Communications on Pure and Applied Mathematics最新文献

We study the Langevin dynamics for spherical $�p$-spin models, focusing on the short time regime described by the Cugliandolo–Kurchan equations. Confirming a prediction of Cugliandolo and Kurchan, we show the asymptotic energy achieved is exactly $��E_{\infty }��\left(�p�\right)��=�2�\sqrt{�\frac{�p�-�1�}{p}�}�$ in the low temperature limit. The upper bound uses hardness results for Lipschitz optimization algorithms and applies for all temperatures. For the lower bound, we prove the dynamics reaches and stays above the lowest energy of any approximate local maximum. In fact the latter behavior holds for any Hamiltonian obeying natural smoothness estimates, even with disorder-dependent initialization and on exponential time-scales.

Given any (not necessarily connected) combinatorial finite graph and any compact smooth 6-manifold $�M^6$ with the third Betti number $��b_3�\ne �0�$, we construct a calibrated 3-dimensional homologically area minimizing surface on $�M$ equipped in a smooth metric $�g$, so that the singular set of the surface is precisely an embedding of this finite graph. Moreover, the calibration form near the singular set is a smoothly $��G�L�(�6�,�R�)�$ twisted special Lagrangian form. The constructions are based on some unpublished ideas of Professor Camillo De Lellis and Professor Robert Bryant.

For the distance function from any closed subset of any complete Finsler manifold, we prove that the singular set is equal to a countable union of delta-convex hypersurfaces up to an exceptional set of codimension two. In addition, in dimension two, the whole singular set is equal to a countable union of delta-convex Jordan arcs up to isolated points. These results are new even in the standard Euclidean space and shown to be optimal in view of regularity.

We prove that every non-degenerate minimal submanifold of codimension two can be obtained as the energy concentration set of a family of critical maps for the (rescaled) Ginzburg–Landau functional. The proof is purely variational, and follows the strategy laid out by Jerrard and Sternberg, extending a recent result for geodesics by Colinet–Jerrard–Sternberg. The same proof applies also to the $��U�\left(�1�\right)�$-Yang–Mills–Higgs and to the Allen–Cahn–Hilliard energies. While for the latter energies gluing methods are also effective, in general dimension our proof is by now the only available one in the Ginzburg–Landau setting, where the weaker energy concentration is the main technical difficulty.

Diameter estimates for Kähler metrics are established which require only an entropy bound and no lower bound on the Ricci curvature. The proof builds on recent PDE techniques for $�L^{\infty }$ estimates for the Monge–Ampère equation, with a key improvement allowing degeneracies of the volume form of codimension strictly greater than one. As a consequence, we solve the long-standing problem of uniform diameter bounds and Gromov–Hausdorff convergence of the Kähler–Ricci flow, for both finite-time and long-time solutions.

We consider a system of infinitely many penduli on an $�m$-dimensional lattice with a weak coupling. For any prescribed path in the lattice, for suitable couplings, we construct orbits for this Hamiltonian system of infinite degrees of freedom which transfer energy between nearby penduli along the path. We allow the weak coupling to be next-to-nearest neighbor or long range as long as it is strongly decaying. The transfer of energy is given by an Arnold diffusion mechanism which relies on the original V. I Arnold approach: to construct a sequence of hyperbolic invariant quasi-periodic tori with transverse heteroclinic orbits. We implement this approach in an infinite dimensional setting, both in the space of bounded $�Z^m$-sequences and in spaces of decaying $�Z^m$-sequences. Key steps in the proof are an invariant manifold theory for hyperbolic tori and a Lambda Lemma for infinite dimensional coupled map lattices with decaying interaction.

To the families of geometric measures of convex bodies (the area measures of Aleksandrov-Fenchel-Jessen, the curvature measures of Federer, and the recently discovered dual curvature measures) a new family is added. The new family of geometric measures, called chord measures, arises from the study of integral geometric invariants of convex bodies. The Minkowski problems for the new measures and their logarithmic variants are proposed and attacked. When the given ‘data’ is sufficiently regular, these problems are a new type of fully nonlinear partial differential equations involving dual quermassintegrals of functions. Major cases of these Minkowski problems are solved without regularity assumptions.

We prove several results for the Coulomb gas in any dimension $��d�\ge �2�$ that follow from isotropic averaging, a transport method based on Newton's theorem. First, we prove a high-density Jancovici–Lebowitz–Manificat law, extending the microscopic density bounds of Armstrong and Serfaty and establishing strictly sub-Gaussian tails for charge excess in dimension 2. The existence of microscopic limiting point processes is proved at the edge of the droplet. Next, we prove optimal upper bounds on the $�k$-point correlation function for merging points, including a Wegner estimate for the Coulomb gas for $��k�=�1�$. We prove the tightness of the properly rescaled $�k$th minimal particle gap, identifying the correct order in $��d�=�2�$ and a three term expansion in $��d�\ge �3�$, as well as upper and lower tail estimates. In particular, we extend the two-dimensional “perfect-freezing regime” identified by Ameur and Romero to higher dimensions. Finally, we give positive charge discrepancy bounds which are state of the art near the droplet boundary and prove incompressibility of Laughlin states in the fractional quantum Hall effect, starting at large microscopic scales. Using rigidity for fluctuations of smooth linear statistics, we show how to upgrade positive discrepancy bounds to estimates on the absolute discrepancy in certain regions.