Mathematische Annalen最新文献

In this paper, we show that for each (kin {mathbb {Z}}^+, p>4), there exists a solution operator ({mathcal {T}}_k) to the (bar{partial }) problem on the Hartogs triangle that maintains the same (W^{k, p}) regularity as that of the data. According to a Kerzman-type example, this operator provides solutions with the optimal Sobolev regularity.

Although exotic blow-ups of the projective plane at n points have been constructed for every (n ge 2), the only examples known by means of rational blowdowns satisfy (n ge 5). It has been an intriguing problem whether it is possible to decrease n. In this paper, we construct the first exotic ({mathbb {C}}{mathbb {P}}^2 # 4 overline{{mathbb {C}}{mathbb {P}}^2}) with this technique. We also construct exotic (3{mathbb {C}}{mathbb {P}}^2 # b^- overline{{mathbb {C}}{mathbb {P}}^2}) for (b^-=9,8,7). All of them are minimal and symplectic, as they are produced from projective surfaces W with Wahl singularities and (K_W) big and nef. In more generality, we elaborate on the problem of finding exotic

from these Kollár–Shepherd-Barron–Alexeev surfaces W, obtaining explicit geometric obstructions on the corresponding configurations of rational curves.

Let X be a smooth Fano variety. We attach a bi-graded associative algebra (textrm{HS}(mathcal {K}u(X))=bigoplus _{i,jin mathbb {Z}} textrm{Hom}(textrm{Id},S_{mathcal {K}u(X)}^{i}[j])) to the Kuznetsov component (mathcal {K}u(X)) whenever it is defined. Then we construct a natural sub-algebra of (textrm{HS}(mathcal {K}u(X))) when X is a Fano hypersurface and establish its relation with Jacobian ring (textrm{Jac}(X)). As an application, we prove a categorical Torelli theorem for Fano hypersurface (Xsubset mathbb {P}^n(nge 2)) of degree d if (textrm{gcd}((n+1),d)=1.) In addition, we give a new proof of the main theorem [15, Theorem 1.2] using a similar idea.

We show that distinct primitive L-functions can achieve extreme values simultaneously on the critical line. Our proof uses a modification of the resonance method and can be applied to establish simultaneous extreme central values of L-functions in families.

We consider the varifold associated to the Allen–Cahn phase transition problem in ({mathbb {R}}^{n+1})(or (n+1)-dimensional Riemannian manifolds with bounded curvature) with integral (L^{q_0}) bounds on the Allen–Cahn mean curvature (first variation of the Allen–Cahn energy) in this paper. It is shown here that there is an equidistribution of energy between the Dirichlet and Potential energy in the phase field limit and that the associated varifold to the total energy converges to an integer rectifiable varifold with mean curvature in (L^{q_0}, q_0 > n). The latter is a diffused version of Allard’s convergence theorem for integer rectifiable varifolds.

For an embedded exact Lagrangian cobordism between Legendrian submanifolds in the 1-jet bundle, we prove that a generating family linear at infinity on the Legendrian at the negative end extends to a generating family linear at infinity on the Lagrangian cobordism after stabilization if and only if the formal obstructions vanish. In particular, a Lagrangian filling with trivial stable Lagrangian Gauss map admits a generating family linear at infinity.

We completely characterise the bounded sets that arise as components of the Fatou and Julia sets of meromorphic functions. On the one hand, we prove that a bounded domain is a Fatou component of some meromorphic function if and only if it is regular. On the other hand, we prove that a planar continuum is a Julia component of some meromorphic function if and only if it has empty interior. We do so by constructing meromorphic functions with wandering compacta using approximation theory.

Ambient prime geodesic theorems provide an asymptotic count of closed geodesics by their length and holonomy and imply effective equidistribution of holonomy. We show that for a smoothed count of closed geodesics on compact hyperbolic 3-manifolds, there is a persistent bias in the secondary term which is controlled by the number of zero spectral parameters. In addition, we show that a normalized, smoothed bias count is distributed according to a probability distribution, which we explicate when all distinct, non-zero spectral parameters are linearly independent.

This article studies the canonical Hilbert energy (H^{s/2}(M)) on a Riemannian manifold for (sin (0,2)), with particular focus on the case of closed manifolds. Several equivalent definitions for this energy and the fractional Laplacian on a manifold are given, and they are shown to be identical up to explicit multiplicative constants. Moreover, the precise behavior of the kernel associated with the singular integral definition of the fractional Laplacian is obtained through an in-depth study of the heat kernel on a Riemannian manifold. Furthermore, a monotonicity formula for stationary points of functionals of the type ({mathcal {E}}(v)=[v]^2_{H^{s/2}(M)}+int _M F(v) , dV), with (Fge 0), is given, which includes in particular the case of nonlocal s-minimal surfaces. Finally, we prove some estimates for the Caffarelli–Silvestre extension problem, which are of general interest. This work is motivated by Caselli et al. (Yau’s conjecture for nonlocal minimal surfaces, arxiv preprint, 2023), which defines nonlocal minimal surfaces on closed Riemannian manifolds and shows the existence of infinitely many of them for any metric on the manifold, ultimately proving the nonlocal version of a conjecture of Yau (Ann Math Stud 102:669–706, 1982). Indeed, the definitions and results in the present work serve as an essential technical toolbox for the results in Caselli et al. (Yau’s conjecture for nonlocal minimal surfaces, arxiv preprint, 2023).

In this paper, we study generalized Schauder theory for the degenerate/singular parabolic equations of the form

When the equation above is singular, it can be derived from Monge–Ampère equations by using the partial Legendre transform. Also, we study the fractional version of Taylor expansion for the solution u, which is called s-polynomial. To prove (C_s^{2+alpha })-regularity and higher regularity of the solution u, we establish generalized Schauder theory which approximates coefficients of the operator with s-polynomials rather than constants. The generalized Schauder theory not only recovers the proof for uniformly parabolic equations but is also applicable to other operators that are difficult to apply the bootstrap argument to obtain higher regularity.