Genealogy

Abstract In this paper, we consider the nonlinear Schrödinger equation (NLS) with a general homogeneous nonlinearity in dimensions up to three. We assume that the degree (i.e., power) of the nonlinearity is such that the equation is mass-subcritical and short-range. We establish global well-posedness (GWP) and scattering for small data in the standard weighted space for a class of homogeneous nonlinearities, including non-gauge-invariant ones. Additionally, we include the case where the degree is less than or equal to the Strauss exponent. When the nonlinearity is not gauge-invariant, the standard Duhamel formulation fails to work effectively in the weighted Sobolev space; for instance, the Duhamel term may not be well-defined as a Bochner integral. To address this issue, we introduce an alternative formulation that allows us to establish GWP and scattering, even in the presence of poor time continuity of the Duhamel term.

Abstract Motivated by the construction based on topological suspension of a family of compact non-Kähler complex manifolds with trivial canonical bundle given by Qin and Wang (Geom Topol 22:2115–2144, 2018), we study toric suspensions of balanced manifolds by holomorphic automorphisms. In particular, we show that toric suspensions of Calabi–Yau manifolds are balanced. We also prove that suspensions associated with hyperbolic automorphisms of hyperkähler manifolds do not admit any pluriclosed, astheno-Kähler or p-pluriclosed Hermitian metric. Moreover, we consider natural extensions for hypercomplex manifolds, providing some explicit examples of compact holomorphic symplectic and hypercomplex non-Kähler manifolds. We also show that a modified suspension construction provides examples with pluriclosed metrics.

Abstract Given a bounded regular domain $$\omega \subset \mathbb {R}^{N-1}$$ ω ⊂ R N - 1 and the half-cylinder $$\Sigma = \omega \times (0,+\infty )$$ Σ = ω × ( 0 , + ∞ ) , we consider the relative overdetermined torsion problem in $$\Sigma $$ Σ , i.e. $$\begin{aligned} {\left\{ \begin{array}{ll} \Delta {u}+1=0 & \hbox { in}\ \Omega ,\\ \partial _\eta u = 0 & \hbox { on}\ {\widetilde{\Gamma }}_\Omega ,\\ u=0 & \hbox { on}\ \Gamma _\Omega ,\\ \partial _{\nu }u =c & \hbox { on}\ \Gamma _\Omega , \end{array}\right. } \end{aligned}$$ Δ u + 1 = 0 in Ω , ∂ η u = 0 on Γ ~ Ω , u = 0 on Γ Ω , ∂ ν u = c on Γ Ω , where $$\Omega \subset \Sigma $$ Ω ⊂ Σ , $$\Gamma _\Omega = \partial \Omega \cap \Sigma $$ Γ Ω = ∂ Ω ∩ Σ , $${\widetilde{\Gamma }}_\Omega = \partial \Omega {\setminus } \Gamma _\Omega $$ Γ ~ Ω = ∂ Ω \ Γ Ω , $$\nu $$ ν is the outer unit normal vector on $$\Gamma _\Omega $$ Γ Ω and $$\eta $$ η is the outer unit normal vector on $${\widetilde{\Gamma }}_\Omega $$ Γ ~ Ω . We build nontrivial solutions to this problem in domains $$\Omega $$ Ω that are the hypograph of certain nonconstant functions $$v: {\overline{\omega }} \rightarrow (0, + \infty )$$ v : ω ¯ → ( 0 , + ∞ ) . Such solutions can be reflected with respect to $$\omega $$ ω , giving nontrivial solutions to the relative overdetermined torsion problem in a cylinder. The proof uses a local bifurcation argument which, quite remarkably, works for most smooth domains $$\omega $$ ω .

Abstract It is shown that a Stallings–Swan theorem holds in a totally disconnected locally compact (= t.d.l.c.) context (cf. Theorem B). More precisely, a compactly generated $${\mathcal{C}\mathcal{O}}$$ C O -bounded t.d.l.c. group G of rational discrete cohomological dimension less than or equal to 1 must be isomorphic to the fundamental group of a finite graph of profinite groups. This result generalises Dunwoody’s rational version of the classical Stallings–Swan theorem to t.d.l.c. groups. The proof of Theorem B is based on the fact that a compactly generated unimodular t.d.l.c. group with rational discrete cohomological dimension 1 has necessarily non-positive Euler–Poincaré characteristic (cf. Theorem H).

