Dongwei Gu, Ngoc Cuong Nguyen
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.
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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.
We consider the Dirichlet problem for positively homogeneous, degenerate elliptic, concave (or convex) Hessian equations. Under natural and necessary conditions on the geometry of the domain, with the $C^{1,1}$ boundary data, we establish the interior $C^{1,1}$-regularity of the unique (admissible) solution, which is optimal even if the boundary data is smooth. Both real and complex cases are studied by the unified (Bellman equation) approach.
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}.
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。
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。
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。
In this note we provide uniform a priori estimates for solutions to degenerate complex Hessian equations on compact hermitian manifolds. Our approach relies on the corresponding a priori estimates for Monge-Ampère equations; it provides an extension as well as a short alternative proof to results of Dinew-Kołodziej, Kołodziej-Nguyen and Guo-Phong-Tong.
Smoothness of bounded solutions to complex Hessian equations on compact Hermitian manifolds
本文在紧致 Hermitian 流形上建立复 Hessian 方程的先验 $C^2$ 估计，并由此得到右端密度依赖于解时的正则性结果，推广了 Székelyhidi--Tosatti 的结论。
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.
We prove the existence of weak solutions of complex $m-$Hessian equations on compact Hermitian manifolds for the nonnegative right hand side belonging to $L^p, p>n/m$ ($n$ is the dimension of the manifold). For smooth, positive data the equation has been recently solved by Szekelyhidi and Zhang. We also give a stability result for such solutions.
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.
The Dirichlet problem for Hessian quotient type curvature equations in Minkowski space
针对非凸区域，在不假设下解和Serrin型条件下建立先验估计，证明闵可夫斯基空间中一类Hessian商型曲率方程Dirichlet问题的存在性。
In this paper, by using the Bochner technique on almost Hermitian manifolds, we obtain a complex Hessian comparison for almost Hermitian manifolds generalizing the Laplacian comparison for almost Hermitian manifolds by Tossati, and reprove a diameter estimate for almost Hermitian manifolds by Gray. Moreover, we obtain a sharp eigenvalue estimate on quasi Kaehler manifolds and a sharp Hessian comparison on nearly Kaehler 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。
Mixed Hessian inequalities on Hermitian manifolds and applications
本文在Hermitian流形上建立了复Hessian算子的弱收敛定理，并证明了混合Hessian不等式，进而证明了复Hessian方程的有界解存在性。
Continuity of solutions to complex Hessian equations on compact Hermitian manifolds
本文在紧致 Hermitian 流形上，通过 Orlicz 空间中的正右端项假设获得 $L^\infty$ 估计，进而证明复 Hessian 方程连续解的存在性。
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 establish integral formulas and sharp two-sided bounds for the Ricci curvature, mean curvature and second fundamental form on a Riemannian manifold with boundary. As applications, sharp gradient and Hessian estimates are derived for the Dirichlet and Neumann eigenfunctions.
This note aims to investigate the regularity of a solution to the Dirichlet problem for the complex Hessian equation, which has a density of the $m$-Hessian measure that belongs to $L^q$, for $q\leq\frac nm$.
In this paper, we solve the Dirichlet problem for Monge-Ampère type equations for $(n-1)$-plurisubharmonic functions on Hermitian manifolds.
In this paper, under suitable settings, we can obtain the existence and uniqueness of solutions to a class of Hessian quotient equations with Dirichlet boundary condition in Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$, which can be seen as a prescribed curvature problem and a continuous work of [12].
We compute the hessian of the natural Hermitian form successively on the Calabi family of a hyperkähler manifold, on the twistor space of a 4-dimensional anti-self-dual Riemannian manifold and on the twistor space of a quaternionic Kähler manifold. We show a strong convexity property of the cycle space of twistor lines on the Calabi family of a hyperkähler manifold. We also prove convexity properties of the 1-cycle space of the twistor space of a 4-dimensional anti-self-dual Einstein manifold of non-positive scalar curvature and of the 1-cycle space of the twistor space of a quaternionic Kähle…
The Dirichlet eigenvalue problems for some concave elliptic Hessian operators
本文研究一类广泛凹椭圆Hessian算子的Dirichlet特征值问题，在温和约束下证明了首个非零特征值及特征函数的存在性与正则性，并推广至Gårding-Dirichlet算子，建立了分岔理论。
Overdetermined problems for fully nonlinear equations with constant Dirichlet boundary conditions in space forms
本文研究空间形式中两类完全非线性方程带常数Dirichlet边界条件的过定问题，证明星形域上Hessian商问题解的径向对称性，并利用Rellich-Pohožaev型恒等式在无星形假设下证明k-Hessian问题的径向对称性。
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 (…
We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex Monge-Ampère, Hessian and inverse Hessian equations. As an application we solve a class of Hessian quotient equations on Kähler manifolds assuming the existence of a suitable subsolution. The method also applies to analogous equations on compact Riemannian manifolds.
