Nikolay Shcherbina
We give a geometric condition on a compact subset of a complex manifold which is necessary and sufficient for the existence of a smooth strictly plurisubharmonic function defined in a neighbourhood of this set.
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We characterise in this work the $q$-plurisubharmonic functions in terms of the theory of viscosity solutions. We show that an upper semicontinuous function is $q$-plurisubharmonic if and only if its complex Hessian has at most $q$ strictly negative eigenvalues in the viscosity sense. This characterisation is then used to prove that the supremum convolution of a (strictly) $q$-plurisubharmonic function is again (strictly) $q$-plurisubharmonic on a maybe different set of definition. Finally, we use the supremum convolution to deduce a new characterisation for the $q$-pseudoconvex subsets in $\m…
We prove that the maximum of two smooth strictly plurisubharmonic functions on an almost complex manifold can be uniformly approximated by smooth strictly plurisubharmonic functions.
We construct a strictly pseudoconvex domain with smooth boundary whose squeezing function is not plurisubharmonic.
This paper has been withdrawn by the author due to a mistake in the section 4.
We study the plurisubharmonic envelopes of functions in the setting of domains in $\mathbb C^n$. In particular we prove a complex analogue of a result of De Philippis and Figalli concerning the optimal regularity of such envelopes in smooth strictly pseudoconvex domains.
We prove that a relatively compact pseudoconvex domain with smooth boundary in an almost complex manifold admits a bounded strictly plurisubharmonic exhaustion function. We use this result for the study of convexity and hyperbolicity properties of these domains and the contact geometry of their boundaries.
We show that every strictly pseudoconvex domain $Ω$ with smooth boundary in a complex manifold $\mathcal{M}$ admits a global defining function, i.e., a smooth plurisubharmonic function $φ\colon U \to \mathbb R$ defined on an open neighbourhood $U \subset \mathcal{M}$ of $\overlineΩ$ such that $Ω= \{φ< 0\}$, $dφ\neq 0$ on $bΩ$ and $φ$ is strictly plurisubharmonic near $bΩ$. We then introduce the notion of the core $\mathfrak{c}(Ω)$ of an arbitrary domain $Ω\subset \mathcal{M}$ as the set of all points where every smooth and bounded from above plurisubharmonic function on $Ω$ fails to be strictl…
We study the minimum sets of plurisubharmonic functions with strictly positive Monge-Ampère densities. We investigate the relationship between their Hausdorff dimension and the regularity of the function. Under suitable assumptions we prove that the minimum set cannot contain analytic subvarieties of large dimension. In the planar case we analyze the influence on the regularity of the right hand side and consider the corresponding free boundary problem with irregular data. We provide sharp examples for the Hausdorff dimension of the minimum set and the related free boundary. We also draw sever…
A version of Gromov's cup product lemma in which one factor is the (1,0)-part of the differential of a continuous plurisubharmonic function is obtained. As an application, it is shown that a connected noncompact complete Kaehler manifold that has exactly one end and admits a continuous plurisubharmonic function that is strictly plurisubharmonic along some germ of a 2-dimensional complex analytic set at some point has the Bochner-Hartogs property; that is, the first compactly supported cohomology with values in the structure sheaf vanishes.
Regularity of geodesics in the spaces of convex and plurisubharmonic functions II
本研究延续凸与多重次调和函数空间测地线的探究，证明光滑严格凸函数间测地线的最优正则性，拓展$C^{1,α}$范畴正则性理论，并探讨环面不变严格多重次调和函数间测地线的正则性。
In this paper, we use Takeuchi's Theorem to show that for every Lipschitz pseudoconvex domain $Ω$ in $\mathbb{CP}^n$ there exists a Lipschitz defining function $ρ$ and an exponent $0<η<1$ such that $-(-ρ)^η$ is strictly plurisubharmonic on $Ω$. This generalizes a result of Ohsawa and Sibony for $C^2$ domains. In contrast to the Ohsawa-Sibony result, we provide a counterexample demonstrating that we may not assume $ρ=-δ$, where $δ$ is the geodesic distance function for the boundary of $Ω$.
