Bulletin of the London Mathematical Society最新文献

We determine when a Legendrian quasipositive 3-braid closure in the standard contact $�R^3$ admits an orientable or nonorientable exact Lagrangian filling. Our main result provides evidence for the orientable fillability conjecture of Hayden and Sabloff, showing that a 3-braid closure is orientably exact Lagrangian fillable if and only if it is quasipositive and the HOMFLY bound on its maximum Thurston–Bennequin number is sharp. Of possible independent interest, we construct explicit Legendrian representatives of quasipositive 3-braid closures with maximum Thurston–Bennequin number.

Given a cooriented contact manifold $��(�M�,�\xi �)�$, it is possible to define a notion of positivity on the group $��\mathrm{Diff}�\left(�M�\right)�$ of diffeomorphisms of $�M$, by looking at paths of diffeomorphisms that are positively transverse to the contact distribution $�\xi$. We show that, in contrast to the analogous notion usually considered on the group of diffeomorphisms preserving $�\xi$, positivity on $��\mathrm{Diff}�\left(�M�\right)�$ is completely flexible. In particular, we show that for the standard contact structure on $�R^{�2�n�+�1�}$ any two diffeomorphisms are connected by a positive path. This result generalizes to compactly supported diffeomorphisms on a large class of contact manifolds. As an application, we answer a question about Legendrians in thermodynamic phase space posed by Entov, Polterovich, and Ryzhik in the context of thermodynamic processes.

Finitely many hypersurfaces are removed from unordered configuration spaces of $�n$ points in $�C$ to obtain a fibration over unordered configuration spaces of $��n�-�1�$ complex points. Fundamental groups of these restricted configuration spaces are computed in small dimensions.

In this paper, we consider a class of higher-order equations and show a sharp upper bound on fractional powers of unbounded linear operators associated with higher-order abstract equations in Hilbert spaces.

Gao and Xie conjectured that the inverse Kazhdan–Lusztig polynomial of any matroid is log-concave. Although these polynomials are not necessarily real-rooted, we conjecture that the Hadamard product of an inverse Kazhdan–Lusztig polynomial of degree $�n$ with $�{�(�1�+�t�)�}^n$ is real-rooted. Using the theory of interlacing polynomials and multiplier sequences, we confirm this conjecture for paving matroids. As a consequence, the log-concavity of inverse Kazhdan–Lusztig polynomials for paving matroids follows from Newton's inequalities.

We classify fibered ribbon pretzel knots up to mutation. The classification is complete, except perhaps for members of Lecuona's “exceptional” family of Lecuona [Algebr. Geom. Topol. 15 (2015), no. 4, 2133–2173]. The result is obtained by combining lattice embedding techniques with Gabai's classification of fibered pretzel knots, and exhibiting ribbon disks, some of which lie outside of known patterns for standard pretzel projections.

In 1994, Brezis et al. studied the Liouville theorem for the planar Ginzburg–Landau equation, and the result shows that the finite energy solutions have bifurcation properties. In 2001, Hang and Lin generalized this result to the static Landau–Lifschitz type equation. Those equations play an important role in the research of superconducting materials and ferromagnetic materials. The Pohozaev identity is the key tool to the argument there. Recent work has shown that the Pohozaev identity in integral forms can also derive the Liouville theorem for integral equations containing Riesz potentials. In this paper, we also deduce this identity and prove the Liouville theorem under certain conditions, and investigate whether the integrable solutions to the integral equations containing $�\log$-Newtonian potentials have bifurcation properties.

We prove a general Schwarz lemma for holomorphic maps between Hermitian manifolds. As a consequence, we show that a compact Kähler manifold with a pluriclosed metric of negative holomorphic sectional curvature is canonically polarized. In particular, such manifolds are projective and admit a Kähler–Einstein metric with negative scalar curvature.

We undertake to determine the homotopy type of gyrations of sphere products and of connected sums, thereby generalising results known in earlier literature as ‘Fico's Lemmata’ which underpin gyrations in their original formulation from geometric topology. We provide applications arising from recasting these results into the modern homotopy theoretic setting.