We are concerned with fully nonlinear elliptic equations on complex manifolds and search for technical tools to overcome difficulties in deriving a priori estimates which arise due to the nontrivial torsion and curvature, as well as the general (non-pseudoconvex) shape of the boundary. We present our methods, which work for more general equations, by considering a specific equation which resembles the complex Monge-Ampere equation in many ways but with crucial differences. Our work is motivated by recent increasing interests in fully nonlinear equations on complex manifolds from geometric prob…
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…
We study weak quasi-plurisubharmonic solutions to the Dirichlet problem for the complex Monge-Amère equation on a general Hermitian manifold with non-empty boundary. We prove optimal subsolution theorems: for bounded and Hölder continuous quasi-plurisubharmonic functions. The continuity of the solution is proved for measures that well dominated by capacity, for example measures with $L^p$, $p>1$ densities, or moderate measures in the sense of Dinh-Nguyen-Sibony.
We derive the solvability and regularity of the Dirichlet problem for fully non-linear elliptic equations possibly with degenerate right-hand side on Hermitian manifolds, through establishing a quantitative version of boundary estimate under a subsolution assumption. In addition, we construct the subsolution when the background manifold is a product of a closed Hermitian manifold with a compact Riemann surface with boundary.
We prove the solvability of a Dirichlet problem for flat hermitian metrics on Hilbert bundles over compact Riemann surfaces with boundary. We also prove a factorization result for flat hermitian metrics on doubly connected domains.
Let $Ω\subset \mathbb C^n$ be a bounded strictly $m$-pseudoconvex domain ($1\leq m\leq n$) and $μ$ a positive Borel measure on $Ω$. We study the Dirichlet problem for the complex Hessian equation $(dd^c u)^m \wedge β^{n - m} = μ$ on $Ω$. First we give a sufficient condition on the "modulus of diffusion" of the measure $μ$ with respect to the $m$-Hessian capacity which guarantees the existence of a continuous solution to the associated Dirichlet problem with a continuous boundary datum. As an application, we prove that if the equation has a continuous $m$-subharmonic subsolution whose modulus o…
The purpose of this paper is to investigate the parabolic deformed Hermitian-Yang-Mills equation with hypercritical phase in a smooth domain $Ω\subset \mathbb{C}^{n}$. By using $J$-functional, we are able to prove the convergence of solutions. As an application, we give an alternative proof of the Dirichlet problem for deformed Hermitian-Yang-Mills equation.
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 the Dirichlet problem of a class of fully nonlinear elliptic equations on Hermitian manifolds and derive a priori $C^2$ estimates which depend on the initial data on manifolds, the admissible subsolutions and the upper bound of the gradients of the solutions. In some special cases, we obtain the gradient estimates, and hence we can solve the corresponding Dirichlet problem with admissible subsolutions. We also study the Hessian quotient equations and $(m-1,m-1)$-Hessian quotient equations on compact Hermitian manifolds without boundary.
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。
We solve the Dirichlet problem for Monge-Ampère equation for $(n-1)$-PSH functions possibly with degenerate right-hand side, through deriving a quantitative version of boundary estimate under the assumption of $(n-1)$-PSH subsolutions. In addition, we confirm the subsolution assumption on a product of a closed balanced manifold with a compact Riemann surface with boundary.
Inverse Hessian Curvature Flow in Minkowski Space II: The Dirichlet problem at infinity
本文研究闵可夫斯基空间中非紧致类空严格凸超曲面的逆σ_k曲率流，证明其存在整体解且归一化后收敛到对应唯一自收缩子，丰富了非紧致情形下自收缩子的相关理论。
In this paper, we consider some generalized holomorphic maps between pseudo-Hermitian manifolds. These maps include the \emph{CR} maps and the transversally holomorphic maps. In terms of some sub-Laplacian or Hessian type Bochner formulas, and comparison theorems in the pseudo-Hermitian version, we are able to establish several Schwarz type results for both the \emph{CR} maps and the transversally holomorphic maps between pseudo-Hermitian manifolds. Finally, we also discuss the \emph{CR} hyperbolicity problem for pseudo-Hermitian manifolds.
We study weak geodesics in the space of potentials for the deformed Hermitian-Yang-Mills equation. The geodesic equation can be formulated as a degenerate elliptic equation, allowing us to employ nonlinear Dirichlet duality theory, as developed by Harvey-Lawson. By exploiting the convexity of the level sets of the Lagrangian angle operator in the highest branch, we are able to construct continuous solutions of the associated Dirichlet problem.
Hermitian Calabi functional in complexified orbits
本研究针对紧致辛流形上与辛形式相容的殆复结构空间，推导了极值殆凯勒度量处埃尔米特卡拉比泛函的黑塞显式公式，证明其在复化轨道临界点上半正定并得到相关推论，同时验证了埃尔米特卡拉比流的弱抛物性。
We derive a priori $C^2$ estimates for a class of complex Monge-Ampere type equations on Hermitian manifolds. As an application we solve the Dirichlet problem for these equations under the assumption of existence of a subsolution; the existence result, as well as the second order boundary estimates, is new even for bounded domains in $\bfC^n$.