We study minimum sets of singular plurisubharmonic functions and their relation to upper contact sets. In particular we develop an algorithm checking when a naturally parametrized curve is such a minimum set. The case of Koch curves is studied in detail. We also study the size of the set of upper non-contact points. We show that this set is always of Lebesgue measure zero thus answering an open problem in the viscosity approach to the complex Monge-Ampère equation. Finally, we prove that similarly to the case of convex functions, strictly plurisubharmonic lower tests yield existence of upper t…
On the finite energy classes of quaternionic plurisubharmonic functions
本文研究了Cegrell类型四元数次调和函数的有限$p$-能量类$E_p$，构造例子证明当$p>0$且$p\ eq 1$时能量估计最优常数严格大于$1$，从而得出无法用变分法求解该类四元数Monge-Ampère方程的结论。
We study the geometric properties of complex manifolds possessing a pair of plurisubharmonic functions satisfying Monge-Ampère type of condition. The results are applied to characterize complex manifolds biholomorphic to $\C^{N}$ viewed as a product of lower dimensional complex euclidean spaces.
This note establishes smooth approximation from above for J-plurisubharmonic functions on an almost complex manifold (X,J). The following theorem is proved. Suppose X is J-pseudoconvex, i.e., X admits a smooth strictly J-plurisubharmonic exhaustion function. Let u be an (upper semi-continuous) J-plurisubharmonic function on X. Then there exists a sequence {u_j} of smooth, strictly J-plurisubharmonic functions point-wise decreasing down to u. On any almost complex manifold (X,J) each point has a fundamental neighborhood system of J-pseudoconvex domains, and so the theorem above establishes loca…
We construct a special plurisubharmonic defining function for a smoothly bounded strictly pseudoconvex domain so that the determinant of the complex Hessian vanishes to high order on the boundary. This construction, coupled with regularity of solutions of complex Monge-Ampere equation and the reflection principle, enables us to give a new proof of the Fefferman mapping theorem.
In this note we investigate the regularity of geodesics in the space of convex and plurisubharmonic functions. In the real setting we prove (optimal) local C^{1,1} regularity. We construct examples which prove that the global C^{1,1} regularity fails both in the real and complex case in contrast to the Kähler manifold setting. Finally we show a necessary and sufficient conditions for existence of a smooth geodesic between two smooth strictly convex functions.
Let $(M,ω)$ be a Kahler manifold. An integrable function on M is called $ω^q$-plurisubharmonic if it is subharmonic on all q-dimensional complex subvarieties. We prove that a smooth $ω^q$-plurisubharmonic function is q-convex. A continuous $ω^q$-plurisubharmonic function admits a local approximation by smooth, $ω^q$-plurisubharmonic functions. For any closed subvariety $Z\subset M$, $\dim Z < q$, there exists a strictly $ω^q$-plurisubharmonic function in a neighbourhood of $Z$ (this result is known for q-convex functions). This theorem is used to give a new proof of Sibony's lemma on integrabi…
In this paper we introduce and study the notion of plurisubharmonic functions in calibrated geometry. These functions generalize the classical plurisubharmonic functions from complex geometry and enjoy many of their important properties. Moreover, they exist in abundance whereas the corresponding pluriharmonics are generally quite scarce. A number of the results established in complex analysis via plurisubharmonic functions are extended to calibrated manifolds. This paper investigates, in depth, questions of: pseudo-convexity and cores, positive phi-currents, Duval-Sibony Duality, and boundari…
Strong localization of invariant metrics
本文研究了复空间域边界点附近不变度量与挤压函数的局部化问题，给出了峰值点处的定量强局部化及反峰值点处的弱局部化，并分析了严格伪凸边界点附近度量的行为。
We present some results dealing with the local geometry of almost complex manifolds. We establish mainly the complete hyperbolicity of strictly pseudoconvex domains, the extension of plurisubharmonic functions through generic submanifolds and the elliptic regularity of some diffeomorphisms in almost complex manifolds with boundary.
Let $D=\{ρ< 0\}$ be a smooth relatively compact domain in an almost complex manifold $(M,J)$, where $ρ$ is a smooth defining function of $D$, strictly $J$-plurisubharmonic in a neighborhood of the closure $\overline{D}$ of $D$. We prove that $D$ has a connected boundary and is Gromov hyperbolic.
