The $2$-nd Hessian type equation on almost Hermitian manifolds

紧致 Hermitian 流形上复 Hessian 方程的粘性解

Viscosity solution to complex Hessian equations on compact Hermitian manifolds

本文证明了紧致 Hermitian 流形上满足行列式支配条件的复 Hessian 方程粘性解的存在性与唯一性，并将右端项不依赖解的情形归结为可解常数的严格单调性。

The Dirichlet problem for a complex Hessian equation on compact Hermitian manifolds with boundary

We solve the classical Dirichlet problem for a general complex Hessian equation on a small ball in $\bC^n$. Then, we show that there is a continuous solution, in pluripotential theory sense, to the Dirichlet problem on compact Hermitian manifolds with boundary that equipped locally conformal Kähler metrics, provided a subsolution.

Sharp Global Bounds for the Hessian on Pseudo-Hermitian Manifolds

We find sharp bounds for the norm inequality on a Pseudo-hermitian manifold, where the L^2 norm of all second derivatives of the function involving horizontal derivatives is controlled by the L^2 norm of the sub-Laplacian. Perturbation allows us to get a-priori bounds for solutions to sub-elliptic PDE in non-divergence form with bounded measurable coefficients. The method of proof is through a Bochner technique. The Heisenberg group is seen to be en extremal manifold for our inequality in the class of manifolds whose Ricci curvature is non-negative.

复Hessian商方程的粘性解

Viscosity solution to complex Hessian quotient equations

本文证明了紧致Hermitian流形上复Hessian方程在粘性意义下存在严格下解时粘性解的存在性，结果涵盖复Hessian商方程，推广了以往需满足行列式支配条件的研究。

关于背景Hermitian形式的复Hessian测度

Complex Hessian measures with respect to a background Hermitian form

本文研究了关于背景Hermitian形式的复Hessian测度，发展了相关势理论，并扩展了复Hessian方程的研究结果。

埃尔米特流形上复赫斯方程的正则化与赫尔德连续性

Regularization and Hölder continuity for complex Hessian equations on Hermitian manifolds

针对紧埃尔米特流形上的复赫斯方程，提出$\u0393$-容许函数的定量正则化方法，证明右端项$L^p$（$p>\frac{n}{m}$）的复$m$-赫斯方程解的赫尔德连续性，并将结果推广至满足特定结构条件的一般类方程。

On uniform estimate of complex elliptic equations on closed Hermitian manifolds

In this paper, we study Hessian equations and complex quotient equations on closed Hermitian manifolds. We directly derive the uniform estimate for the admissible solution. As an application, we solve general Hessian equations on closed Kähler manifolds.

紧致Hermitian流形上复Hessian方程的一致\(L^\infty\)估计

Uniform $L^\infty$ estimates for complex hessian equations on compact Hermitian manifolds

研究紧致Hermitian流形上复Hessian方程，通过开发多重势方法，结合弱比较原理、容量理论及非线性迭代方案，在\(f \in L^p\)且\(f \ge 0\)（\(p>1\)）时给出一致\(L^\infty\)估计，并得到相关弱解结果，扩展了该理论。

Fully nonlinear elliptic equations with gradient terms on compact almost Hermitian manifolds

In this paper, we establish second order estimates for a general class of fully nonlinear equations with linear gradient terms on compact almost Hermitian manifolds. As an application, we first prove the existence of solutions for the Monge-Ampère equation with linear gradient terms for $(n-1)$-plurisubharmonic functions, originated from Gaudochon conjecture, in the almost Hermitian setting. Second, we solve the Monge-Ampère equation and Hessian equations with linear gradient terms. Third, we give the $C^{\infty}$ a priori estimates for the deformed Hermitian-Yang-Mills equation with supercrit…

Parabolic complex Monge-Ampère equations on compact Hermitian manifolds

We prove the long-time existence and convergence of solutions to a general class of parabolic equations, not necessarily concave in the Hessian of the unknown function, on a compact Hermitian manifold. The limiting function is identified as the solution of an elliptic complex Monge-Ampère equation.

Fully non-linear elliptic equations on compact almost Hermitian manifolds

In this paper, we establish a priori estimates for solutions of a general class of fully non-linear equations on compact almost Hermitian manifolds. As an application, we solve the complex Hessian equation and the Monge--Ampère equation for $(n-1)$-plurisubharmonic equations in the almost Hermitian setting.

