Weisong Dong, Chang Li
We derive second order estimates for $χ$-plurisubharmonic solutions of complex Hessian equations with right hand sides depending on gradients on compact Hermitian manifolds.
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In this paper, we derive an \emph{a priori} second order estimate for solutions which are in $Γ_{k+1}$ cone to a class of complex Hessian equations with both sides of the equation depending on the gradient on compact Hermitian manifolds.
In this paper, we derive the second order estimate to the $2$-nd Hessian type equation on a compact almost Hermitian manifold.
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.
In this paper, using the technical tools in \cite{TW5}, we solve the complex Hessian equation on closed Hermitian manifolds, which generalizes the the Kähler case results in \cite{HMW} and \cite{DK}.
Second order estimates for $χ$-semi convex solutions of Hessian equations on Hermitian manifolds
本文在Hermitian流形上，针对方程两边含梯度项的复Hessian方程，利用半凸性假设建立修正凹性不等式，从而获得可容许解的二阶估计。
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。
Second order estimates for equations with sums of Hessian operators on Hermitian manifolds
本文基于动态 plurisubharmonic 条件，为满足条件的可允许解建立二阶估计，利用复Hessian算子之和的凹性不等式推导。
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。
In this paper we study the Dirichlet problem for a class of Hessian type equation with its structure as a combination of elementary symmetric functions on Hermitian manifolds. Under some conditions with the initial data on manifolds and admissible subsolutions, we derive a priori estimates for this complex mixed Hessian equation and solvability of the corresponding Dirichlet problem.
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.
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。
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。
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.
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.
Let $(X,ω)$ be a compact Kähler manifold of dimension $n$ and fix $m\in \mathbb{N}$ such that $1\leq m \leq n$. We prove that any $(ω,m)$-sh function can be approximated from above by smooth $(ω,m)$-sh functions. A potential theory for the complex Hessian equation is also developed which generalizes the classical pluripotential theory on compact Kähler manifolds. We then use novel variational tools due to Berman, Boucksom, Guedj and Zeriahi to study degenerate complex Hessian equations.
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。
Weak solutions to Hessian type equations on compact Riemannian manifolds
本文研究无边界紧黎曼流形上Hessian型方程弱解的存在性问题。
The Eigenvalue Problem for the Complex Hessian Operator on $m$-Pseudoconvex Manifolds
本文研究强$m$-伪凸流形上复Hessian算子的特征值问题，证明了第一特征函数的$C^{1,1}$正则性与唯一性，并给出第一特征值的变分公式，进而推导出分岔型定理和几何界。
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 (…
In this paper, we derive $C^2$ estimates for a class of mixed Hessian type equations with Dirichlet boundary condition, and obtain the existence theorem of admissible solutions for the classical Dirichlet problem of these mixed Hessian type equations.
The author extends previous results to general classes of equations under weaker assumptions obtained in 2016 by Bao, Dong and Jiao concerning the study of the regularity of solutions for the first initial-boundary value problem for parabolic Hessian equations on Riemannian manifolds.
In this paper, we consider a class of Hessian type equations which include the $(n-1)$ Monge-Ampère equation on Riemannian manifolds. The \emph{a priori} $C^2$ estimates and the existence of solutions are established.
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。
In this paper, we consider the Dirichlet problem for a class of Hessian quotient equations on Riemannian manifolds. Under the assumption of an admissible subsolution, we solve the existence and the uniquness for the Dirichlet problem in a domain without any geometric restrictions on the boundary, based on the a priori estimates for the solutions to the Hessian quotient type equations.
In this paper, we consider Hessian equations with its structure as a combination of elementary symmetric functions on closed Kähler manifolds. We provide a sufficient and necessary condition for the solvability of these equations, which generalize the results of Hessian equations and Hessian quotient equations.
In this paper, complex Hessian equation over Kähler manifold was studied. Under the condition that the underline Kähler manifold has non-negative holomorphic bisectional curvature, the existence and regularity of the solution was proved.
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…
In this paper, we prove the $C^{1, 1}$-regularity of the plurisubharmonic envelope of a $C^{1,1}$ function on a compact Hermitian manifold. We also present examples to show this regularity is sharp.