A complex manifold $X$ is \emph{weakly complete} if it admits a continuous plurisubharmonic exhaustion function $ϕ$. The minimal kernels $Σ_X^k, k \in [0,\infty]$ (the loci where are all $\mathcal{C}^k$ plurisubharmonic exhaustion functions fail to be strictly plurisubharmonic),introduced by Slodkowski-Tomassini, and the Levi currents, introduced by Sibony, are both concepts aimed at measuring how far $X$ is from being Stein. We compare these notions, prove that all Levi currents are supported by all the $Σ_X^k$'s, and give sufficient conditions for points in $Σ_X^k$ to be in the support of so…
In this paper, we explore the existence of pluricomplex Green functions for Stein manifolds from a functional analysis point of view. For a Stein manifold $M$, we will denote by $O(M)$ the Fréchet space of analytic functions on $M$ equipped with the topology of uniform convergence on compact subsets. In the first section, we examine the relationship between the existence of pluricomplex Green functions and the diametral dimension of $O(M)$. This led us to consider negative plurisubharmonic functions on $M$ with a nontrivial relatively compact sublevel set (semi-proper). In section 2, we charac…
Let $M$ be a compact, pseudoconvex-oriented, $(2n+1)$-dimensional, abstract CR manifold of hypersurface type, $n\geq 2$. We prove the following: (i) If $M$ admits a strictly CR-plurisubharmonic function on $(0,q_0)$-forms, then the complex Green operator $G_q$ exists and is continuous on $L^2_{0,q}(M)$ for degrees $q_0\le q\le n-q_0$. In the case that $q_0=1$, we also establish continuity for $G_0$ and $G_n$. Additionally, the $\bar\partial_b$-equation on $M$ can be solved in $C^\infty(M)$. (ii) If $M$ satisfies "a weak compactness property" on $(0,q_0)$-forms, then $G_q$ is a continuous opera…
We study the Parabolic complex Monge-Ampére equation in a bounded strictly pseudoconvex domain in \mathbb{C}^n, with the boundary condition u=φand the initial condition u=u_0. In this paper, we consider the case where φis smooth and u_0 is an arbitrary plurisubharmonic function in a neighbourhood of Ωsatisfying u_0=φ(.,0) on the boundary of Ω.
We introduce and study the notion of plurisubharmonic functions in calibrated geometry. These functions generalize the classical plurisubharmonic functions from complex geometry and enjoy their important properties. Moreover, they exist in abundance whereas the corresponding pluriharmonics are generally quite scarce. A number of the results established in complex analysis via plurisubharmonic functions are extended to calibrated manifolds. In particular, the notion of pseudo-convexity for a calibrated manifold (X,ϕ) is introduced and studied. Analogues of totally real submanifolds are also int…
Let $(V,ω)$ be a compact Kähler manifold such that $V$ admits a cover by Zariski-open Stein sets with the property that $ω$ has a strictly plurisubharmonic exhaustive potential on each element of the cover. If $X\subset V$ is an analytic subvariety, we prove that any $ω|_X$-plurisubharmonic function on $X$ extends to a $ω$-plurisubharmonic function on $V$. This result generalizes a previous result of ours on the extension of singular metrics of ample line bundles. It allows one to show that any transcendental Kähler class in the real Neron-Severi space $NS_{\mathbb R}(V)$ has this extension pr…
On the Rigidity of Analytic Mappings in Complex Analysis and Geometry
本文研究复几何中解析映射和亚调和函数的刚性性质，通过不同条件证明了映射的拓扑度、纤维结构和极小值的存在性，揭示了李群作用的微分拓扑障碍。
Maximal Plurisubharmonic Functions and Fujii-Seo Determinants in Hilbert spaces
本文提出利用 Fujii-Seo 正则化确定性描述极值全纯亚调和函数，定义 Fujii-Seo 确定性密度并研究其与 Levi 形的谱端点关系，证明极值性与该密度为零等价，建立全局退化条件和比较原理。
Geodesic connectivity and rooftop envelopes in the Cegrell classes
本文在Cegrell类中证明复Monge-Ampère方程可比奇异性解的唯一性，解决Cegrell提出的未决问题，并借此推广测地连通性判据及屋顶包络的幂等性条件。
Let $w_0$ be a bounded, $C^3$, strictly plurisubharmonic function defined on $B_1\subset \mathbb{C}^n$. Then $w_0$ has a neighborhood in $L^{\infty}(B_1)$ with the following property: for any continuous, plurisubharmonic function $u$ in this neighborhood solving $1-\eps \le MA(u)\le 1+\eps$, one has $u\in W^{2,p}(B_{\frac{1}{2}})$, as long as $\eps>0$ is small enough depending only on $n$ and $p$. This partially generalizes Caffarelli's interior $W^{2,p}$ estimates for real Monge-Ampere to the complex version.
Hypersurface Convexity and Extension of Kähler Forms
本文推广了S. Nemirovski的结果，刻画了射影或Stein流形中紧集的补为正除子并的等价条件，关联了严格多重次调和函数的$dd^c$形式的Hodge延拓性质。
We establish a lower estimate for the Kobayashi-Royden infinitesimal pseudometric on an almost complex manifold $(M,J)$ admitting a bounded strictly plurisubharmonic function. We apply this result to study the boundary behaviour of the metric on a strictly pseudoconvex domain in $M$ and to give a sufficient condition for the complete hyperbolicity of a domain in $(M,J)$. Finally we obtain the regularity up to the bounday of $J$-holomorphic discs attached to a totally real submanifold in $M$.