The initial value problem for a third-order dispersive flow into compact almost Hermitian manifolds

We present a time-local existence theorem of the initial value problem for a third-order dispersive evolution equation for open curves on compact almost Hermitian manifolds arising in the geometric analysis of vortex filaments. This equation causes the so-called loss of one-derivative since the target manifold is not supposed to be a Kähler manifold. We overcome this difficulty by using a gauge transformation of a multiplier on the pull-back bundle to eliminate the bad first order terms essentially.

厄米流形上完全非线性椭圆方程的稳定性与梯度估计

Stability and gradient estimates for fully nonlinear elliptic equations on hermitian manifolds

本文针对紧致厄米流形上的凹完全非线性椭圆方程，提出一种不依赖Hou-Ma-Wu估计和Liouville定理的梯度估计直接证明方法，并结合稳定性估计与比较论证，同时给出紧致Kähler流形上复Hessian估计的独立证明。

Hessian estimate for semiconvex solutions to the sigma-2 equation

We derive a priori interior Hessian estimates for semiconvex solutions to the sigma-2 equation. An elusive Jacobi inequality, a transformation rule under the Legendre-Lewy transform, and a mean value inequality for the still nonuniformly elliptic equation without area structure are the key to our arguments. Previously, this result was known for almost convex solutions.

二阶估计用于退化复k-Hessian和克里斯托费尔-闵科夫斯基方程

Second-order estimates for degenerate complex $k$-Hessian and Christoffel-Minkowski equations

本文在不同退化条件下研究复k-Hessian和克里斯托费尔-闵科夫斯基方程的正则性，证明了当k≥5时复k-Hessian方程具有近C^{1,1}正则性，而克里斯托费尔-闵科夫斯基方程具有C^{1,1}正则性。

Viscosity solutions to complex Hessian equations

We study viscosity solutions to complex hessian equations. In the local case, we consider $Ω$ a bounded domain in $\mathbb{C}^n,$ $β$ the standard Kähler form in $\mathcal{C}^n$ and $1\leq m\leq n.$ Under some suitable conditions on $F, g$, we prove that the equation $(dd^c φ)^m\wedgeβ^{n-m}=F(x,φ)β^n,\ \f=g$ on $\pO$ admits a unique viscosity solution modulo the existence of subsolution and supersolution. If moreover, the datum are Hölder continuous then so is the solution. In the global case, let $(X,ω)$ be a compact hermitian homogeneous manifold where $ω$ is an invariant hermitian metric (…

The deformed Hermitian-Yang-Mills equation on almost Hermitian manifolds

In this paper, we consider the deformed Hermitian-Yang-Mills equation on closed almost Hermitian manifolds. In the case of hypercritical phase, we derive a priori estimates under the existence of an admissible $\mathcal{C}$-subsolution. As an application, we prove the existence of solutions for the deformed Hermitian-Yang-Mills equation under the condition of existence of a supersolution.

复Hessian流的Cauchy-Dirichlet问题：从先验估计到多复势理论

The Cauchy-Dirichlet Problem for Complex Hessian Flows: From A Priori Estimates to Pluripotential Theory

研究Hermitian流形和有界严格m-伪凸域上抛物复Hessian方程的Cauchy-Dirichlet问题，通过先验估计建立光滑解的存在唯一性，并发展退化右端项的势理论框架。

关于紧致殆 Hermitian 流形上完全非线性椭圆方程的一个注记

A remark for fully non-linear elliptic equations on compact almost Hermitian manifolds

本文推广 sub-slope 定义至殆 Hermitian 流形，证明一类完全非线性椭圆方程解的存在性，并应用于复 Hessian 商方程和形变 Hermitian-Yang-Mills 方程。

完全非线性椭圆方程的上斜率和子解

Sup-slopes and sub-solutions for fully nonlinear elliptic equations

本文通过引入解析斜率不变量，给出闭黎曼或埃尔米特流形上一般完全非线性椭圆方程可解的充要条件，解决了Li-Urbas公开问题，并应用于求解非常数J-方程。

迈向火星II的Floer理论——Floer Hessian场几乎可延拓

Towards a Floer theory for Mars II -- Floer Hessian field almost extends

本文证明非局部哈密顿作用量泛函的线性化梯度流是Fredholm算子，并建立非局部椭圆正则性结果，为变分检测扭曲Zeeman系统的碰撞周期轨道奠定基础。

Monge-Ampère type equations on almost Hermitian manifolds

In this paper, we consider the Monge-Ampère type equations on compact almost Hermitian manifolds. We derive $C^{\infty}$ a priori estimates under the existence of an admissible $\mathcal{C}$-subsolution. Finally, we obtain an existence result if there exists an admissible supersolution.

