--- title: Poletsky Discs in Analytic and Pluripotential Theory url: https://www.emergentmind.com/topics/poletsky-discs type: topic --- # Poletsky Discs in Analytic and Pluripotential Theory Searching arXiv for recent and foundational papers on Poletsky discs, analytic discs, projective hulls, and related Poletsky theory. Poletsky discs are analytic discs used to encode extremal plurisubharmonic or quasiplurisubharmonic behavior by testing a point against families of discs centered at that point. In a concrete measure-theoretic form, for a point \(p\), an open set \(U\), and \(\epsilon>0\), a Poletsky disc is a \(J\)-holomorphic disc \(u\in \mathcal O(\mathbb D,M,p)\) such that there exists an exceptional set \(E\subset[0,2\pi)\) with \(|E|<\epsilon\) and \(u(e^{it})\in U\) for \(t\notin E\); in the envelope formalism, one takes infima of disc functionals over analytic discs satisfying a center condition \(f(0)=x\) [1710.07059, 1304.1704]. ## 1. Analytic-disc formalism A basic ambient class is \[ \mathcal A_X=\{f:\overline{\mathbb D}\to X: f \text{ continuous on }\overline{\mathbb D},\ f|_{\mathbb D}\text{ holomorphic}\}, \] and, for \(W\subset X\), \[ \mathcal A_X^W=\{f\in \mathcal A_X: f(\mathbb T)\subset W\}. \] Given a disc functional \(H\), its envelope over a class \(\mathcal C\subset \mathcal A_X\) is \[ E_{\mathcal C}H(x)=\inf\{H(f): f\in \mathcal C,\ f(0)=x\}. \] The center condition \(f(0)=x\) is essential: the envelope computes the best value attainable at \(x\) by testing against all discs centered at \(x\). The standard Poisson disc functional is \[ H_\varphi(f)=\int_{\mathbb T}\varphi\circ f\, d\sigma. \] In the Lárusson–Poletsky setting used as the affine model for later projective results, one has \[ \sup\{u(x):u\in \operatorname{PSH}(X),\ u|_W\le \varphi\} = \inf\left\{\int_{\mathbb T}\varphi\circ f\, d\sigma : f\in \mathcal A_X^W,\ f(0)=x\right\}. \] This is the prototypical Poletsky envelope formula: plurisubharmonic subextensions are recovered as envelopes of disc functionals [1304.1704]. The same disc-centered viewpoint also underlies the notion of a Poletsky sequence. For a compact \(K\) in a complex space \(X\), a sequence \(f_j:\overline{\mathbb D}\to X\) with \(f_j(0)=x\) is a Poletsky sequence for \((K,x)\) when \[ \bigl|\{t\in [0,2\pi]: \operatorname{dist}_X(f_j(e^{it}),K)<1/j\}\bigr| > 2\pi-\frac1j. \] The boundary circle is therefore not required to lie entirely near \(K\); only a set of angles of asymptotically full measure must map close to \(K\) [1201.0653]. ## 2. Extremal functions and disc formulas The projective-space formulation replaces ordinary plurisubharmonicity by \(\omega\)-plurisubharmonicity relative to the Fubini–Study form \(\omega\). On \(\mathbb P^n\), an upper semicontinuous function \(u\) is \(\omega\)-plurisubharmonic if \[ dd^c u + \omega \ge 0, \] and the class is denoted \(\operatorname{PSH}(\mathbb P^n,\omega)\). For a domain \(W\subset \mathbb P^n\) and an upper semicontinuous \(\varphi\) on \(W\), the largest \(\omega\)-plurisubharmonic function bounded above by \(\varphi\) on \(W\) admits the disc formula \[ \sup\{ u(x) ; u \in \operatorname{PSH}(\mathbb P^n,\omega),\ u|_W \leq \varphi \} = \inf \left\{ -\frac{1}{2\pi}\int_{\mathbb D} \log|\zeta|\, f^*\omega + \int_{\mathbb T} \varphi\circ f\, d\sigma\ ;\ f \in \mathcal A_{\mathbb P^n}^W,\ f(0)=x \right\}. \] The corresponding projective disc functional is \[ H_{\omega,\varphi}(f)= -\frac{1}{2\pi}\int_{\mathbb D}\log|\zeta|\, f^*\omega + \int_{\mathbb T}\varphi\circ f\, d\sigma. \] The first term is the correction reflecting the curvature/current \(\omega\); it is the interior mass term that distinguishes the projective quasiplurisubharmonic setting from the affine Poisson functional [1304.1704]. When \(\varphi=0\), one obtains the projective global extremal function \[ \Lambda_W(x) = \inf\left\{ -\frac{1}{2\pi}\int_{\mathbb D}\log|\zeta|\, f^*\omega \ ;\ f\in \mathcal A_{\mathbb P^n}^W,\ f(0)=x \right\}. \] A central identity behind this formula is \[ \pi^*\omega = dd^c \log\|\cdot\| \quad \text{on } \mathbb C^{n+1}\setminus\{0\}, \] where \(\pi:\mathbb C^{n+1}\setminus\{0\}\to \mathbb P^n\) is the standard projection. The projective problem is lifted to \(\mathbb C^{n+1}\setminus\{0\}\) through logarithmically homogeneous functions, \[ \widetilde\varphi(z)=\varphi(\pi(z))+\log\|z\|, \] and the correspondence \[ u\in \operatorname{PSH}(\mathbb P^n,\omega) \quad \Longleftrightarrow \quad u\circ \pi + \log\|\cdot\| \in \operatorname{PSH}(\mathbb C^{n+1}\setminus\{0\}) \] transfers the Poletsky-type subextension problem upstairs before descending it back to projective space [1304.1704]. ## 3. Projective