Annali di Matematica Pura ed Applicata最新文献

An odd generalized metric (E_{-}) on a Lie group G of dimension n is a left-invariant generalized metric on a Courant algebroid (E_{H, F}) of type (B_{n}) over G with left-invariant twisting forms (Hin Omega ^{3}(G)) and (Fin Omega ^{2}(G)). Given an odd generalized metric (E_{-}) on G we determine the affine space of left-invariant Levi-Civita generalized connections of (E_{-}). Given in addition a left-invariant divergence operator (delta ) we show that there is a left-invariant Levi-Civita generalized connection of (E_{-}) with divergence (delta ) and we compute the corresponding Ricci tensor (textrm{Ric}^{delta }) of the pair ((E_{-}, delta )). The odd generalized metric (E_{-}) is called odd generalized Einstein with divergence (delta ) if (textrm{Ric}^{delta }=0). As an application of our theory, we describe all odd generalized Einstein metrics of arbitrary left-invariant divergence on all 3-dimensional unimodular Lie groups.

We provide several families of compact complex curves embedded in smooth complex surfaces such that no neighborhood of the curve can be embedded in an algebraic surface. Different constructions are proposed, by patching neighborhoods of curves in projective surfaces, and blowing down exceptional curves. These constructions generalize examples recently given by S. Lvovski. One of our non algebraic argument is based on an extension theorem of S. Ivashkovich.

Let X be a ruled surface over a nonsingular curve C of genus (gge 0). The main goal of this paper is to construct simple prioritary vector bundles of any rank r on X and to give effective bounds for the dimension of their module of global sections.

In this paper we obtain necessary conditions and sufficient conditions on the initial data for the solvability of fractional semilinear heat equations with power nonlinearities in the Heisenberg group (mathbb {H}^N). Using these conditions, we can prove that (1+2/Q) separates the ranges of exponents of nonlinearities for the global-in-time solvability of the Cauchy problem (so-called the Fujita-exponent), where (Q=2N+2) is the homogeneous dimension of (mathbb {H}^N), and identify the optimal strength of the singularity of the initial data for the local-in-time solvability. Furthermore, our conditions lead sharp estimates of the life span of solutions with nonnegative initial data having a polynomial decay at the space infinity.

We consider the Cauchy problem for a damped Euler–Maxwell system with no ionic background. For smooth enough data satisfying suitable so-called dispersive conditions, we establish the global in time existence and uniqueness of a strong solution that decays uniformly in time. Our method is inspired by the works of D. Serre and M. Grassin dedicated to the compressible Euler system.

The main purpose of this note is to study the compactness of extremals for the singular Moser-Trudinger inequality. More precisely, let (Omega subset {mathbb {R}}^n), (nge 2), be a bounded open smooth domain and (0in Omega ), (W^{1,n}_{0}(Omega )) be the standard Sobolev space. For (epsilon in [0,1)), Csato-Roy-Nguyen [J. Diff. Equ. 270:843–882, 2021] proved that the following singular Moser-Trudinger inequality

can be achieved by a nonnegative function (u_epsilon in W^{1,n}_{0}(Omega )) with (int _{Omega }|nabla u_epsilon |^n dxle 1). Here (alpha _{n}=n omega _{n-1}^{1/(n-1)}) with (omega _{n-1}) being the surface area of the ((n-1))-dimensional unit sphere.

Relying on above result, by blow-up analysis, we consider the compactness of function family ({u_epsilon }_{0<epsilon <1}) and prove, up to a subsequence, (u_epsilon rightarrow u_0) in (W^{1,n}_0(Omega )cap C^0({overline{Omega }})cap C_{textrm{loc}}^{1}({overline{Omega }}{setminus }{0})) as (epsilon rightarrow 0), where (u_0) is an extremal function of the following supremum

We establish the existence of a solution to a nonlinear competitive Schrödinger system whose scalar potential tends to a positive constant at infinity with an appropriate rate. This solution has the property that all components are invariant under the action of a group of linear isometries and each component is obtained from the previous one by composing it with some fixed linear isometry. We call it a pinwheel solution. We describe the asymptotic behavior of the least energy pinwheel solutions when the competing parameter tends to zero and to minus infinity. In the latter case the components are segregated and give rise to an optimal pinwheel partition for the Schrödinger equation, that is, a partition formed by invariant sets that are mutually isometric through a fixed isometry.

In this work we discuss stability and nondegeneracy properties of some special families of minimal hypersurfaces embedded in (mathbb {R}^mtimes mathbb {R}^n) with (m,nge 2). These hypersurfaces are asymptotic at infinity to a fixed Lawson cone (C_{m,n}). In the case (m+nge 8), we show that such hypersurfaces are strictly stable and we provide a full classification of their bounded Jacobi fields, which in turn allows us to prove the non-degeneracy of such surfaces. In the case (m+nle 7), we prove that such hypersurfaces have infinite Morse index.

The so-called Dao numbers are a sort of measure of the asymptotic behaviour of full properties of certain product ideals in a Noetherian local ring R with infinite residue field and positive depth. In this paper, we answer a question of H. Dao on how to bound such numbers. The auxiliary tools range from Castelnuovo–Mumford regularity of appropriate graded structures to reduction numbers of the maximal ideal. In particular, we substantially improve previous results (and answer questions) by the authors. Finally, as an application of the theory of Dao numbers, we provide new characterizations of when R is regular; for instance, we show that this holds if and only if the maximal ideal of R can be generated by a d-sequence (in the sense of Huneke) if and only if the third Dao number of any (minimal) reduction of the maximal ideal vanishes.

We investigate the metric and cohomological properties of higher dimensional analogues of Inoue surfaces, that were introduced by Endo and Pajitnov. We provide a solvmanifold structure and show that in the diagonalizable case, they are formal and have invariant de Rham cohomology. Moreover, we obtain an arithmetic and cohomological characterization of pluriclosed and astheno-Kähler metrics and show they give new examples in all complex dimensions.