Abstract In this article, we extend our earlier unique continuation results of Pan and Zhang (2025, Comm. Anal. Geom. 33, 1975–2014) for the Schrödinger-type inequality |∂¯u|≤V|u| $ |\bar \partial u| \le V|u|$ StartAbsoluteValue ModifyingAbove partial differential With bar u EndAbsoluteValue less than or equals upper V StartAbsoluteValue u EndAbsoluteValue on a domain in Cn $\mathbb C^n$ double struck upper C Superscript n , by removing the smoothness assumption on solutions u=(u1,…,uN) $u = (u_1, \ldots , u_N)$ u equals left parenthesis u 1 comma ellipsis comma u Subscript upper N Baseline right parenthesis . More specifically, we establish the unique continuation property for Wloc1,1 $W_{loc}^{1,1}$ upper W Subscript l o c Superscript 1 comma 1 solutions when the potential V∈Llocp $V\in L_{loc}^p $ upper V element of upper L Subscript l o c Superscript p , p>2n $ p>2n$ p greater than 2 n , and for Wloc1,2n+ϵ $W_{loc}^{1,2n+\epsilon }$ upper W Subscript l o c Superscript 1 comma 2 n plus epsilon solutions when V∈Lloc2n $V\in L_{loc}^{2n}$ upper V element of upper L Subscript l o c Superscript 2 n with N=1 $N=1$ upper N equals 1 or n=2 $n = 2$ n equals 2 . Although the unique continuation property fails in general if V∈Llocp,p<2n $V\in L_{loc}^{p}, p<2n$ upper V element of upper L Subscript l o c Superscript p Baseline comma p less than 2 n , we show that the property still holds for Wloc1,1 $W_{loc}^{1,1}$ upper W Subscript l o c Superscript 1 comma 1 solutions when V $V $ upper V is a small constant multiple of 1|z| $ \frac {1}{|z|}$ StartFraction 1 Over StartAbsoluteValue z EndAbsoluteValue EndFraction . The key idea that allows us to lower the regularity assumption is a careful analysis of how regularity and flatness are transferred to the sliced functions along complex radial directions.
Abstract We show that if X is a regular complex projective threefold and Y is a very general surface of sufficiently high degree containing a thick infinitesimal neighborhood Z of a smooth curve C , then the restriction map r : Pic X → Pic Y $r: \textrm{Pic}\ X \to \textrm{Pic}\ Y$ r colon Pic upper X right arrow Pic upper Y is an isomorphism. Our proof uses Moishezon’s Noether–Lefschetz theorem and some general tools of Hartshorne and Polini. We give the complete picture for all d when X = P 3 $X = {\mathbb P}^3$ upper X equals double struck upper P cubed , C is a line, and m = 2 $m=2$ m equals 2 , using a classification of Urabe to handle the difficult case of quartic surfaces.
Abstract Joyce structures are a class of geometric structures that first arose in relation to Donaldson–Thomas theory. There is a special class of examples, called class S [ A 1 ] $S[A_1]$ upper S left bracket upper A 1 right bracket , whose underlying manifold parameterizes Riemann surfaces of some fixed genus equipped with a meromorphic quadratic differential with poles of fixed orders. We study two Joyce structures of this type using the isomonodromic systems associated with the Painlevé II and III 3 $_3$ Subscript 3 equations. We give explicit formulas for the Plebański functions of these Joyce structures and compute several associated objects, including their tau functions, which we explicitly relate to the corresponding Painlevé tau functions. We show that the behavior of the Joyce structure near the zero section can be studied analytically through poles of Painlevé equations. The systematic treatment gives a blueprint for the study of more general Joyce structures associated with meromorphic quadratic differentials on the Riemann sphere.
Abstract In this article, we consider a generalization of the conjugate Hardy H 2 $H^2$ upper H squared spaces and give some properties of the minimal norm of the generalization and some relations between the norm of the generalization and the minimal L 2 $L^2$ upper L squared integrals. As applications, we give some monotonicity results for the conjugate Hardy H 2 $H^2$ upper H squared kernels and the Bergman kernels on planar regions, and some relations between the conjugate Hardy H 2 $H^2$ upper H squared kernels and the Bergman kernels on planar regions.
We give a new definition of matrix Schwarzian derivative, which is simpler than the Lagrange Schwarzian derivative and also provides an alternative to other definitions which appear in the literature. Some basic properties are discussed, in particular, analogs of M & ouml;bius invariance and the result of a change of independent variable, these being the two properties of the scalar Schwarzian derivative often considered to account for its universality. We then use our new definition of matrix Schwarzian derivative to construct new Schwarzian matrix ordinary and partial differential equation hierarchies: a Schwarzian matrix second Painlev & eacute; hierarchy and a Schwarzian matrix Korteweg-de Vries hierarchy, respectively. In addition, we define a new matrix second Painlev & eacute; hierarchy.