Overdetermined problems for sum Hessian equations in the space forms
本文研究空间形式中规范化Hessian和方程及其商方程的过定问题，证明在星形或Dirichlet值上界条件下区域为测地球，并推广至无星形假设情形，方法为辅助函数与Rellich–Pohožaev恒等式。
$C^2$ estimates for $k$-Hessian equations and a rigidity theorem
本文通过证明半凸条件下 $k$-Hessian 算子的凹性不等式，建立相关方程解的内部估计与 Liouville 型结果，并简化 $k$-曲率方程全局曲率估计的证明。
Let L be an ample bundle over a compact complex manifold X. Fix a Hermitian metric in L whose curvature defines a Kähler metric on X. The Hessian of Mabuchi energy is a fourth-order elliptic operator D on functions which arises in the study of scalar curvature. We quantise D by the Hessian E(k) of balancing energy, a function appearing in the study of balanced embeddings. E(k) is defined on the space of Hermitian endomorphisms of H^0(X, L^k), endowed with the L^2-innerproduct. We first prove that the leading order term in the asymptotic expansion of E(k) is D. We next show that if Aut(X,L) is …
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.
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。
We consider a class of fully nonlinear second order elliptic equations on Hermitian manifolds closely related to the general notion of $\bfG$-plurisubharmonicity of Harvey-Lawson and an equation treated by Székelyhidi-Tosatti-Weinkove in the proof of Gauduchon conjecture. Under fairly general assumptions we derive interior estimates and establish the existence of smooth solutions for the Dirichlet problem as well as for equations on closed manifolds.
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.
In this paper we propose new insights and ideas to set up quantitative boundary estimates for solutions to Dirichlet problem of a class of fully non-linear elliptic equations on compact Hermitian manifolds with real analytic Levi flat boundary. With the quantitative boundary estimates at hand, we can establish the gradient estimate and give a unified approach to investigate the existence and regularity of solutions of Dirichlet problem with sufficiently smooth boundary data, which include the geodesic equation in the space of Kähler metrics as a special case. Our method can also be applied to …
The paper introduces a new differential-geometric system which originates from the theory of $m$-Hessian operators. The core of this system is a new notion of invariant differentiation on multidimensional surfaces. This novelty gives rise to the following absolute geometric invariants: invariant derivatives of the surface position vector, an invariant connection on a surface via subsurface, curvature matrices of a hypersurface and its normal sections, $p$-curvatures and $m$-convexity of a hypersurface, etc. Our system also produces a new interpretation of the classic geometric invariants and o…
Hölder Regularity of Dirichlet Problem For The Complex Monge-Ampère Equation
本文研究复 Monge-Ampère 方程狄利克雷问题的解的 Hölder 正则性，证明在特定条件下解的全局 Hölder 连续性。
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。
We study complex Monge-Ampere equations on Hermitian manifolds, extending classical existence results of Yau and Aubin in the Kahler case, and those of Caffarelli, Kohn, Nirenberg and Spruck for the Dirichlet problem in $C^n$. As an application we generalize existing results on the Donaldson conjecture on geodesics in the space of Kahler metrics to the Hermitian setting.
Let $Ω$ be a bounded domain (with smooth boundary) on the hyperbolic plane $\mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$. In this paper, by using a priori estimates, we can establish Pogorelov type estimates of $k$-convex solutions to a class of Hessian quotient equations defined over $Ω\subset\mathscr{H}^{n}(1)$ and with the vanishing Dirichlet boundary condition.
In this paper we study the Dirichlet problem for fully nonlinear second-order equations on a riemannian manifold. As in a previous paper we define equations via closed subsets of the 2-jet bundle. Basic existence and uniqueness theorems are established in a wide variety of settings. However, the emphasis is on starting with a constant coefficient equation as a model, which then universally determines an equation on every riemannian manifold which is equipped with a topological reduction of the structure group to the invariance group of the model. For example, this covers all branches of the ho…
Let $M$ be a closed Riemann surface, $N$ a Riemannian manifold of Hermitian non-positive curvature, $f:M\to N$ a continuous map, and $E$ the function on the Teichmüller space of $M$ that assigns to a complex structure on $M$ the energy of the harmonic map homotopic to $f$. We show that $E$ is a plurisubharmonic function on the Teichmüller space of $M$. If $N$ has strictly negative Hermitian curvature, we characterize the directions in which the complex Hessian of $E$ vanishes.
A Born Structure on the Tangent Bundle of a Hessian Manifold
本文研究Hessian流形切丛上的几何结构，探讨了仿射联络与黎曼度量构成的有序对定义Hessian结构的条件，和其诱导的殆Born结构可积条件之间的等价性，关联了两类几何结构的研究。
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