Heming Jiao, Jinxuan Liu
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.
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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 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 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.
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.
Second order estimates for Hessian equations of parabolic type on Riemannian manifolds
本文建立了黎曼流形上一般Hessian型完全非线性抛物型方程第一初边值问题解的二阶估计。本文中使用的技术可以在非常一般的条件下对广泛的完全非线性偏微分方程起作用。
Estimates for a class of Hessian type fully nonlinear parabolic equations on Riemannian manifolds
本文给出了黎曼流形上一类Hessian型完全非线性抛物型方程第一初边值问题解的梯度和二阶导数的先验估计.这些先验估计是在接近最优的条件下导出的。特别地，在黎曼流形的边界上没有几何限制。作为应用，得到了第一初边值问题无穷时间光滑解的存在性。
The first initial boundary value problem for Hessian equations of parabolic type on Riemannian manifolds
本文研究了黎曼流形上一类完全非线性抛物型方程的第一初边值问题。像往常一样，建立先验C^2估计是我们的主要部分。基于这些估计，在近似最优的条件下，证明了经典解的存在性.
A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds
研究了黎曼流形上一类完全非线性方程障碍问题解的先验估计。本文利用关伯介绍的新技巧，在相当一般的条件下，得到了其奇异扰动问题的先验二阶估计的新结果。通过逼近，证明了C^{1，1}黏性解的存在性.
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.
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。
Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds
本文给出了黎曼流形上一类完全非线性椭圆型方程解的二阶导数的先验估计。我们考虑的方程自然出现在几何问题和其他应用中，如最优运输。在文献中有一些基本的假设，以确保方程是椭圆的，并且一旦估计到二阶导数，就可以应用Evans-Krylov定理。然而，在以前的工作需要额外的假设，这是更多的技术性质，以克服各种困难。在本文中，我们能够消除这些技术假设中的大部分。事实上，我们在几乎最优的条件下推导出了估计，并证明了狄利克雷问题的存在性结果，即使对于欧几里得空间中的有界域来说，这些结果也是新的。此外，我们的方法可以应用到其他类型的非线性椭圆和抛物型方程，包括那些在复流形。
In this paper, we derive the second order estimate to the $2$-nd Hessian type equation on a compact almost Hermitian manifold.
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。
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。
A class of sum Hessian quotient equations arising in conformal geometry
本文研究黎曼流形上带修正Schouten张量的和型Hessian商方程的(n-1)-容许解的存在性，通过建立一致先验估计，利用度理论证明其可解性。
In this paper we study the {\it a priori} gradient estimates for admissible solutions to Neumann boundary value problem of fully nonlinear Hessian equations on Riemannian manifolds. We firstly derive an interior gradient estimates for admissible solutions, then we obtain boundary gradient estimates based on the interior gradient estimates we have got.
$\mathrm{C}^2$ 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 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.
Complex Hessian Equation on Kähler Manifold
研究了Kähler流形上的复Hessian方程.在下划Kähler流形具有非负全纯双截曲率的条件下，证明了解的存在性和正则性.
Energy identity of approximate biharmonic maps to Riemannian manifolds and its application
本文研究了具有双张力场的黎曼流形上的近似双调和映射序列在四维弱收敛情形下的性质，其中双张力场是有界于$L^p$（$p>\frac43$）中的。我们证明了一个能量恒等式，它用$\mathbb R^4$上的100个非平凡双调和映射上的Hessian能量之和来解释Hessian能量的损失.作为推论，我们得到了双调和映射在无穷远时刻热流的一个能量恒等式。
Asymptotics of the principal eigenvalue of an elliptic operator on closed and orientable Riemannian manifolds: small diffusion
本文研究闭定向黎曼流形上椭圆算子主特征值在小扩散极限下的渐近性，在Morse函数假设下，确定其极限值由函数临界点及对应黎曼Hessian的相关参数完全刻画。
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.
Hessian equations on closed Hermitian manifolds
本文利用\cite{TW5}中的技术工具，求解了闭Hermite流形上的复Hessian方程，推广了\cite{HMW}和\cite{DK}中Kähler情形的结果.
We derive second order estimates for $χ$-plurisubharmonic solutions of complex Hessian equations with right hand sides depending on gradients on compact Hermitian manifolds.
Hessian metrics, CD(K,N)-spaces, and optimal transportation of log-concave measures
研究了将一个概率测度$μ= e^{-V} \ dx$推进到另一个概率测度$ν= e^{-W} \ dx$上的最优运输映射$\nabla Φ: \mathbb{R}^d \mapsto \mathbb{R}^d$.按照E. Calabi在$\mathbb{R}^d$上引入了黎曼度量$g = D^2 Φ$，并研究了度量测度空间$M=(\mathbb{R}^d, g, μ)$的谱性质.我们证明，特别是，$M$承认一个非负的Bakry-{\displaystyle {\frac} merry张量提供$V$和$W$是凸的。若目标测度$ν$是凸集$Ω$上的Lebesgue测度，且$μ$是对数凹的，则证明了$M$是$CD(K,N)$空间.这些结果的应用包括$\| D^2 Φ\|$的一些全局无量纲先验估计。借助于黎曼流形上的比较技巧和概率集中论，我们证明了$M$的一些直径估计.
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.
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。
Complex Hessian Equations on Compact Kahler Manifold
这篇论文已经撤回了作者由于一个关键的错误证明引理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.
