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

We show that analytic analogs of Brunn–Minkowski-type inequalities fail for functional intrinsic volumes on convex functions. This is demonstrated both through counterexamples and by connecting the problem to results of Colesanti, Hug, and Saorín Gómez. By restricting to a smaller set of admissible functions, we then introduce a family of variational functionals and establish Wulff-type inequalities for these quantities. In addition, we derive inequalities for the corresponding family of mixed functionals, thereby generalizing an earlier Aleksandrov–Fenchel-type inequality by Klartag and recovering a special case of a Pólya–Szegő-type inequality by Klimov, which was also recently investigated by Bianchi, Cianchi, and Gronchi.

In this article, we show that generally almost regular flows, introduced by Bamler and Kleiner, in closed 3-manifolds will either go extinct in finite time or flow to a collection of smooth embedded minimal surfaces, possibly with multiplicity. Using a perturbative argument, then we construct piecewise almost regular flows that either go extinct in finite time or flow to a stable minimal surface, possibly with multiplicity. We apply these results to construct minimal surfaces in 3-manifolds in a variety of circumstances, mainly novel from the point of the view that the arguments are via parabolic methods.

We construct an explicit infinite family of pairwise non-isomorphic infinite simple groups of type $�F_{\infty }$ (in particular, they are finitely presented) that act faithfully on the circle by orientation-preserving homeomorphisms, but that admit neither non-trivial piecewise affine nor piecewise projective actions on the projective line. Our examples are certain forest-skein groups which, informally, are a mixture of Richard Thompson's groups with Vaughan Jones' planar algebras.

We show that Wise's power alternative is stable under certain group constructions, use this to prove the power alternative for new classes of groups and recover known results from a unified perspective. For groups acting on trees, we introduce a dynamical condition that allows us to deduce the power alternative for the group from the power alternative for its stabilisers of points. As an application, we reduce the power alternative for Artin groups to the power alternative for free-of-infinity Artin groups, under some conditions on their parabolic subgroups. We also introduce a uniform version of the power alternative and prove it, among other things, for a large family of two-dimensional Artin groups. As a corollary, we deduce that these Artin groups have uniform exponential growth. Finally, we prove that the power alternative is stable under taking relatively hyperbolic groups. We apply this to show that various examples, including all free-by-$�Z$ groups and a natural subclass of hierarchically hyperbolic groups, satisfy the uniform power alternative.

In this article, we prove that Ramsey's theorem for pairs and two colors is a $��\forall �{\Pi }_4^0�$ conservative extension of $��{\mathrm{RCA}}_0�+�B�{\Sigma }_2^0�$, where a $��\forall �{\Pi }_4^0�$ formula consists of a universal quantifier over sets followed by a $�{\Pi }_4^0$ formula. The proof is an improvement of a result by Patey and Yokoyama and a step toward the resolution of the longstanding question of the first-order part of Ramsey's theorem for pairs. For this, we introduce a new general technique for proving $�{\Pi }_4^0$-conservation theorems.

Let $��f�:�N�\to �(�M�,�g�)�$ be a two-sided, complete, stable, minimal, immersed hypersurface. In this paper, we establish various vanishing theorems for the space of $�L^2$-harmonic forms and spinors (when $�M$ is additionally spin) under suitable positive curvature assumptions on the ambient manifold. Our results in the setting of forms extend to higher dimensions and more general ambient Riemannian manifolds previous vanishing theorems due to Tanno [J. Math. Soc. Japan 48 (1996), no. 4, 761–768] and Zhu [Nonlinear Anal. 75 (2012), no. 13, 5039–5043]. In the setting of spin manifolds, our results allow to conclude, for instance, that any oriented, complete, stable, minimal, immersed hypersurface of $�R^m$ or $�S^m$ carries no non-trivial $�L^2$-harmonic spinors. Finally, analogous results are proved for strongly stable constant mean curvature hypersurfaces.

We develop a probabilistic approach to the celebrated Jacobian conjecture, which states that any Keller map (i.e. any polynomial mapping $��F�:�C^n�\to �C^n�$ whose Jacobian determinant is a non-zero constant) has a compositional inverse which is also a polynomial. The Jacobian conjecture may be formulated in terms of a problem involving labellings of rooted trees; we give a new probabilistic derivation of this formulation using multi-type branching processes. Thereafter, we develop a simple and novel approach to the Jacobian conjecture in terms of a problem involving shuffling subtrees of $�d$-Catalan trees, that is, planar $�d$-ary trees. We also show that, if one can construct a certain Markov chain on large $�d$-Catalan trees which updates its value by randomly shuffling certain nearby subtrees, and in such a way that the stationary distribution of this chain is uniform, then the Jacobian conjecture is true. Finally, we use the local limit theory of large random trees to show that the subtree shuffling conjecture is true in a certain asymptotic sense, and thereafter use our machinery to prove an approximate version of the Jacobian conjecture, stating that inverses of Keller maps have small power series coefficients for their high-degree terms.

We develop a new approach to the study of spectral asymmetry. Working with the operator $��curl�:�=�\ast �d�$ on a connected oriented closed Riemannian 3-manifold, we construct, by means of microlocal analysis, the asymmetry operator — a scalar pseudodifferential operator of order $��-�3�$. The latter is completely determined by the Riemannian manifold and its orientation, and encodes information about spectral asymmetry. The asymmetry operator generalises and contains the classical eta invariant traditionally associated with the asymmetry of the spectrum, which can be recovered by computing its regularised operator trace. Remarkably, the whole construction is direct and explicit.