The Jensen envelope is plurisubharmonic on all manifolds
Finnur Larusson, Ragnar Sigurdsson
备注：
The paper, while correct, has been withdrawn, as the main result turned out to be an easy consequence of Edwards' Theorem. This was belatedly discovered by the authors and later pointed out by an anonymous referee
The Jensen envelope $Jϕ$ of an upper semicontinuous function $ϕ$ on a complex manifold X is defined at $x\in X$ as the infimum of $μ(ϕ)$ over all Jensen measures $μ$ centred at x. The Poisson envelope $Pϕ$ is defined by using only the boundary measures of analytic discs centred at x. One of the main open problems in the theory of disc functionals is whether the Poisson envelope is plurisubharmonic on an arbitrary manifold. This is equivalent to the two envelopes being equal, so plurisubharmonicity of $Jϕ$ is a necessary condition for $Pϕ$ to be plurisubharmonic. We prove that the Jensen envelope is plurisubharmonic, with no assumptions on the manifold X. Hence $Jϕ$ is the largest plurisubharmonic function smaller than $ϕ$. We also show that the Poisson envelope is plurisubharmonic if and only if boundary measures of analytic discs are dense among Jensen measures.
In this paper, we extend some recent results of Guedj-Lu-Zeriahi about psh envelopes of bounded functions on bounded domains in $\mathbb{C}^n$. We also present a result on the regularity of psh envelopes.
Given a negative plurisubharmonic function $ϕ$ in a bounded pseudoconvex domain of ${\mathbb C}^n$, we introduce and study its residual function $g_ϕ$ determined by the asymptotic behavior of $ϕ$ near its singularity points, both inside the domain and on its boundary. For certain choices of $ϕ$, the function $g_ϕ$ coincides with different versions of pluricomplex Green functions. The considerations are motivated by a problem on when two given plurisubharmonic functions can be connected by a plurisubharmonic geodesic.
On bounded B-regular domains, we study envelopes of plurisubharmonic functions bounded from above by a function $ϕ$ such that $ϕ^*=ϕ_*$ on the closure of the domain. For $ϕ$ satisfying certain additional criteria limiting its behavior at the singularities, we establish a set where the Perron-Bremermann envelope $P ϕ$ is guaranteed to be continuous. This result is a generalization of a classic result in pluripotential theory due to J. B. Walsh. As an application, we show that the complex Monge--Ampère equation \[ (dd^cu)^n = μ\] being uniquely solvable for continuous boundary data implies that …
We develop a new approach to $L^{\infty}$-a priori estimates for degenerate complex Monge-Ampère equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. Our method allows one to obtain new and efficient proofs of several fundamental results in Kähler geometry as we explain in this article. In a sequel we shall explain how this approach also applies to the hermitian setting producing new relative a priori bounds, as well as existence results.
We develop a new approach to $L^{\infty}$-a priori estimates for degenerate complex Monge-Ampère equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. In a prequel \cite{GL21a} we have shown how this method allows one to obtain new and efficient proofs of several fundamental results in Kähler geometry. In \cite{GL21b} we have studied the behavior of Monge-Ampère volumes on hermitian manifolds. We extend here the techniques of \cite{GL21a} to the hermitian setting and use the bounds established in \cite{GL21b}, producing new …
In \cite{GL21a} we have developed a new approach to $L^{\infty}$-a priori estimates for degenerate complex Monge-Ampère equations, when the reference form is closed. This simplifying assumption was used to ensure the constancy of the volumes of Monge-Ampère measures. We study here the way these volumes stay away from zero and infinity when the reference form is no longer closed. We establish a transcendental version of the Grauert-Riemenschneider conjecture, partially answering conjectures of Demailly-Păun \cite{DP04} and Boucksom-Demailly-Păun-Peternell \cite{BDPP13}. Our approach relies on a…
Dipartimento di Matematica, Università degli Studi di Roma Tor Vergata（罗马托尔维加塔大学）；Department of Mathematics and Physics, University of Stavanger（斯塔万格大学）
We prove that quasi-plurisubharmonic envelopes with prescribed analytic singularities in suitable big cohomology classes on compact Kähler manifolds have the optimal $C^{1,1}$ regularity on a Zariski open set. This also proves regularity of certain pluricomplex Green's functions on Kähler manifolds. We then go on to prove the same regularity for envelopes when the manifold is assumed to have boundary. As an application, we answer affirmatively a question of Ross--Witt-Nyström concerning the Hele-Shaw flow on an arbitrary Reimann surface.
