Journal of the London Mathematical Society-Second Series最新文献

We consider a nematic liquid crystal flow with partially free boundary in a smooth bounded domain in $�R^2$. We prove regularity estimates and the global existence of weak solutions enjoying partial regularity properties, and a uniqueness result.

For a log Calabi Yau pair ($��X�,�D�$) with $��X�\setminus �D�$ smooth affine, satisfying either a maximal degeneracy assumption or contains a Zariski dense torus, we prove under the condition that D is the support of a nef divisor that the structure constants defining a trace form on the mirror algebra constructed by Gross–Siebert are given by the naive curve counts defined by Keel–Yu. As a corollary, we deduce that the equality of the mirror algebras constructed by Gross–Siebert and Keel–Yu in the case $��X�\setminus �D�$ contains a Zariski dense torus. In addition, we use this result to prove a mirror conjecture proposed by Mandel for Fano pairs satisfying the maximal degeneracy assumption.

In this article, we prove that for a definable set in an o-minimal structure with connected link (at 0 or infinity), the inner distance of the link is equivalent to the inner distance of the set restricted to the link. With this result, we obtain several consequences. We present also several relations between the local and the global Lipschitz geometry of singularities. For instance, we prove that two sets in Euclidean spaces, not necessarily definable in an o-minimal structure, are outer lipeomorphic if and only if their stereographic modifications are outer lipeomorphic if and only if their inversions are outer lipeomorphic.

For large classes of even-dimensional Riemannian manifolds $��(�M�,�g�)�$, we construct and analyze conformally invariant random fields. These centered Gaussian fields $��h�=�h_g�$, called co-polyharmonic Gaussian fields, are characterized by their covariance kernels k which exhibit a precise logarithmic divergence: $��|�k��(�x�,�y�)��-�\log �\frac{1}{�d�(�x�,�y�)�}�|�\le �C�$. They share a fundamental quasi-invariance property under conformal transformations. In terms of the co-polyharmonic Gaussian field $�h$, we define the Liouville Quantum Gravity measure, a random measure on $�M$, heuristically given as

In this paper, we study the hyperbolic and parabolic strip deformations of ideal (possibly once-punctured) hyperbolic polygons whose vertices are decorated with horoballs. We prove that the interiors of their arc complexes parametrise the open convex set of all uniformly lengthening infinitesimal deformations of the decorated hyperbolic metrics on these surfaces, motivated by the work of Danciger–Guéritaud–Kassel. We also give a version of this result for the undecorated ideal polygons and once-punctured ideal polygons.

This paper is concerned with the analytical construction of piecewise smooth solutions containing a single shock wave for the radially symmetric relativistic Euler equations with polytropic gases. We derive meticulously the a priori $�C^1$-estimates on the Riemann invariants of the governing system under some assumptions on the piecewise initial data. Based on these estimates, we show that the long time of existence of smooth solutions in the angular region bounded by a characteristic curve and a shock curve. The piecewise smooth initial conditions ensured the existence of smooth solutions in the angular region are discussed. Moreover, it is verified that the existence time is proportional to the initial discontinuous position.

We construct a canonical family of elements in the reduced exterior powers of unit groups of global fields and investigate their detailed arithmetic properties. We then show that these elements specialise to recover the classical theory of cyclotomic elements in real abelian fields and also have connections to the theory of non-commutative Euler systems for $��Z_p��\left(�1�\right)��$ over general number fields.

We describe surjective linear isometries and linear isometry groups of a large class of Lipschitz-free spaces that includes, for example, Lipschitz-free spaces over any graph. We define the notion of a Lipschitz-free rigid metric space whose Lipschitz-free space only admits surjective linear isometries coming from surjective dilations (i.e., rescaled isometries) of the metric space itself. We show that this class of metric spaces is surprisingly rich and contains all 3-connected graphs as well as geometric examples such as nonabelian Carnot groups with horizontally strictly convex norms. We prove that every metric space isometrically embeds into a Lipschitz-free rigid space that has only three more points.