Let X be a complex manifold of dimension 2, which admits a strictly plurisubharmonic function r which is proper as a function with values in the intervall ]inf r, sup r[. We prove that the concave end of X can be compactified, if and only if, the first cohomology of X is Hausdorff.
In the study of complex flag manifolds, flag domains and their cycle spaces, a key point is the fact that the cycle space $\mathcal M_D$ of a flag domain $D$ is a Stein manifold. That fact has a long history. The earliest approach relied on construction of a strictly plurisubharmonic function on $\mathcal M_D$, starting with a $q$--convex exhaustion function on $D$, where $q$ is the dimension of a particular maximal compact subvariety of $D$ (we use the normalization that 0--convex means Stein). Construction of that exhaustion function on $D$ required that $D$ be measurable. In that case the e…
Let $Φ$ be a strictly plurisubharmonic and radial function on the unit disk ${\cal D}\subset {\complex}$ and let $g$ be the \K metric associated to the \K form $ω=\frac{i}{2}\partial\bar\partialΦ$. We prove that if $g$ is $g_{eucl}$-balanced of height 3 (where $g_{eucl}$ is the standard Euclidean metric on ${\complex}={\real}^2$), and the function $h(x)=e^{-Φ(z)}$, $x=|z|^2$, extends to an entire analytic function on ${\real}$, then $g$ equals the hyperbolic metric. The proof of our result is based on a interesting characterization of the function $f(x)=1-x$.
Let $X$ be a complex manifold and $M\subset X$ a compact, smooth, pseudoconvex CR manifold of dimension $2n-1$. (Here $n\ge 3$ or, in case $n=2$, it is made the extra assumption that $\dib_b$ has closed range on functions.) Assume that there exists a strictly CR-plurisubharmonic function in a neighborhood of $M$ in $X$. In this situation, there are here proved \begin{enumerate} \item[(i)] The global existence of $C^\infty$ solutions to the tangential Cauchy-Riemann operator $\dib_b$. \item[(ii)] The global hypoellipticity of the Kohn-Laplacian $\Box_b$, under the additional condition of "weak …
Let $w_0$ be a bounded, $C^3$, strictly plurisubharmonic function defined on $B_1\subset \mathbb{C}^n$. Then $w_0$ has a neighborhood in $L^{\infty}(B_1)$. Suppose that we have a function $ϕ$ in this neighborhood with $1-ε\le MA(u)\le 1+ε$ and there exists a function $u$ solving the linearized complex Monge-Ampere equation: $det(ϕ_{k\bar{l}})ϕ^{I\bar{j}}u_{I\bar{j}}=0$. Then one has an estimate on $|u|_{C^α(B_{\frac{1}{2}})}$ for some $α>0$ depending on $n$, as long as $ε$ is small depending on $n$. This partially generalizes Caffarelli's estimate for linearized real Monge-Ampere equation to t…
A note on Lang's conjecture for quotients of bounded domains
本文针对万有覆叠空间具有有界严格多重次调和函数的射影流形（包括有界域的紧自由商空间），验证了Lang关于复射影流形Kobayashi双曲性等价于自身及所有子簇为一般型的猜想。
Bergman Kernels and Hankel Operators on Weighted Fock Spaces in Several Complex Variables
本文在多复变量加权Fock空间中，通过正则化曲率假设建立Bergman核估计，并构造\bar\partial解算子，进而用加权IDA空间刻画Hankel算子的有界性与紧性。
Global regularity for the $\bar\partial$-Neumann problem on pseudoconvex manifolds
本文为拟凸域上的 $\bar\partial$-Neumann 问题建立了 $(p,q)$-形式全局正则性的一般充分条件，主要利用严格多重次调和函数或正全纯双截曲率，以及满足弱紧性估计的横截向量场族，并给出应用实例。
About top-degree $L^2$- and $L^{2,\mathrm{loc}}$-Dolbeault cohomologies of complex spaces with pseudoconvex boundary
本文证明在具有 $\mathscr{C}^3$ 光滑边界的伪凸相对紧域中，$L^2$- 和 $L^{2,\mathrm{loc}}$-Dolbeault $(n,q)$-上同调群在 $q>0$ 时消失，并推广到 Nakano 半正定全纯向量丛取值的情形，进而证明由严格多重次调和函数定义的域中两类上同调群的等价性。
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