The parabolic Monge-Ampère equation on compact almost Hermitian manifolds

We prove the long time existence and uniqueness of solutions to the parabolic Monge-Ampère equation on compact almost Hermitian manifolds. We also show that the normalization of solution converges to a smooth function in $C^{\infty}$ topology as $t\rightarrow\infty$. Up to scaling, the limit function is a solution of the Monge-Ampère equation. This gives a parabolic proof of existence of solutions to the Monge-Ampère equation on almost Hermitian manifolds.

椭圆与抛物型和 Hessian 方程的 Neumann 问题

Neumann Problems for Elliptic and Parabolic Sum Hessian Equations

本文研究与 Hessian 方程的 Neumann 边值问题，在几乎凸且一致 $(k-1)$-凸区域中建立 $(k-1)$-容许解的先验 $C^2$ 估计与存在性，并推广至一致凸区域及抛物型情形。

A subsolution theorem for the Monge-Ampère equation over an almost Hermitian manifold

Let $Ω\subseteq M$ be a bounded domain with a smooth boundary $\partialΩ$, where $(M,J,g)$ is a compact, almost Hermitian manifold. The main result of this paper is to consider the Dirichlet problem for a complex Monge-Ampère equation on $Ω$. Under the existence of a $C^{2}$-smooth strictly $J$-plurisubharmonic ($J$-psh for short) subsolution, we can solve this Dirichlet problem. Our method is based on the properties of subsolutions which have been widely used for fully nonlinear elliptic equations over Hermitian manifolds.

Schroedinger flow into almost Hermitian manifolds

We present a short-time existence theorem of solutions to the initial value problem for Schroedinger maps of a closed Riemannian manifold to a compact almost Hermitian manifold. The classical energy method cannot work for this problem since the almost complex structure of the target manifold is not supposed to be parallel with respect to the Levi-Civita connection. In other words, a loss of one derivative arises from the covariant derivative of the almost complex structure. To overcome this difficulty, we introduce a bounded pseudodifferential operator acting on sections of the pullback bundle…

A third order dispersive flow for closed curves into almost Hermitian manifolds

We discuss a short-time existence theorem of solutions to the initial value problem for a third order dispersive flow for closed curves into a compact almost Hermitian manifold. Our equations geometrically generalize a physical model describing the motion of vortex filament. The classical energy method cannot work for this problem since the almost complex structure of the target manifold is not supposed to be parallel with respect to the Levi-Civita connection. In other words, a loss of one derivative arises from the covariant derivative of the almost complex structure. To overcome this diffic…

近厄米几何中的指定Gauduchon（高东雄）标量曲率问题

The prescribed Gauduchon scalar curvature problem in almost Hermitian geometry

该研究针对近厄米流形的指定Gauduchon标量曲率问题，通过共形变化推导将其转化为带指数非线性的半线性偏微分方程，用上下解方法证明解的存在性与Gauduchon度相关常数的符号有关，负号时给出充要条件，还复现了三类相关Yamabe问题。

Bott-Chern and $\bar\partial$ Harmonic forms on Almost Hermitian 4-manifolds

We prove that on a compact almost Hermitian 4-manifold the space of $\bar\partial$-harmonic $(1,1)$-forms always has dimension $h_{\bar\partial}^{1,1} = b_- +1$ or $b_-$, whilst the space of Bott-Chern harmonic $(1,1)$-forms always has dimension $h_{BC}^{1,1} = b_- +1$. We also perform calculations of $h^{2,1}_{BC}$ and $h^{1,2}_{BC}$ on the Kodaira-Thurston manifold, thereby providing a full account of when $h^{p,q}_{BC}$ is or is not invariant of the choice of almost Hermitian metric. Finally, we introduce a decomposition of the space of $L^2$ functions on all torus bundles over $S^1$, which…

$\bar{\partial}$-Harmonic Maps Between Almost Hermitian Manifolds

In this paper we study an energy of maps between almost Hermitian manifolds for which pseudo-holomorphic maps are global minimizers. We derive its Euler-Lagrange equation, the $\bar{\partial}$-harmonic map equation, and show that it coincides with the harmonic map equation up to first order terms. We prove results analogous to the those that hold for harmonic maps, including obstructions to the long time existence of the associated parabolic flow, an Eells-Sampson type result under appropriate conditions on the target manifold, and a bubbling result for finite time singularities on surfaces. W…