Per Ahag, Rafal Czyz
By proving an estimate of the sublevel sets for $(ω,m)$-subharmonic functions we obtain a Sobolev type inequality that is then used to characterize the degenerate complex Hessian equations for such functions with bounded $(p,m)$-energy.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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.
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.
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.
Let $(X,ω)$ be an $n$-dimensional compact Kähler manifold. We study degenerate complex Hessian equations of the form $(ω+dd^cφ)^m\wedge ω^{n-m}=F(x,φ)ω^n.$ Under some natural conditions on $F$, this equation has a unique continuous solution. When $(X,ω)$ is rational homogeneous we further show that the solution is Hölder continuous.
A viscosity approach is introduced for the Dirichlet problem associated to complex Hessian type equations on domains in $\C^n$. The arguments are modelled on the theory of viscosity solutions for real Hessian type equations developed by Trudinger. As consequence we solve the Dirichlet problem for the Hessian quotient and special Lagrangian equations. We also establish basic regularity results for the solutions.
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。
We show a second order a priori estimate for solutions to the complex $k$-Hessian equation on a compact Kähler manifold provided the $(k$-$1)$-st root of the right hand side is $\mathcal C^{1,1}$. This improves an estimate of Hou-Ma-Wu. An example is provided to show that the exponent is sharp.
Degenerate complex Hessian equations with arbitrary measure in bounded domains
本文研究了在有界域中退化复Hessian方程的解的存在性，通过引入广义Cegrell类和m极大函数，解决了正Radon测度支配的方程问题。
Degenerate Complex Hessian type equations on compact Hermitian manifolds and Applications
本文研究了在紧致Hermitian流形上退化复Hessian方程的理论及其应用，提出了解决这些方程的新方法，并在Kähler情况下取得了新的结果。
In this paper, we studied integrals involving both real and complex Hessian operators over bounded domain. Poincare type inequalities were proved in both cases which generalized a early results of Trudinger and Wang.
We prove a Liouville type theorem for entire maximal $m$-subharmonic functions in $\mathbb C^n$ with bounded gradient. This result, coupled with a standard blow-up argument, yields a (non-explicit) a priori gradient estimate for the complex Hessian equation on a compact Kähler manifold. This terminates the program, initiated by Hou, Ma and Wu, of solving the non-degenerate Hessian equation on such manifolds in full generality. We also obtain, using our previous work, continuous weak solutions in the degenerate case for the right hand side in some $L^p, $ with sharp bound on $p$.
Construction of Twice Differentiable Functions with Continuous Laplacian and Bounded Hessian
本文构造了$\mathbb{R}^n$中具有连续Laplacian和有界Hessian的二次可微函数的例子，并将该构造方法推广应用于更高阶可微性、Monge-Ampère方程及超曲面平均曲率方程。
Hessian type equations for $m-ω-$subharmonic functions on bounded domains in $\mathbb C^n$
研究复\(n\)维空间中有界域上\(m - ω -\)次调和函数的黑塞型方程，借助文献结果证明此类方程有界解的存在性。
Bounded solution to Hessian type equations for $(ø,m)-β$-subharmonic functions on a ball in $\mathbb C^n$
本文研究$\u2102^n$中球面上$(ø,m)-β$-次调和函数的Hessian型方程，借助已有最新结果证明了该类方程有界解的存在性。
We prove the subsolution theorem for the complex Hessian equations in a smoothly bounded strongly $m$-pseudoconvex domain, $1 < m < n$, in $\bC^n$.
It is noted that using complex Hessian equations and the concavity inequalities for elementary symmetric polynomials implies a generalized form of Hodge index inequality. Inspired by this result, using Gårding's theory for hyperbolic polynomials, we obtain a mixed Hodge-index type theorem for classes of type $(1,1)$. The new feature is that this Hodge-index type theorem holds with respect to mixed polarizations in which some satisfy particular positivity condition, but could be degenerate and even negative along some directions.
