A. Scalari, A. Tumanov
Let M be a real analytic strictly pseudoconvex manifold of higher codimension in complex space, and let M' be the cartesian product of two or more compact real analytic strictly convex hypersurfaces. We prove that a germ of a biholomorphic map taking an open set in M to M' extends as a biholomorphic map along any path in M.
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Hölder regularity of Siciak-Zaharjuta extremal functions on compact Hermitian manifolds
研究了紧致Hermitian流形上极值函数的Hölder连续性，证明其局部性质及加权极值函数的Hölder连续性，扩展了Nguyen在紧致Kähler流形上的结果。
Rigidity of Extremal and cscK Bergman Metrics on Pseudoconvex Domains
本文证明有界伪凸域上Bergman度量的极值性、常数量曲率与Kähler--Einstein条件等价，并推广到无界情形，核心工具是ACH边界处的唯一延拓定理。
We generalize Lempert's and Poletsky's works on the description of extremal discs for the Kobayashi metric to a higher order setting.
We consider analytic functions from a reproducing kernel Hilbert space. Given that such a function is of order $ε$ on a set of discrete data points, relative to its global size, we ask how large can it be at a fixed point outside of the data set. We obtain optimal bounds on this error of analytic continuation and describe its asymptotic behavior in $ε$. We also describe the maximizer function attaining the optimal error in terms of the resolvent of a positive semidefinite, self-adjoint and finite rank operator.
We prove a disc formula for the weighted Siciak-Zahariuta extremal function $V_{X,q}$ for an upper semicontinuous function $q$ on an open connected subset $X$ in $\C^n$. This function is also known as the weighted Green function with logaritmic pole at infinity and weighted global extremal function.
We prove that every $J$-holomorphic variational vector field can be realized as derivation $\frac{d}{dt}_{|t=0}f_t$ where $(f_t)$ is a one parametric family of $J$-holomorphic discs. Furthermore, we discuss properness of an extremal $J$-holomorphic disc in a bounded pseudoconvex domain.
We prove two disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of complex affine space. We use these formulas to characterize the polynomial hull of an arbitrary compact subset of complex affine space in terms of analytic discs. Similar results in previous work of ours required the subsets to be connected.
We study extremal discs for the Kobayashi metric. Inspired by work of Lempert on strongly convex domains, we present results on strongly pseudoconvex domains. We also consider a useful biholomorphic invariant, inspired by the Kobayashi (and Carathéodory) metric, and prove several new results about biholomorphic equivalence of domains. Some useful results about automorphism groups of complex domains are also established.
We use our disc formula for the Siciak-Zahariuta extremal function to characterize the polynomial hull of a connected compact subset of complex affine space in terms of analytic discs.
We show that if the Segre varieties of a strictly pseudoconvex hypersurface in $\mathbb{C}^2$ are extremal discs for the Kobayashi metric, then that hypersurface has to be locally spherical. In particular, this gives yet another characterization of the unit sphere in terms of two important invariant families of objects coinciding.
We discuss a new approach to realization of the well-known Weierstrass's programme on efficient continuation of an analytic element corresponding to a~multivalued analytic function with finite number of branch points. Our approach is based on the use of Hermite--Padé polynomials.
Let D be a convex domain with smooth boundary in complex space and let f be a continuous function on the boundary of D. Suppose that f holomorphically extends to the extremal discs tangent to a convex subdomain of D. We prove that f holomorphically extends to D. The result partially answers a conjecture by Globevnik and Stout of 1991.
Using a recent result of Lárusson and Poletsky regarding plurisubharmonic subextensions we prove a disc formula for the quasiplurisubharmonic global extremal function for domains in complex projective space. As a corollary we get a characterization of the projective hull for connected compact sets in complex projective space by the existence of analytic discs.
We study a disc formula for the relative extremal function for Borel sets in complex manifolds.
We establish plurisubharmonicity of envelopes of certain classical disc functionals on locally irreducible complex spaces, thereby generalizing the corresponding results for complex manifolds. We also find new formulae expressing the Siciak-Zaharyuta extremal function of an open set in a locally irreducible affine algebraic variety as the envelope of certain disc functionals, similarly to what has been done for open sets in C^n by Lempert and by Larusson and Sigurdsson.
In the paper we discuss three different notions of extremal holomorphic mappings: weak $m$-extremals, $m$-extremals and $m$-complex geodesics. We discuss relations between them in general case and in the special cases of unit ball, classical Cartan domains, symmetrised bidisc and tetrablock. In particular we show that weak $3$-extremal maps in the symmetrised bidisc are rational thus giving the (partial) answer to a problem posed in a recent paper by J. Agler, Z. Lykova and N. J. Young.
In this paper we characterize the unit disc, the bidisc and the symmetrized bidisc \[ G =\{(z+w,zw):|z|<1,\ |w|<1\} \] in terms of the possession of small classes of analytic maps into the unit disc that suffice to solve all Carathéodory extremal problems in the domain.
We generalize the Poletsky disc envelope formula for the function $\sup \{u \in \PSH(X,ω) ; u\leq ϕ\}$ on any complex manifold $X$ to the case where the real (1,1)-current $ω=ω_1-ω_2$ is the difference of two positive closed (1,1)-currents and $ϕ$ is the difference of an $ω_1$-upper semicontinuous function and a plurisubharmonic function.
We establish disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of complex affine space, generalizing Lempert's formula for the convex case. This function is also known as the pluricomplex Green function with logarithmic growth or a logarithmic pole at infinity. We extend Lempert's formula for this function from the convex case to the connected case.
For each closed, positive (1,1)-current ωon a complex manifold X and each ω-upper semicontinuous function ϕon X we associate a disc functional and prove that its envelope is equal to the supremum of all ω-plurisubharmonic functions dominated by ϕ. This is done by reducing to the case where ωhas a global potential. Then the result follows from Poletsky's theorem, which is the special case ω=0. Applications of this result include a formula for the relative extremal function of an open set in X and, in some cases, a description of the ω-polynomial hull of a set.
Admissibility of Hörmander--Bernhardsson extremal zeros
本文研究了 Hörmander--Bernhardsson 极端函数的零点序列可接受性，证明了其平方零点序列在热追踪展开中的可接受性，并给出了谱ζ函数和正则化乘积的特殊值信息。
Extremal rate of convergence in discrete hyperbolic and parabolic dynamics
本文研究上半平面全纯自映射的动力学行为，利用 Herglotz 表示、Koenigs 函数共形性及双曲距离等工具，刻画了轨道以最慢速率逼近 Denjoy--Wolff 点的双曲与抛物自映射的极值收敛速率。
An extremal $k$-packing is a collection of $k$ mutually disjoint metric discs, embedded in a surface, whose radius is maximal for the given topology. We study compact non-orientable surfaces of genus $g\ge 3$ containing extremal $k$-packings.
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