Open Subset Research Articles

On the extension problem for semiconcave functions with fractional modulus

Consider a locally Lipschitz function u on the closure of a possibly unbounded open subset Ω of Rn with nonempty boundary. Suppose u is (locally) semiconcave on Ω¯ with a fractional semiconcavity modulus. Is it possible to extend u in a neighborhood of any boundary point retaining the same semiconcavity modulus? We show that this is indeed the case and we give two applications of this extension property. First, we derive an approximation result for semiconcave functions on closed domains. Then, we use the above extension property to study the propagation of singularities of semiconcave functions at boundary points.

Floer Theory of Higher Rank Quiver 3-folds

We study threefolds Y fibred by A_m-surfaces over a curve S of positive genus. An ideal triangulation of S defines, for each rank m, a quiver Q(Delta _m), hence a CY_3-category mathcal {C}(W) for any potential W on Q(Delta _m). We show that for omega in an open subset of the Kähler cone, a subcategory of a sign-twisted Fukaya category of (Y,omega ) is quasi-isomorphic to (mathcal {C},W_{[omega ]}) for a certain generic potential W_{[omega ]}. This partially establishes a conjecture of Goncharov (in: Algebra, geometry, and physics in the 21st century, Birkhäuser/Springer, Cham, 2017) concerning ‘categorifications’ of cluster varieties of framed {mathbb {P}}GL_{m+1}-local systems on S, and gives a symplectic geometric viewpoint on results of Gaiotto et al. (Ann Henri Poincaré 15(1):61–141, 2014) on ‘theories of class {mathcal {S}}’.

Automorphism group of the moduli space of parabolic bundles over a curve

We find the automorphism group of the moduli space of parabolic bundles on a smooth curve (with fixed determinant and system of weights). This group is generated by: automorphisms of the marked curve, tensoring with a line bundle, taking the dual, and Hecke transforms (using the filtrations given by the parabolic structure). A Torelli theorem for parabolic bundles with arbitrary rank and generic weights is also obtained. These results are extended to the classification of birational equivalences which are defined over “big” open subsets (3-birational maps, i.e. birational maps giving an isomorphism between open subsets with complement of codimension at least 3).Finally, an analysis of the stability chambers for the parabolic weights is performed in order to determine precisely when two moduli spaces of parabolic vector bundles with different parameters (curve, rank, determinant and weights) can be isomorphic.

Unique Continuation at the Boundary for Harmonic Functions in C1 Domains and Lipschitz Domains with Small Constant

AbstractLet be a domain, or more generally, a Lipschitz domain with small local Lipschitz constant. In this paper it is shown that if u is a function harmonic in and continuous in , which vanishes in a relatively open subset ; moreover, the normal derivative vanishes in a subset of with positive surface measure; then u is identically zero. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Two topologies on the lattice of Scott closed subsets

For a poset P, let σ(P) and Γ(P) respectively denote the lattice of its Scott open subsets and Scott closed subsets ordered by inclusion, and set ΣP=(P,σ(P)). In this paper, we discuss the lower Vietoris topology and the Scott topology on Γ(P) and give some sufficient conditions to make the two topologies equal. We build an adjunction between σ(P) and σ(Γ(P)) and prove that ΣP is core-compact iff ΣΓ(P) is core-compact iff ΣΓ(P) is sober, locally compact and σ(Γ(P))=υ(Γ(P)) (the lower Vietoris topology). This answers a question in [17]. Brecht and Kawai [2] asked whether the consonance of a topological space X implies the consonance of its lower powerspace, we give a partial answer to this question in the last part of this paper.

Robust Exponential Mixing and Convergence to Equilibrium for Singular-Hyperbolic Attracting Sets

We extend results on robust exponential mixing for geometric Lorenz attractors, with a dense orbit and a unique singularity, to singular-hyperbolic attracting sets with any number of (either Lorenz- or non-Lorenz-like) singularities and finitely many ergodic physical/SRB invariant probability measures, whose basins cover a full Lebesgue measure subset of the trapping region of the attracting set. We obtain exponential mixing for any physical probability measure supported in the trapping region and also exponential convergence to equilibrium, for a $$C^2$$ open subset of vector fields in any d-dimensional compact manifold ( $$d\ge 3$$ ).

Expansiveness and Hyperbolicity in Convex Billiards

We say that a convex planar billiard table \(B\) is \(C^{2}\)-stably expansive on a fixed open subset \(U\) of the phase space if its billiard map \(f_{B}\) is expansive on the maximal invariant set \(\Lambda_{B,U}=\bigcap_{n\in\mathbb{Z}}f^{n}_{B}(U)\), and this property holds under \(C^{2}\)-perturbations of the billiard table. In this note we prove for such billiards that the closure of the set of periodic points of \(f_{B}\) in \(\Lambda_{B,U}\) is uniformly hyperbolic. In addition, we show that this property also holds for a generic choice among billiards which are expansive.