Abstract We prove that horn maps associated to quadratic semi-parabolic fixed points of Hénon maps, first introduced by Bedford, Smillie, and Ueda, satisfy a weak form of the Ahlfors island property. As a consequence, two natural definitions of their Julia set (the non-normality locus of the family of iterates and the closure of the set of the repelling periodic points) coincide. As another consequence, we also prove that there exist small perturbations of semi-parabolic Hénon maps for which the Hausdorff dimension of the forward Julia set $$J^+$$ J + is arbitrarily close to 4.

Abstract We prove that every closed, connected, orientable surface S of negative Euler characteristic admits a pair of finite-degree covers which are length isospectral over S but generically not simple length isospectral over S. To do this, we first characterize when two finite-degree covers of a connected, orientable surface of negative Euler characteristic are isomorphic in terms of which curves have simple elevations. We also construct hyperbolic surfaces X and Y with the same full unmarked length spectrum but so that for each k, the sets of lengths associated to curves with at most k self-intersections differ.

Abstract Gaiotto, Moore and Neitzke introduced spectral networks to understand the framed G-local systems over punctured surfaces for G a split Lie group via a procedure called abelianization. We generalize this construction to groups G of the form $${{\,\textrm{GL}\,}}_2(A)$$ GL 2 ( A ) , where A is a unital associative ring, and to some of its subgroups. This relies on a precise analysis of the two-fold ramified coverings associated with spectral networks and triangulations and on a matrix reinterpretation of their path lifting rules; along the way we provide another proof of the Laurent phenomenon brought to light by Berenstein and Retakh. The partial abelianization enables us to gives parametrizations of the moduli spaces of decorated G-local systems and of framed G-local systems over punctured surfaces. For $$(A, \sigma )$$ ( A , σ ) a Hermitian involutive $${\mathbb {R}}$$ R -algebra the group $$G={{\,\textrm{Sp}\,}}_2(A, \sigma )$$ G = Sp 2 ( A , σ ) is a classical Hermitian Lie group of tube type, and we are able to identify and parametrize the moduli space of maximal framed G-local systems.

Abstract We establish precise nonvanishing results for asymptotic syzygies of smooth projective varieties. This refines Ein–Lazarsfeld’s asymptotic nonvanishing theorem. Combining with the author’s previous asymptotic vanishing result, we completely determine the asymptotic shapes of the minimal free resolutions of the graded section modules of a line bundle on a smooth projective variety as the positivity of the embedding line bundle grows.

Abstract Structural stability of holomorphic functions has been the subject of much research in the last fifty years. Due to various technicalities, however, most of that work has focused on so-called finite-type functions (functions whose set of singular values has finite cardinality). Recent developments in the field go beyond this setting. In this paper we extend Eremenko and Lyubich’s result on natural families of entire maps to the case where the set of singular values is not the entire complex plane, showing under this assumption that the set $$M_f$$ M f of entire functions quasiconformally equivalent to f admits the structure of a complex manifold (of possibly infinite dimension). Moreover, we will consider functions with wandering domains—another hot topic of research in complex dynamics. Given an entire function f with a simply connected wandering domain U, we construct an analogue of the multiplier of a periodic orbit, called a distortion sequence, and show that, under some hypotheses, the distortion sequence moves analytically as f moves within appropriate parameter families.

In this paper we prove symmetry of nonnegative solutions of the integral equation $$\begin{aligned} u (\zeta ) = \int \limits _{{\mathbb {H}}^n} |\zeta ^{-1} \xi |^{-(Q-\alpha )} u(\xi )^{p} d\xi \quad 1< p \le \frac{Q+\alpha }{Q-\alpha },\ 0< \alpha

Abstract We consider a compact manifold $$(M,\mathfrak {F})$$ ( M , F ) with a foliation $$\mathfrak {F}$$ F , and a smooth affine connection $$\nabla $$ ∇ on the tangent bundle of the foliation $$T\mathfrak {F}$$ T F . We introduce and study a foliated completeness problem. Namely, under which conditions on $$\nabla $$ ∇ the leaves are complete? We consider different natural geometric settings: the first one is the case of a totally geodesic lightlike foliation of a compact Lorentzian manifold, and the second one is the case where the leaves have particular affine structures. In the first case, we characterize the completeness, and obtain in particular that if a compact Lorentzian manifold admits a null Killing field V such that the distribution orthogonal to V is integrable, then it defines a (totally geodesic) foliation with complete leaves. In the second case, we give a completeness result for a specific affine structure called “the unimodular affine lightlike geometry”, and characterize the completeness for a natural relaxation of the geometry. On the other hand, we study the global completeness of a compact Lorentzian manifold in the presence of a null Killing field. We give two non-complete examples, starting from dimension 3: one is a locally homogeneous manifold, and the other is a 3D example where the Killing field dynamics is equicontinuous.