hulls and Poletsky sequences For a compact \(K\subset \mathbb P^n\), the projective hull \(\widehat K_{\mathbb P^n}\) is the set of points \(x\in \mathbb P^n\) such that there exists \(C(x)<\infty\) with \[ |P(x)| \le C(x)^d \sup_K |P| \] for every holomorphic section \(P\) of \(\mathcal O_{\mathbb P^n}(d)\) and every integer \(d>0\). The associated extremal function satisfies \[ \widehat K_{\mathbb P^n}=\{x\in\mathbb P^n: V_K(x)<+\infty\}, \qquad V_K(x)=\log C_K(x). \] In homogeneous coordinates, if \[ S_K = S\cap \pi^{-1}(K)\subset \mathbb C^{n+1}, \] then \[ \widehat K_{\mathbb P^n}=\pi(\widehat{S_K}\setminus\{0\}), \] which reduces projective hull questions to polynomial convexity in \(\mathbb C^{n+1}\) [1201.0653]. The projective-hull analogue of Poletsky’s theorem states that \(x\in \widehat K_{\mathbb P^n}\) if and only if there exists a Poletsky sequence \[ f_j:\overline{\mathbb D}\to \mathbb P^n \] for \((K,x)\) with the bounded lifting property: there are liftings \(F_j:\overline{\mathbb D}\to \mathbb C^{n+1}_*\) and a constant \(C>0\) such that \[ \sup_{t\in[0,2\pi]} |F_j(e^{it})| \le C\,|F_j(0)|. \] The bounded lifting condition is essential; without it, analytic discs in projective space can approach a point of \(K\) along a projective line and make their boundaries arbitrarily close to \(K\), regardless of whether the center lies in the projective hull [1201.0653]. For connected compact sets, stronger full-boundary approximation statements are available. If \(K\subset \mathbb P^n\) is compact and connected and \(\Lambda>0\), then \(x\in \widehat K(\Lambda)\) is equivalent to the following quantitative disc condition: for every \(\varepsilon>0\) and every neighborhood \(U\) of \(K\), there exists \[ f\in \mathcal A_{\mathbb P^n}^U,\qquad f(0)=x, \] such that \[ -\frac{1}{2\pi}\int_{\mathbb D}\log|\zeta|\,f^*\omega < \Lambda+\varepsilon. \] In this form, projective hull membership is encoded by discs whose boundaries lie in every neighborhood of \(K\) and whose Fubini–Study mass term remains quantitatively controlled [1304.1704]. ## 4. Direct construction on compact almost complex manifolds On a smooth, connected compact manifold \(M\) equipped with an almost complex structure \(J\), a map \(u:(M',J')\to(M,J)\) is \((J',J)\)-holomorphic if \[ J(u(p))\circ d_pu = d_pu\circ J'(p). \] For a point \(p\in M\), the disc space \[ \mathcal O(\mathbb D,M,p) \] consists of smooth maps \(u:\mathbb D\to M\) which are \(J\)-holomorphic in a neighborhood of \(\overline{\mathbb D}\) and satisfy \(u(0)=p\). In this setting, a Poletsky disc associated to \(p\), \(\epsilon>0\), and an open set \(U\subset M\) is precisely such a disc with boundary in \(U\) outside an exceptional set \(E\subset[0,2\pi)\) of measure \(<\epsilon\) [1710.07059]. A direct existence theorem holds for compact almost complex manifolds with a regular almost complex structure and the doubly tangent property: for every \(p\in M\), every \(\epsilon>0\), and every open set \(U\subset M\), there exist \(u\in\mathcal O(\mathbb D,M,p)\) and \(E\subset[0,2\pi)\) such that \(|E|<\epsilon\) and \[ u(e^{it})\in U \qquad \text{for } t\notin E. \] The stronger approximation theorem says that for any \(C^2\)-map \[ \lambda:\partial\mathbb D\to M, \] there exist \(u\in\mathcal O(\mathbb D,M,p)\) and \(E\subset[0,2\pi)\), \(|E|<\epsilon\), such that \[ \mathrm{dist}\bigl(u(e^{it}),\lambda(e^{it})\bigr)<\epsilon \qquad \text{for } t\in [0,2\pi)\setminus E. \] This gives a direct construction of Poletsky discs via local arc approximation and a Runge-type theorem by A. Gournay, rather than deriving existence from the disc-envelope formula [1710.07059]. The proof combines two ingredients. First, a local approximation theorem along boundary arcs produces \(J\)-holomorphic maps near chosen arcs: \[ \|u-\varphi\|_{C^{1,\alpha}(\Gamma)}<\epsilon \] for a prescribed \(C^2\)-map \(\varphi\) on a smooth arc \(\Gamma\). Second, Gournay’s Runge-type theorem globalizes such local data on a compact Riemann surface. The resulting construction is geometric: one approximates a boundary map on finitely many disjoint arcs covering almost all of \(\partial\mathbb D\), inserts a small \(J\)-holomorphic disc at the center, connects the pieces continuously, and then applies global approximation to obtain a single disc centered at \(p\) [1710.07059]. ## 5. Poletsky–Stessin spaces and disc-related pluripotential theory Poletsky–Stessin Hardy spaces belong to the same pluripotential lineage but are not disc spaces in the literal sense. On a hyperconvex domain \(D\subset \mathbb C^n\), with a negative continuous plurisubharmonic exhaustion \(u\), one defines \[ B_u(r)=\{z\in D:u(z)