Mathematische Annalen最新文献

A taming symplectic structure provides an upper bound on the area of an approximately pseudoholomorphic curve in terms of its homology class. We prove that, conversely, an almost complex manifold with such an area bound admits a taming symplectic structure. This confirms a speculation by Gromov. We also characterize the cone of taming symplectic structures numerically, prove that complex 2-cycles can be approximated by coarsely holomorphic curves, and provide a lower energy bound for such curves.

With the aim of quantifying turbulent behaviors of vortex filaments, we study the multifractality and intermittency of the family of generalized Riemann’s non-differentiable functions

These functions represent, in a certain limit, the trajectory of regular polygonal vortex filaments that evolve according to the binormal flow. When (x_0) is rational, we show that (R_{x_0}) is multifractal and intermittent by completely determining the spectrum of singularities of (R_{x_0}) and computing the (L^p) norms of its Fourier high-pass filters, which are analogues of structure functions. We prove that (R_{x_0}) has a multifractal behavior also when (x_0) is irrational. The proofs rely on a careful design of Diophantine sets that depend on (x_0), which we study by using the Duffin–Schaeffer theorem and the Mass Transference Principle.

This paper is to provide a criterion of the uniqueness of periodic solutions for a Rayleigh-Liénard system with state-dependent impulses. Notice that such results of a planar system with state-dependent impulses are few. Moreover, the Rayleigh-Liénard system with state-dependent impulses has wide applications, such as a simple pendulum and a spring vibrator. Further, we obtain the uniqueness of periodic solutions of the simple pendulum with state-dependent impulses and the spring vibrator with state-dependent impulses by the criterion.

We introduce and study the family of uniformly super McDuff (hbox {II}_1) factors. This family is shown to be closed under elementary equivalence and also coincides with the family of (hbox {II}_1) factors with the Brown property introduced in Atkinson et al. (Adv. Math. 396, 108107, 2022). We show that a certain family of existentially closed factors, the so-called infinitely generic factors, are uniformly super McDuff, thereby improving a recent result of Chifan et al. (Embedding Universality for (hbox {II}_1) Factors with Property (T). arXiv preprint, 2022). We also show that Popa’s family of strongly McDuff (hbox {II}_1) factors are uniformly super McDuff. Lastly, we investigate when finitely generic (hbox {II}_1) factors are uniformly super McDuff.

This paper focuses on the existence of normalized solutions for the following Kirchhoff equation:

where (a,b,c>0), (mu in {mathbb {R}}) and (2<q<6), (lambda in {mathbb {R}}) will arise as a Lagrange multiplier that is not a priori given. By using new analytical techniques, the paper establishes several existence results for the case (mu >0):

1. (1)

   The existence of two solutions, one being a local minimizer and the other of mountain-pass type, under explicit conditions on c when (2<q<frac{10}{3}).

2. (2)

   The existence of a mountain-pass type solution under explicit conditions on c when (frac{10}{3}le q<frac{14}{3}).

3. (3)

   The existence of a ground state solution for all (c>0) when (frac{14}{3}le q<6).

Furthermore, the paper presents the first non-existence result for the case (mu le 0) and (2<q<6). In particular, refined estimates of energy levels are proposed, suggesting a new threshold of compactness in the (L^2)-constraint. This study addresses an open problem for (2<q<frac{10}{3}) and fills a gap in the case (frac{10}{3}le q<frac{14}{3}). We believe that our approach can be applied to a broader range of nonlinear terms with Sobolev critical growth, and the underlying ideas have potential for future development and applicability.

Given an elliptic operator (L= - {{,textrm{div},}}(A nabla cdot )) subject to mixed boundary conditions on an open subset of (mathbb {R}^d), we study the relation between Gaussian pointwise estimates for the kernel of the associated heat semigroup, Hölder continuity of L-harmonic functions and the growth of the Dirichlet energy. To this end, we generalize an equivalence theorem of Auscher and Tchamitchian to the case of mixed boundary conditions and to open sets far beyond Lipschitz domains. Yet, we prove the consistency of our abstract result by encompassing operators with real-valued coefficients and their small complex perturbations into one of the aforementioned equivalent properties. The resulting kernel bounds open the door for developing a harmonic analysis for the associated semigroups on rough open sets.

In this paper, we mainly consider nonnegative weak solution to the (D^{1,p}(mathbb {R}^{N}))-critical quasi-linear static Schrödinger–Hartree equation with p-Laplacian (-Delta _{p}) and nonlocal nonlinearity:

where (1<p<frac{N}{2}), (Nge 3) and (uin D^{1,p}(mathbb {R}^N)). First, we establish regularity and the sharp estimates on asymptotic behaviors for any positive solution u (and (|nabla u|)) to more general equation (-Delta _p u=V(x)u^{p-1}) with (Vin L^{frac{N}{p}}(mathbb {R}^{N})). Then, as a consequence, we can apply the method of moving planes to prove that all the nontrivial nonnegative solutions are radially symmetric and strictly decreasing about some point (x_0in mathbb {R}^N). The radial symmetry and sharp asymptotic estimates for more general nonlocal quasi-linear equations were also included.

Comparison theorems are foundational to our understanding of the geometric features implied by various curvature constraints. This paper considers manifolds with a positive lower bound on either scalar, 2-Ricci, or Ricci curvature, and contains a variety of theorems which provide sharp relationships between this bound and notions of width. Some inequalities leverage geometric quantities such as boundary mean curvature, while others involve topological conditions in the form of linking requirements or homological constraints. In several of these results open and incomplete manifolds are studied, one of which partially addresses a conjecture of Gromov in this setting. The majority of results are accompanied by rigidity statements which isolate various model geometries—both complete and incomplete—including a new characterization of round lens spaces, and other models that have not appeared elsewhere. As a byproduct, we additionally give new and quantitative proofs of several classical comparison statements such as Bonnet-Myers’ and Frankel’s Theorem, as well as a version of Llarull’s Theorem and a notable fact concerning asymptotically flat manifolds. The results that we present vary significantly in character, however a common theme is present in that the lead role in each proof is played by spacetime harmonic functions, which are solutions to a certain elliptic equation originally designed to study mass in mathematical general relativity.

We prove that a k-regulous function defined on a two-dimensional non-singular affine variety can be extended to an ambient variety. Additionally we derive some results concerning sums of squares of k-regulous functions; in particular we show that every positive semi-definite regular function on a non-singular affine variety can be written as a sum of squares of locally Lipschitz regulous functions.

The goal of the article is to show that an n-dimensional complex torus embedded in a complex manifold of dimensional (n+d), with a split tangent bundle, has a neighborhood biholomorphic to a neighborhood of the zero section in its normal bundle, provided the latter has locally constant and diagonalizable transition functions and satisfies a non-resonant Diophantine condition.