Mathematische Annalen最新文献

We consider some algebraic aspects of the dynamics of an automorphism on a family of polarized abelian varieties parameterized by the complex unit disk. When the action on the cohomology of the generic fiber has no cyclotomic factor, we prove that such a map can be made regular only if the family of abelian varieties does not degenerate. As a contrast, we show that families of translations are always regularizable. We further describe the closure of the orbits of such maps, inspired by results of Cantat and Amerik–Verbitsky.

We give Martin representation of nonnegative functions caloric with respect to the fractional Laplacian in Lipschitz open sets. The caloric functions are defined in terms of the mean value property for the space-time isotropic (alpha )-stable Lévy process. To derive the representation, we first establish the existence of the parabolic Martin kernel. This involves proving new boundary regularity results for both the fractional heat equation and the fractional Poisson equation with Dirichlet exterior conditions. Specifically, we demonstrate that the ratio of the solution and the Green function is Hölder continuous up to the boundary.

This paper generalizes D. Burago and S. Ivanov’s work (Duke Math J 162:1205–1248, 2013) on filling volume minimality and boundary rigidity of almost real hyperbolic metrics. We show that regions with metrics close to a negatively curved symmetric metric are strict minimal fillings and hence boundary rigid. This includes perturbations of complex, quaternionic and Cayley hyperbolic metrics.

If U is a (C^{infty }) function with compact support in the plane, we let u be its restriction to the unit circle ({mathbb {S}}), and denote by (U_i,,U_e) the harmonic extensions of u respectively in the interior and the exterior of ({mathbb {S}}) on the Riemann sphere. About a hundred years ago, Douglas [9] has shown that

thus giving three ways to express the Dirichlet norm of u. On a rectifiable Jordan curve (Gamma ) we have obvious analogues of these three expressions, which will of course not be equal in general. The main goal of this paper is to show that these 3 (semi-)norms are equivalent if and only if (Gamma ) is a chord-arc curve.

In this paper we show that if a compact set (E subset {mathbb {R}}^d), (d ge 3), has Hausdorff dimension greater than (frac{(4k-1)}{4k}d+frac{1}{4}) when (3 le d<frac{k(k+3)}{(k-1)}) or (d- frac{1}{k-1}) when (frac{k(k+3)}{(k-1)} le d), then the set of congruence class of simplices with vertices in E has nonempty interior. By set of congruence class of simplices with vertices in E we mean

where (2 le k <d). This result improves the previous best results in the sense that we now can obtain a Hausdorff dimension threshold which allow us to guarantee that the set of congruence class of triangles formed by triples of points of E has nonempty interior when (d=3) as well as extending to all simplices. The present work can be thought of as an extension of the Mattila–Sjölin theorem which establishes a non-empty interior for the distance set instead of the set of congruence classes of simplices.

Non-compact hyperkähler spaces arise frequently in gauge theory. The 4-dimensional hyperkähler ALE spaces are a special class of complete non-compact hyperkähler spaces. They are in one-to-one correspondence with the finite subgroups of SU(2) and have interesting connections with representation theory and singularity theory, captured by the McKay Correspondence. The 4-dimensional hyperkähler ALE spaces are first classified by Peter Kronheimer via a finite-dimensional hyperkähler reduction. In this paper, we give a new gauge-theoretic construction of these spaces. More specifically, we realize each 4-dimensional hyperkähler ALE space as a moduli space of solutions to a system of equations for a pair consisting of a connection and a section of a vector bundle over an orbifold Riemann surface, modulo a gauge group action. The construction given in this paper parallels Kronheimer’s original construction and hence can also be thought of as a gauge-theoretic interpretation of Kronheimer’s construction of these spaces.

Let (overline{{mathcal {H}}}_{g,n}) denote the Hodge bundle over (overline{{mathfrak {M}}}_{g,n}), and ({mathbb {P}}overline{{mathcal {H}}}_{g,n}) its associated projective bundle. Let ({mathcal {H}}_{g,n}) and ({mathbb {P}}{mathcal {H}}_{g,n}) be respectively the restriction of (overline{{mathcal {H}}}_{g,n}) and ({mathbb {P}}overline{{mathcal {H}}}_{g,n}) to the smooth part ({mathfrak {M}}_{g,n}) of (overline{{mathfrak {M}}}_{g,n}). The Hodge norm provides us with a Hermtian metric on ({mathscr {O}}(-1)_{{mathbb {P}}{mathcal {H}}_{g,n}}). Let (Theta ) denote the curvature form of this metric. In this paper, we show that if (overline{{mathcal {N}}}) is a subvariety of ({mathbb {P}}overline{{mathcal {H}}}_{g,n}) that intersects ({mathcal {H}}_{g,n}), then the integral of the top power of (Theta ) over the smooth part of (overline{{mathcal {N}}}cap {mathbb {P}}{mathcal {H}}_{g,n}) equals the self-intersection number of the tautological divisor (c_1({mathscr {O}}(-1)_{{mathbb {P}}overline{{mathcal {H}}}_{g,n}})cap overline{{mathcal {N}}}) in (overline{{mathcal {N}}}). This implies that the volume of a linear subvariety of ({mathbb {P}}{mathcal {H}}_{g,n}) whose local coordinates do not involve relative periods can be computed by the intersection number of its closure in ({mathbb {P}}overline{{mathcal {H}}}_{g,n}) with some power of any divisor representing the tautological line bundle. We also genralize this statement to the bundles ({mathbb {P}}overline{{mathcal {H}}}^{(k)}_{g,n}), (k in {mathbb {Z}}_{ge 2}), of k-differentials with poles of order at most ((k-1)) over (overline{{mathfrak {M}}}_{g,n}). To obtain these results, we use the existence of an appropriate desingularization of (overline{{mathcal {N}}}) and a deep result of Kollár (Subadditivity of the Kodaira Dimension: Fiber of General Type, Algebraic Geometry, Sendai, 1985, Advanced studies in Pure Math. (1987)) on variation of Hodge structure.

Given an isometry invariant valuation on a complex space form we compute its value on the tubes of sufficiently small radii around a set of positive reach. This generalizes classical formulas of Weyl, Gray and others about the volume of tubes. We also develop a general framework on tube formulas for valuations in Riemannian manifolds.

For the non-rotating gaseous stars modeled by the compressible Euler–Poisson system with general pressure law, Lin and Zeng (Comm Pure Appl Math 75: 2511–2572, 2022) proved a turning point principle, which gives the sharp linear stability/instability criteria for the non-rotating gaseous stars. In this paper, we prove that the sharp linear stability criterion for the non-rotating stars also implies nonlinear orbital stability against general perturbations provided the global weak solutions exist. If the perturbations are further restricted to be spherically symmetric, then nonlinear stability holds true unconditionally in the sense that the existence of global weak solutions near the non-rotating star can be proved.

We study the continuity set (the set of all continuous points) of pullback random attractors from a parametric space into the space of all compact subsets of the state space with Hausdorff metric. We find a general theorem that the continuity set is an IOD-type (a countable intersection of open dense sets) with the local similarity under appropriate conditions of random dynamical systems, and we further show that any IOD-type set in the parametric space has the continuous cardinality, which affirmatively answers the unsolved question about the cardinality of the continuity set of attractors in the literature. Applying to the random nonautonomous nonlocal parabolic equations on an unbounded domain driven by colored noise, we establish the existence and IOD-type continuity of pullback random attractors in time, sample-translation and noise-size, moreover, we prove that the continuity set of the pullback random attractor on the plane of time and sample-translation is composed of diagonal rays whose number of bars is the continuous cardinality.