In this paper, we establish the second order estimates of solutions to the first initial-boundary value problem for general Hessian type fully nonlinear parabolic equations on Riemannian manifolds. The techniques used in this article can work for a wide range of fully nonlinear PDEs under very general conditions.
We show the $C^0$ estimate for solutions to Hessian quotient equations on hyperKähler with torsion manifolds without any additional assumption on its hypercomplex structure by a clever use of the cone condition directly.
On Polynomial Complex Hessian Equations I: Non-Linear Non-Spectral Elliptic Hessian Equations on Riemannian Manifolds
本文研究紧致黎曼流形上的一般完全非线性椭圆Hessian方程，通过弱化标准对称性假设，在推导先验估计后利用连续性方法求解。
In this paper, we derive a priori estimates for the gradient and second order derivatives of solutions to a class of Hessian type fully nonlinear parabolic equations with the first initial-boundary value problem on Riemannian manifolds. These a priori estimates are derived under conditions which are nearly optimal. Especially, there are no geometric restrictions on the boundary of the Riemannian manifolds. And as an application, the existence of smooth solutions to the first initial-boundary value problem even for infinity time is obtained.
In this paper, we are concerned with the first initial boundary value problem for a class of fully nonlinear parabolic equations on Riemannian manifolds. As usual, the establishment of the a priori C^2 estimates is our main part. Based on these estimates, the existence of classical solutions is proved under conditions which are nearly optimal.
This paper has been withdrawn by the author due to a crucial error in the proof of Lemma 2.2.
In this paper, we consider a class of Hessian quotient equations in the warped product manifold $\overline{M}=I\times_λM$. Under some sufficient conditions, we obtain an existence result for the star-shaped compact hypersurface $Σ$ in $\overline{M}$ using standard degree theory based on a priori estimates for solutions to the Hessian quotient equations.
We are concerned with a priori estimates for the obstacle problem of a wide class of fully nonlinear equations on Riemannian manifolds. We use new techniques introduced by Bo Guan and derive new results for a priori second order estimates of its singular perturbation problem under fairly general conditions. By approximation, the existence of a C^{1,1} viscosity solution is proved.
We study the long time behavior of the Hesse-Koszul flow on compact Hessian manifolds. When the first affine Chern class is negative, we prove that the flow converges to the unique Hesse-Einstein metric. We also derive a convergence result for a twisted Hesse-Koszul flow on any compact Hessian manifold. These results give alternative proofs for the existence of the unique Hesse-Einstein metric by Cheng-Yau and Caffarelli-Viaclovsky as well as the real Calabi theorem by Cheng-Yau, Delanoë and Caffarelli-Viaclovsky.
We solve the Dirichlet problem for $k$-Hessian equations on compact complex manifolds with boundary, given the existence of a subsolution. Our method is based on a second order a priori estimate of the solution on the boundary with a particular gradient scale. The scale allows us to apply a blow-up argument to obtain control on all necessary norms of the solution.
Alternative proof of the gradient estimate for complex Hessian equations
本文针对紧致及带边界的Kähler流形上的复Hessian方程，提出一种无需爆破论证的梯度估计新证明，并明确其与右端项的关系。
We consider degenerate Monge-Ampère equations on compact Hessian manifolds. We establish compactness properties of the set of normalized quasi-convex functions and show local and global comparison principles for twisted Monge-Ampère operators. We then use the Perron method to solve Monge-Ampère equations whose RHS involves an arbitrary probability measure, generalizing works of Cheng-Yau, Delanoë, Caffarelli-Viaclovsky and Hultgren-Önnheim. The intrinsic approach we develop should be useful in deriving similar results on mildly singular Hessian varieties, in line with the Strominger-Yau-Zaslow…
We obtain a priori local pointwise second derivative estimates for solutions $u$ to a class of augmented Hessian equations on Riemannian manifolds, in terms of the $C^1$ norm and certain $W^{2,p}$ norms of $u$. We consider the case that no structural assumptions are imposed on either the augmenting term or the right hand side of the equation, and the case where these terms are convex in the gradient variable. In the latter case, under an additional ellipticity condition we prove that the dependence on any $W^{2,p}$ norm can be dropped. Our results are derived using integral estimates.
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.