Small combination of slices, dentability and stability results of small diameter properties in Banach spaces

In this work we study three different versions of small diameter properties of the unit ball in a Banach space and its dual. The related concepts for all closed bounded convex sets of a Banach space was initiated and developed in [9], [4], [12], [13] was extensively studied in the context of dentability, huskability, Radon Nikodym Property and Krein Milman Property in [14]. We introduce the Ball Huskable Property (BHP), namely, the unit ball has relatively weakly open subsets of arbitrarily small diameter. We compare this property to two related properties, BSCSP namely, the unit ball has convex combination of slices of arbitrarily small diameter and BDP namely, the closed unit ball has slices of arbitrarily small diameter. We show BDP implies BHP which in turn implies BSCSP and none of the implications can be reversed. We prove similar results for the w⁎-versions. We prove that all these properties are stable under lp sum for 1≤p≤∞,c0 sum and Lebesgue Bochner spaces. Finally, we explore the stability of these with properties in the light of three space property. We show that BHP is a three space property provided X/Y is finite dimensional and same is true for BSCSP when X has BSCSP and X/Y is strongly regular ([14]).

Strict pure strategy Nash equilibrium in large finite-player games when the action set is a manifold

We present results on the relationship between non-atomic games (in distributional form) and approximating games with a large but finite number of players. Specifically, in a setting with differentiable payoff functions, we show that: (1) The set of all non-atomic games has an open dense subset such that any finite-player game that is sufficiently close (in terms of distributions of players’ characteristics) to a game in this subset and has sufficiently many players has a strict pure strategy Nash equilibrium (Theorem 1), and (2) any equilibrium distribution of any non-atomic game is the limit of equilibrium distributions defined from strict pure strategy Nash equilibria of finite-player games (Theorem 2). This supplements our paper Carmona and Podczeck (2020b), where analogous results are established for the case where the action set of players is a subset of some Euclidean space, with non-empty interior, and payoff functions are such that equilibrium actions are in the interior of the action set. The goal of the present paper is to remove these assumptions.

Fukaya–Seidel categories of Hilbert schemes and parabolic category $\mathcal{O}$

We realise Stroppel’s extended arc algebra \[13, 51] in the Fukaya–Seidel category of a natural Lefschetz fibration on the generic fibre of the adjoint quotient map on a type $A$ nilpotent slice with two Jordan blocks, and hence obtain a symplectic interpretation of certain parabolic two-block versions of Bernstein–Gel’fand–Gel’fand category $\\mathcal{O}$. As an application, we give a new geometric construction of the spectral sequence from annular to ordinary Khovanov homology. The heart of the paper is the development of a cylindrical model to compute Fukaya categories of (affine open subsets of) Hilbert schemes of quasi-projective surfaces, which may be of independent interest.

Solutions and eigenvalues of Laplace's equation on bounded open sets

We obtain solutions for Laplace's and Poisson's equations on bounded open subsets of \(R^n\) (\(n\geq 2)\), via Hammerstein integral operators involving kernels and Green's functions, respectively. The new solutions are different from the previous ones obtained by the well-known Newtonian potential kernel and the Newtonian potential operator. Our results on eigenvalue problems of Laplace's equationare different from the previous results that use the Newtonian potential operator and require \(n\geq 3\). As a special case of the eigenvalue problems, we provide a result under an easily verifiable condition on the weight function when \(n\geq 3\). This result cannot be obtained by using the Newtonian potential operator. For more information see https://ejde.math.txstate.edu/Volumes/2021/87/abstr.html

Open and surjective mapping theorems for differentiable maps with critical points

Let $\Omega$ be an open subset of $\mathbb R^n (n \ge 2)$, and let $F : \Omega \rightarrow \mathbb R^n$ be a continuously differentiable map with countably many critical points. We show that $F$ is an open map. Let $G :\mathbb R^n \rightarrow \mathbb R^n (n \ge 1)$ be a continuously differentiable map such that $G(x) \rightarrow \infty$ as $x \rightarrow \infty$. Then it is proved that $G$ is surjective if and only if each connected component of the complement of the set of critical values of $G$ contains at least one image of $G$. Several applications of both theorems especially to complex analysis are presented.

Generalized nonlocal Robin Laplacian on arbitrary domains

In this paper, we prove that it is always possible to define a realization of the Laplacian \(\Delta _{\kappa ,\theta }\) on \(L^2(\Omega )\) subject to nonlocal Robin boundary conditions with general jump measures on arbitrary open subsets of \({\mathbb {R}}^N\). This is made possible by using a capacity approach to define an admissible pair of measures \((\kappa ,\theta )\) that allows the associated form \({\mathcal {E}}_{\kappa ,\theta }\) to be closable. The nonlocal Robin Laplacian \(\Delta _{\kappa ,\theta }\) generates a sub-Markovian \(C_0\)-semigroup on \(L^2(\Omega )\) which is not dominated by the Neumann Laplacian semigroup unless the jump measure \(\theta \) vanishes.