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。
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。
Let $Ω$ be a $m$-hyperconvex domain of $\mathbb{C}^n$ and $β$ be the standard Kähler form in $\mathbb{C}^n$. We introduce finite energy classes of $m$-subharmonic functions of Cegrell type, $\mathcal{E}_m^p, p>0$ and $\mathcal{F}_m$. Using a variational method we show that the degenerate complex Hessian equation $(dd^cφ)^m\wedge β^{n-m}=μ$ has a unique solution in $\mathcal{E}_m^1$ if and only if every function in $\mathcal{E}_m^1$ is integrable with respect to $μ$. If $μ$ has finite total mass and does not charge $m$-polar sets, then the equation has a unique solution in $\mathcal{F}_m$.
There are two parts of this paper. First, we discovered an explicit formula for the complex Hessian of the weighted log-Bergman kernel on a parallelogram domain, and utilised this formula to give a new proof about the strict convexity of the Mabuchi functional along a smooth geodesic. Second, when a C^{1,1}-geodesic connects two non-degenerate energy minimizers, we also proved this strict convexity, by showing that such a geodesic must be non-degenerate and smooth.
A variational approach to the eigenvalue problem for complex Hessian operators
该研究采用变分方法，证明了m阶扭曲复Hessian算子第一特征值及有限能量m-次调和特征函数的存在性，在额外假设下得到特征函数的Hölder连续性，并将结果应用于右端依赖未知函数的退化复Hessian方程可解性问题。
Many aspects of pluripotential theory are generalized to quaternionic $m$-subharmonic functions. We introduce quaternionic version of notions of the $m$-Hessian operator, $m$-subharmonic functions, $m$-Hessian measure, $m$-capapcity, the relative $m$-extremal function and the $m$-Lelong number, and show various propositions for them, based on $d_0$ and $ d_1$ operators, the quaternionic counterpart of $\partial$ and $\overline{\partial}$, and quaternionic closed positve currents. The definition of quaternionic $m$-Hessian operator can be extended to locally bounded quaternionic $m$-subharmonic…
We generalize and strenghten Kołodziej's stability theorem. In particular we give sharp stability exponent and treat the case with more singular right hand side of the Monge-Ampère equation.
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 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。
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…
We give a sharp estimate of the modulus of continuity of the solution to the Dirichlet problem for the complex Hessian equation of order $m$ ($1 \leq m \leq n$) with a continuous right hand side and a continuous boundary data in a bounded strongly $m$-pseudoconvex domain $\Om \Subset \C^n$. Moreover when the right hand side is in $L^p(\Om) $, for some $p > n/m$ and the boundary value function is $C^{1,1}$ we prove that the solution is Hölder continuous.
Let $Ω\Subset \mathbb C^n$ be a bounded strongly $m$-pseudoconvex domain ($1\leq m\leq n$) and $μ$ a positive Borel measure with finite mass on $Ω$. Then we solve the Hölder continuous subsolution problem for the complex Hessian equation $(dd^c u)^m \wedge β^{n - m} = μ$ on $Ω$. Namely, we show that this equation admits a unique Hölder continuous solution on $Ω$ with a given Hölder continuous boundary values if it admits a Hölder continuous subsolution on $Ω$. The main step in solving the problem is to establish a new capacity estimate showing that the $m$-Hessian measure of a Hölder continuou…
Stability and strong convergence for complex Hessian equations with $L^1$ data
研究有界超凸域上右边项在\(L^1(\Omega)\)的复\(m -\)Hessian 方程，核心方法未提及，主要贡献是得到 Hessian 能量意义下弱解的强稳定性结果。
Let $1\leq m\leq n$ be two fixed integers. Let $Ω\Subset \mathbb C^n$ be a bounded $m$-hyperconvex domain and $\mathcal A \subset Ω\times ]0,+ \infty[$ a finite set of weighted poles. We define and study properties of the $m$-subharmonic Green function of $Ω$ with prescribed behaviour near the weighted set $A$. In particular we prove uniform continuity of the exponential Green function in both variables $(z,\mathcal A)$ in the metric space $\bar Ω\times \mathcal F$, where $\mathcal F$ is a suitable family of sets of weighted poles in $Ω\times ]0,+ \infty[$ endowed with the Hausdorff distance. …
A Pluripotential-Theoretic Approach to Finiteness of Polarized Calabi--Yau Manifolds
本文提出多势理论框架，通过m-Hessian容量等估计连接复势理论与几何代数有界性，以研究极化Calabi--Yau流形的有限性，并指出关键待发展步骤。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。