK metric. Finally, we prove that weak minimizers of the $K$-energy in $(\mathcal{E}^1, d_1)$ are smooth.

math.DG↗

arXiv2017

On the constant scalar curvature Kähler metrics, apriori estimates

In this paper, we derive apriori estimates for constant scalar curvature Kähler metrics on a compact Kähler manifold. We show that higher order derivatives can be estimated in terms of a $C^0$ bound for the Kähler potential. We also discuss some local versions of these estimates which can be of independent interest.

math.DG↗

arXiv2016

A rigorous treatment of moist convection in a single column

We study a single column model of moist convection in the atmosphere. We state the conditions for it to represent a stable steady state. We then evolve the column by subjecting it to an upward displacement which can release instability, leading to a time dependent sequence of stable steady states. We propose a definition of measure valued solution to describe the time dependence and prove its existence.

math.AP↗

arXiv2016

Semi-geostrophic System with Variable Coriolis parameter

We prove short time existence and uniqueness of smooth solutions ( in $C^{k+2,α}$ with $k\geq 2$) to the 2-D semi-geostrophic system and semi-geostrophic shallow water system with variable Coriolis parameter $f$ and periodic boundary conditions, under the natural convexity condition on the initial data. The dual space used in analysis of the semi-geostrophic system with constant $f$ does not exist for the variable Coriolis parameter case, and we develop a time-stepping procedure to overcome this difficulty.

math.AP↗

arXiv2015

Semigeostrophic equations in physical space with free upper boundary

We define various notions of Lagrangian solution in physical space for 3-d incompressible geostrophic system with free upper boudary under different conditions for initial data,then prove their existence via the minimization with respect to a geostrophic functional.As a byproduct of our proof,we obtain the existence of measure-valued dual space solutions when the initial measure $ν_0\in\mathcal{P}_2(\mathbf{R}^3)$ and is supported on $\{-\frac{1}δ\leq x_3\leq-δ\}$

math.AP↗

arXiv2015

Long time existence of smooth solutions to semigeostrophic equations on a torus

In this work,we show the long time existence of smooth solu- tions to semigeostrophic equations on a torus when the initial dual density is bounded between two positive constants and smooth.The key ingredient is a more precise estimate on C 2 norm of Monge-Amp`re equations established e in [1] .

math.AP↗