Diameter two properties in some vector-valued function spaces

We introduce a vector-valued version of a uniform algebra, called the vector-valued function space over a uniform algebra. The diameter two properties of the vector-valued function space over a uniform algebra on an infinite compact Hausdorff space are investigated. Every nonempty relatively weakly open subset of the unit ball of a vector-valued function space $$A(K, (X, \tau ))$$ over an infinite dimensional uniform algebra has diameter two, where $$\tau $$ is a locally convex Hausdorff topology on a Banach space X compatible to a dual pair. Under the assumption of X equipped with the norm topology being uniformly convex and the additional condition that $$A\otimes X\subset A(K, X)$$ , it is shown that Daugavet points and $$\Delta $$ -points on A(K, X) over a uniform algebra A are the same, and they are characterized by the norm-attainment at a limit point of the Shilov boundary of A. In addition, a sufficient condition for the convex diametral local diameter two property of A(K, X) is also provided. Similar results also hold for an infinite dimensional uniform algebra.

Hyperbolicity for Log Smooth Families with Maximal Variation

Abstract We prove that the base space of a log smooth family of log canonical pairs of log general type is of log general type as well as algebraically degenerate, when the family admits a relative good minimal model over a Zariski open subset of the base and the relative log canonical model is of maximal variation.

Effective Bertini theorem and formulas for multiplicity and the local Łojasiewicz exponent

The classical Bertini theorem on generic intersection of an algebraic set with hyperplanes states the following: Let X be a nonsingular closed subvariety ofPkn, where k is an algebraically closed field. Then there exists a hyperplaneH⊂Pknnot containing X and such that the schemeH∩Xis regular at every point. Furthermore, the set of hyperplanes with this property forms an open dense subset of the complete linear system|H|considered as a projective space. We show that one can effectively indicate a finite family of hyperplanes H such that at least one of them satisfies the assertion of the Bertini theorem, provided the characteristic of the field k is equal to zero. As an application of the method used in the proof we give effective formulas for the multiplicity and the Łojasiewicz exponent of polynomial mappings.

Equivariant realizations of Hermitian symmetric space of noncompact type

Let \(M=G/K\) be a Hermitian symmetric space of noncompact type. We provide a way of constructing K-equivariant embeddings from M to its tangent space \(T_oM\) at the origin by using the polarity of the K-action. As an application, we reconstruct the K-equivariant holomorphic embedding so called the Harish-Chandra realization and the K-equivariant symplectomorphism constructed by Di Scala-Loi and Roos under appropriate identifications of spaces. Moreover, we characterize the holomorphic/symplectic embedding of M by means of the polarity of the K-action. Furthermore, we show a special class of totally geodesic submanifolds in M is realized as either linear subspaces or bounded domains of linear subspaces in \(T_oM\) by the K-equivariant embeddings. We also construct a K-equivariant holomorphic/symplectic embedding of an open dense subset of the compact dual \(M^*\) into its tangent space at the origin as a dual of the holomorphic/symplectic embedding of M.

A Nullstellensatz for ideals of $C^\infty$ functions in dimension 2

Suppose that an ideal $J$ of $C^\infty$ functions on an open subset of $\mathbf{R}^2$ is a Łojasiewicz ideal. We describe the set of $C^\infty$ functions vanishing on the zeros of $J$ explicitly using $J$ in an open neighborhood of each point in zeros of $J$, it can be obtained by taking real radical and closure starting from $J$ repeatedly for a finite number of times. This gives an another affirmative answer to Bochnak's conjecture in dimension 2, which is first done by Risler.

Uniformization of branched surfaces and Higgs bundles

Given a compact connected Riemann surface [Formula: see text] of genus [Formula: see text], and an effective divisor [Formula: see text] on [Formula: see text] with [Formula: see text], there is a unique cone metric on [Formula: see text] of constant negative curvature [Formula: see text] such that the cone angle at each point [Formula: see text] is [Formula: see text] [R. C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988) 222–224; M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793–821]. We describe the Higgs bundle on [Formula: see text] corresponding to the uniformization associated to this conical metric. We also give a family of Higgs bundles on [Formula: see text] parametrized by a nonempty open subset of [Formula: see text] that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchin’s results in [N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987) 59–126] for the case [Formula: see text].

Unique continuation property for biharmonic hypersurfaces in spheres

We prove a unique continuation theorem for non-minimal biharmonic hypersurfaces of spheres, based on Aronszajn’s 1957 article. Under the right hypotheses, this result shows that, for these immersions, CMC on an open subset implies globally CMC. We then deduce new rigidity theorems to support the Conjecture that biharmonic submanifolds of Euclidean spheres must be of constant mean curvature.