We study a class of fully nonlinear elliptic equations on closed Hermitian manifolds. Under the assumption of cone condition, we derive the $L^\infty$ estimate directly.
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。
Remarks on Hessian quotient equations on Riemannian manifolds
针对 Delanoë 和 Urbas 提出的相关问题，本文在二维黎曼流形上利用新测试函数与商算子凹性，证明实 Hessian 商方程具有无障碍二阶先验估计，并揭示其与复几何中 $J$-方程障碍的差异。
Pogorelov type interior $C^2$ estimate for sum Hessian quotient equations on Riemannian manifolds
针对黎曼流形上一类和Hessian商方程，建立了测地球中心处的内部$C^2$估计以及带零Dirichlet边界的Pogorelov型估计。
The interpolation method for the complex Hessian equation on compact Kähler manifolds
本文在紧致Kähler流形上研究复Hessian方程，通过约束双截曲率与解的振荡，利用插值论证而非爆破技术获得梯度估计，并应用于右端$L^1$逼近1的情形。
Sharp Hessian inequalities and monotonicity for Green's functions of positively curved Einstein manifolds
该研究针对正曲率爱因斯坦流形的格林函数，证明了其尖锐黑塞不等式及相关单调性公式，扩展了Park的结果，并给出几何应用。
We derive a priori estimates for second order derivatives of solutions to a wide calss of fully nonlinear elliptic equations on Riemannian manifolds. The equations we consider naturally appear in geometric problems and other applications such as optimal transportation. There are some fundamental assumptions in the literature to ensure the equations to be elliptic and that one can apply Evans-Krylov theorem once estimates up to second derivatives are derived. However, in previous work one needed extra assumptions which are more technical in nature to overcome various difficulties. In this paper…
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。
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。
$\mathrm{C}^2$ estimates for general $p$-Hessian equations on closed Riemannian manifolds
本文针对闭黎曼流形上一般 $p$-Hessian 方程，引入伪解概念，在无曲率、凸性或 MTW 条件下证明了 $p\in\{2,n-1,n\}$ 时的 $C^1$ 与 $C^2$ 估计，并给出相关先验估计结论。
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.
Let $(X, ω)$ be a compact connected Hermitian manifold of dimension $n$. We consider the Bott-Chern cohomology and let $[χ] \in H^{1,1}_{\text{BC}}(X; \mathbb{R})$. We study the deformed Hermitian-Yang-Mills equation, which is the following nonlinear elliptic equation $\sum_{i} \arctan (λ_i) = h(x)$, where $λ_i$ are the eigenvalues of $χ$ with respect to $ω$.
In this paper, we solve the Dirichlet problem for Monge-Ampère type equations for $(n-1)$-plurisubharmonic functions on Hermitian manifolds.
We study generalized complex Monge-Ampère type equations on closed Hermitian manifolds. We derive {\em a priori} estimates and then prove the existence of admissible solutions. Moreover, the gradient estimate is improved.
We study a class of fully nonlinear elliptic equations on closed Hermitian manifolds. We derive $C^\infty$ {\em a priori} estimates, and then prove the existence of admissible solutions. In the approach, a new Hermitian metic is constructed to launch the method of continuity.
Under structural conditions which are almost optimal, we derive a quantitative version of boundary estimate then prove existence of solutions to Dirichlet problem for a class of fully nonlinear elliptic equations on Hermitian manifolds.
On non-Kähler manifolds the notion of harmonic maps is modified to that of Hermitian harmonic maps in order to be compatible with the complex structure. The resulting semilinear elliptic system is {\it not} in divergence form. The case of noncompact complete preimage and target manifolds is considered. We give conditions for existence and uniqueness of Hermitian-harmonic maps and solutions of the corresponding parabolic system, which observe the non-divergence form of the underlying equations. Numerous examples illustrate the theoretical results and the fundamental difference to harmonic maps.
Starshaped compact hypersurfaces in warped product manifolds II: a class of Hessian type equations
本文通过导出全局曲率估计，证明了一类星形紧致超曲面的存在性，将既有结果从欧几里得空间推广至 Riemannian 扭曲积，并利用类似论证建立了相关完全非线性椭圆偏微分方程容许解的内部二阶先验估计。
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