Bulletin of the London Mathematical Society最新文献

In this paper, we consider mixed curvature $�C_{�a�,�b�}$, which is a convex combination of Ricci curvature and holomorphic sectional curvature introduced by Chu–Lee–Tam [Trans. Amer. Math. Soc. 375 (2022), no. 11, 7925-7944]. We prove that if a compact complex manifold $�M$ admits a Kähler metric with quasi-positive mixed curvature and $��3�a�+�2�b�⩾�0�$, then it is projective. If $��a�,�b�⩾�0�$, then $�M$ is rationally connected. As a corollary, the same result holds for $�k$-Ricci curvature. We also show that any compact Kähler manifold with quasi-positive 2-scalar curvature is projective. Lastly, we generalize the result to the Hermitian case. In particular, any compact Hermitian threefold with quasi-positive real bisectional curvature has vanishing Hodge number $�h^{�2�,�0�}$. Furthermore, if it is Kählerian, then it is projective.

For a compact relative Kähler fibration over a compact Kähler manifold with negative holomorphic sectional curvature, if the relative Kähler form on each fiber also exhibits the negative holomorphic sectional curvature, we can construct Kähler metrics with the negative holomorphic sectional curvature on the total space. Additionally, if this form induces a Griffiths negative Hermitian metric on the relative tangent bundle, and the base admits a Kähler metric with the negative holomorphic bisectional curvature, we can also construct Kähler metrics with the negative holomorphic bisectional curvature on the total space. As an application, for a non-trivial fibration where both the fibers and base have Kähler metrics with negative holomorphic bisectional curvature, and the fibers are one-dimensional, we can explicitly construct Kähler metrics of the negative holomorphic bisectional curvature on the total space, thus resolving a question posed by To and Yeung for the case where the fibers have dimension one.

In this paper, we present stability results for various reverse isoperimetric problems in $�R^2$. Specifically, we prove the stability of the reverse isoperimetric inequality for $�\lambda$-convex bodies — convex bodies with the property that each of their boundary points $�p$ supports a ball of radius $��1�/�\lambda �$ so that the body lies inside the ball in a neighborhood of $�p$. For convex bodies with smooth boundaries, $�\lambda$-convexity is equivalent to having the curvature of the boundary bounded below by $��\lambda �>�0�$. Additionally, within this class of convex bodies, we establish stability for the reverse inradius inequality and the reverse Cheeger inequality. Even without its stability version, the sharp reverse Cheeger inequality is new in dimension 2.

This paper establishes the noncommutative maximal inequality for discrete spherical averages over a lacunary sequence. Our main result extends the work of Hughes [J. Anal. Math. 138 (2019), no. 1, 1-21] to the noncommutative setting and, meanwhile, strengthens the recent study of Chen and Hong [arXiv: 2410.06035 (2024)] to the case of $�Z^4$.

Let $�p$ be an odd prime, and suppose that $�f_1$ and $�f_2$ are weight two newforms sharing the same irreducible Galois representation modulo $�p$. We establish a transition formula relating the $�\lambda$-invariants $��\lambda �\left(�f_1�\right)�$ and $��\lambda �\left(�f_2�\right)�$ in the case where their underlying modular varieties have the same dimension. For abelian extensions $��F�/�Q�$, this essentially removes the $��\mu �\left(�f_i�\right)�=�0�$ condition present in the earlier work of Greenberg–Vatsal and Emerton–Pollack–Weston.

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.