Hessian matrix estimates of heat-type equations via Bismut-Stroock Hessian formula
该研究借助Bismut型Hessian公式推导黎曼流形上热型方程的显式全局Hessian矩阵估计，并将其应用于得到新型反向弱Harnack不等式与特征函数逐点Hessian估计。
An Upper Bound for Hessian Matrices of Positive Solutions of Heat Equations
证明了黎曼流形上热方程对数正解的Hessian的整体和局部上界。度量要么是固定的，要么是在里奇流下演化的。这些上界补充了众所周知的全球下界。
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.
Critical functions on Riemannian Manifold wich is locally Euclidean
研究了局部欧氏的紧致黎曼流形上极值函数的存在性。
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.
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…
Behavior of bounded solutions of quasilinear elliptic equations on Riemannian manifolds
讨论了满足特殊条件的黎曼流形上拟线性椭圆型方程的有界解。
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.
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。
Degenerate complex Hessian equations on compact Kähler manifolds
设$(X,ω)$是维数为$n$的紧致Kähler流形，固定$m\in \mathbb{N}$使得$1\leq m \leq n$ .我们证明了任何$(ω,m)$ -sh函数都可以用光滑的$(ω,m)$ -sh函数来逼近.复Hessian方程的一个位势论也得到了发展，它推广了紧致Kähler流形上的经典多元位势论。然后，我们使用新的变分工具，由于伯曼，Boucksom，Guedj和Zeriahi研究退化复杂的海森方程。
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
我们发现了伪厄米流形上范数不等式的精确界，其中涉及水平导数的函数的所有二阶导数的L^2范数由次拉普拉斯算子的L^2范数控制。摄动使我们能够得到非散度型的亚椭圆型偏微分方程解的先验界。证明的方法是通过Bochner技术。在里奇曲率为非负的流形类中，我们的不等式证明了海森堡群是一个极值流形。
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。
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。
In this paper, we propose a coupled system of complex Hessian equations which generalizes the equation for constant scalar curvature Kähler (cscK) metrics. We show this system can be realized variationally as the Euler-Lagrange equation of a Hessian version of the Mabuchi K-energy in an infinite dimensional space of $k$-Hessian potentials, which can be seen as an infinite dimensional Riemannian manifold with negative sectional curvature. Finally, we prove an a priori $C^0$-estimate for this system which depends on the Entropy, which generalizes a fundamental result of Chen and Cheng for cscK m…
This article is devoted to the study of several estimations for a positive solution to a nonlinear weighted parabolic equation on a weighted Riemannian manifold. We therefore derive new Li-Yau type and Hamilton type gradient estimates yielding several consequences. We also derive Hessian estimate and some corollaries for the same equation. Among the applications of our estimates discussed here are Harnack type inequalities, Liouville type theorems and a local time reversed Harnack inequality.
JKO schemes with general transport costs
本文通过用一般传输代价替换Wasserstein距离来修改JKO格式，证明当代价函数的混合Hessian定义黎曼度量时，该格式收敛到对应的黎曼Fokker-Planck方程，并应用于Hessian流形上的Bregman散度代价。
This paper is a survey of some recent results on the validity and the failure of global $W^{2,p}$ regularity properties of smooth solutions of the Poisson equation $Δu = f$ on a complete Riemannian manifold $(M,g)$. We review different methods developed to obtain a-priori $L^p$-Hessian estimates of the form $\| \Hess(u) \|_{L^p} \leq C_1 \| u \|_{L^p} + C_2 \| f \|_{L^p}$ under various geometric conditions on $M$ both in the case of real valued functions and for manifold valued maps. We also present explicit and somewhat implicit counterexamples showing that, in general, this integral inequali…
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
本文研究了闭Hermitian流形上的Hessian方程和复商方程。我们直接导出了可采纳解的一致估计。作为应用，我们求解了闭Kähler流形上的一般Hessian方程.
Generalized Bochner formulas and Ricci lower bounds for sub-Riemannian manifolds of rank two
研究一类新的秩为2的次黎曼流形，它包括黎曼流形、Webster-Tanaka挠为零的CR流形、黎曼流形上的正交丛和阶为2的分次幂零李群.这些流形允许一个正则水平联络和一个正则次拉普拉斯算子。我们在这些流形上构造了一个类似的黎曼Ricci张量，并证明了次拉普拉斯算子的Bochner型公式。因此，有可能在这些空间上形成所谓曲率维数不等式的次黎曼模拟。满足这个不等式的次黎曼流形与里奇曲率是下界的黎曼流形具有许多共同的性质
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。
We prove that polyharmonic maps of arbitrary order from complete nonparabolic Riemannian manifolds to arbitrary Riemannian manifolds must be harmonic if certain smallness and integrability conditions hold.
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。
Stochastic Partial Differential Equations on Evolving Surfaces and Evolving Riemannian Manifolds
在随机偏微分方程分析的变分背景下，我们建立了黎曼流形、移动曲面、一般演化黎曼流形（在适当的假设下）和随机度量黎曼流形上的随机偏微分方程.主要考虑线性随机偏微分方程，建立了各种解的存在唯一性定理。
A note on the PDE approach to the $L^\infty$ estimates for complex Hessian equations on transverse Kähler manifolds
本文改编 Guo-Phong-Tong 等人的 PDE 方法，证明横截 Kähler 流形上复 Monge-Ampère 方程的 $L^\infty$ 估计，并由此给出 Q-Gorenstein T-簇上 Calabi-Yau 锥度量正则性的纯 PDE 证明。
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。
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