Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.39425

The functional analytic approach establishes the Corona theorem in the theory of SCV

S. R. Patel
View original source ↗Read source document ↗Export metadata ↓

Abstract

We give an affirmative solution of the original Corona problem (1962) for certain bounded pseudoconvex domains or polydomains in $\mathbb{C}^n$, including the balls and polydiscs, by using the Gleason solvability property of such domains for $H^\infty(G)$. In fact, we establish an equivalency of the Corona problem and Gleason's problem (1964) in the theory of several complex variables. Indeed, we extend our recent works on both problems based on the functional analytic approaches, and extend recent work of Clos, as well as apply the usual Banach algebra method. As an application, we establish the I. J. Schark's theorem for these $G$'s. We also obtain a welcome extension of a result of Krantz and Li (1995) for a bounded, strictly pseudoconvex domain in $\mathbb{C}^n$ with $C^3$ boundary.

Explore related subjects

math.CVmath.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

S. R. Patel. 2026-10-29. The functional analytic approach establishes the Corona theorem in the theory of SCV. https://arxiv.org/abs/2610.39425

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Thompson's groups and Teichmüller modular groups of generalized Cantor sets

Thompson's groups, which are denoted by $F, T$ and $V$, were introduced by R. Thompson. It is known that they are related to various fields in mathematics. In this paper, we discuss Thompson's groups $F, T$ and $V$ in Teichmüller modular groups of Teichmüller spaces of generalized Cantor sets with certain conditions by using a quasiconformal geometric method. Moreover, we show that groups corresponding to $F$ and $T$ act properly discontinuously on such Teichmüller spaces. We also show the same results for the asymptotic Teichmüller space. E. de Faria, F. P. Gardiner and W. J. Harvey show similar results on Thompson's group $F$ and asymptotic Teichmüller spaces. So, our results are improvement of them. We also discuss the actions of Thompson's groups on other spaces which we call \lq\lq pre-Tiechmüller spaces" of generalized Cantor sets. Especially, we show that the action of a group which is isomorphic to Thompson's group $V$ on the pre-Teichüller space is not properly discontinuous while groups isomorphic to $F$ and $T$ act properly discontinuously. Finally, we construct infinitely many distinct Teichmüller spaces of generalized Cantor sets where our results can apply.

math.CV↗
arXiv2026

Spectrum of Complete Kähler Metrics on Bounded Hyperconvex Domains

Let $Ω\subset\mathbb{C}^{n}$ be a bounded domain, and let $ρ:Ω\to[-1,0)$ be a smooth strictly plurisubharmonic exhaustion function. We consider the logarithmic potential $g=-\log(-ρ)$ and the associated complete Kähler metric $ω=dd^{c} g$. We prove that if $ρ$ satisfies the finite weighted Monge-Ampère mass condition with weight $(-ρ)^{\varepsilon}$ for every $\varepsilon>0$, then the spectrum and the essential spectrum of the Laplace-Beltrami operator of $(Ω,ω)$ both coincide with $[n^{2},\infty)$. In particular, the bottom of the spectrum satisfies $λ_{0}(Δ_ω,Ω)=n^{2}$, and $n^{2}$ is not an $L^{2}$-eigenvalue. For each $α> \frac{n}{2}$, we set $f=(-ρ)^α$ and prove that $f$ belongs to $W^{1,2}(Ω,ω)$ if and only if the Monge-Ampère mass of $ρ$ weighted by $(-ρ)^{2α-n}$ is finite. The integral identities underlying this equivalence, applied to the complex powers $(-ρ)^{α+it}$, give, after normalization, a singular Weyl sequence at every $n^{2}+4t^{2}$, $t\in\mathbb{R}$, as $α\downarrow\frac{n}{2}$. Combining our theorem with Cegrell's finite-mass exhaustion theorem, we obtain a complete Kähler metric with spectrum and essential spectrum equal to $[n^{2},\infty)$ on every bounded hyperconvex domain.

math.CV↗