Inventiones mathematicae

Li, Dong; Yang, Kai

doi: 10.1007/s00222-026-01419-3pmid: N/A

The linearized operator ℒ associated with the three-dimensional cubic nonlinear Schrödinger equation is shown to possess no embedded eigenvalues in the essential spectrum. This result rigorously confirms the key spectral hypothesis underpinning the construction of center-stable manifolds by Schlag (Ann. Math. (2) 169(1):139–227, 2009), and thereby provides the final analytical link for an unconditional stability theory near the ground state soliton. Unlike the one-dimensional case, the 3D model lacks integrability; the non-self-adjointness of ℒ and the non-explicit profile of the ground state render classical techniques insufficient to exclude eigenvalues in the continuum. The proof rests on the introduction of a weight-modulated positivity trap to exploit the hyperbolic instability of the fundamental mode, combined with a novel constrained shooting method and delicate comparison arguments for higher angular momenta. Beyond the cubic model, the machinery introduced here provides a robust analytical framework for addressing the spectral coercivity conjectures central to the Merle–Raphaël’s log-log blow-up on mass-critical NLS.

Tatsuoka, Alison

doi: 10.1007/s00222-026-01420-wpmid: N/A

We show that there exist infinitely many pairwise non-isotopic splitting 3-spheres for two unlinked, unknotted 2-spheres in S4\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$S^{4}$\end{document}. This answers a question posed by Hughes, Kim, and Miller.

Bringmann, Bjoern

doi: 10.1007/s00222-026-01414-8pmid: N/A

We discuss the (1+1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$(1+1)$\end{document}-dimensional wave maps equation with values in a compact Lie group. The corresponding Gibbs measure is given by a Brownian motion on the Lie group, which plays a central role in stochastic geometry. Our main theorem is the almost sure global well-posedness and invariance of the Gibbs measure for the wave maps equation. It is the first result of this kind for any geometric wave equation. Our argument relies on a novel finite-dimensional approximation of the wave maps equation which involves the so-called Killing renormalization. The main part of this article then addresses the global convergence of our approximation and the almost invariance of the Gibbs measure under the corresponding flow. The proof of global convergence requires a carefully crafted Ansatz which includes modulated linear waves, modulated bilinear waves, and mixed modulated objects. The interactions between the different objects in our Ansatz are analyzed using an intricate combination of analytic, geometric, and probabilistic ingredients. In particular, geometric aspects of the wave maps equation are utilized via orthogonality, which has previously been used in the deterministic theory of wave maps at critical regularity. The proof of almost invariance of the Gibbs measure under our approximation relies on conservative structures, which are a new framework for the approximation of Hamiltonian equations, and delicate estimates of the energy increment.

Johansson, Kurt; Viklund, Fredrik

doi: 10.1007/s00222-026-01417-5pmid: N/A

Let D\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$D$\end{document} be a Jordan domain of unit capacity. We study the partition function of a planar Coulomb gas in D\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$D$\end{document} with a hard wall along η=∂D\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\eta = \partial D$\end{document}, Zn(D)=1n!∫Dn∏1⩽k<ℓ⩽n|zk−zℓ|2∏k=1nd2zk.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ Z_{n}(D) =\frac{1}{n!}\int _{D^{n}}\prod _{1\leqslant k < \ell \leqslant n}|z_{k}-z_{\ell }|^{2} \prod _{k=1}^{n} d^{2}z_{k}. $$\end{document} We are interested in how the geometry of η\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\eta $\end{document} is reflected in the large n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$n$\end{document} behavior of Zn(D)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$Z_{n}(D)$\end{document}. We prove that η\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\eta $\end{document} is a Weil-Petersson quasicircle if and only if limn→∞logZn(D)Zn(D)=−112IL(η),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \lim _{n \to \infty } \log \frac{Z_{n}(D)}{Z_{n}(\mathbb{D})} = -\frac{1}{12}I^{L}( \eta ), $$\end{document} where IL\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$I^{L}$\end{document} is the Loewner energy, D\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\mathbb{D}$\end{document} is the unit disc, and logZn(D)=logπn/n!\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\log Z_{n}(\mathbb{D}) = \log \pi ^{n}/n!$\end{document}. We next consider piecewise analytic η\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\eta $\end{document} with m\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$m$\end{document} corners of interior opening angles παp,p=1,…,m\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\pi \alpha _{p}, p=1,\ldots , m$\end{document}. Our main result is the asymptotic formula limn→∞1lognlogZn(D)Zn(D)=−16∑p=1m(αp+1αp−2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \lim _{n\to \infty }\frac{1}{\log n} \log \frac{Z_{n}(D)}{Z_{n}(\mathbb{D})} =-\frac{1}{6}\sum _{p=1}^{m} \left (\alpha _{p}+\frac{1}{\alpha _{p}}-2 \right ) $$\end{document} which is consistent with physics predictions. The starting point of our analysis is an exact expression for logZn(D)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\log Z_{n}(D)$\end{document} in terms of a Fredholm determinant involving the truncated Grunsky operator for D\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$D$\end{document}. The proof of the main result is based on careful asymptotic analysis of the Grunsky coefficients. As further applications of our method we also study the Loewner energy and the related Fekete-Pommerenke energy, a quantity appearing in the analysis of Fekete points, for equipotentials approximating the boundary of a domain with corners. We formulate several conjectures and open problems.