Communications on Pure and Applied Mathematics最新文献

{"title":"Inhomogeneous turbulence for the Wick Nonlinear Schrödinger equation","authors":"Zaher Hani,&nbsp;Jalal Shatah,&nbsp;Hui Zhu","doi":"10.1002/cpa.22198","DOIUrl":"10.1002/cpa.22198","url":null,"abstract":"<p>We introduce a simplified model for wave turbulence theory—the Wick <i>nonlinear Schrödinger equation</i>, of which the main feature is the absence of all self-interactions in the correlation expansions of its solutions. For this model, we derive several wave kinetic equations that govern the effective statistical behavior of its solutions in various regimes. In the homogeneous setting, where the initial correlation is translation invariant, we obtain a wave kinetic equation similar to the one predicted by the formal theory. In the inhomogeneous setting, we obtain a wave kinetic equation that describes the statistical behavior of the wavepackets of the solutions, accounting for both the transport of wavepackets and collisions among them. Another wave kinetic equation, which seems new in the literature, also appears in a certain scaling regime of this setting and provides a more refined collision picture.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 11","pages":"4100-4162"},"PeriodicalIF":3.1,"publicationDate":"2024-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140807360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The threshold energy of low temperature Langevin dynamics for pure spherical spin glasses","authors":"Mark Sellke","doi":"10.1002/cpa.22197","DOIUrl":"10.1002/cpa.22197","url":null,"abstract":"<p>We study the Langevin dynamics for spherical <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>-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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>E</mi>\u0000 <mi>∞</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>p</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>=</mo>\u0000 <mn>2</mn>\u0000 <msqrt>\u0000 <mfrac>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <mi>p</mi>\u0000 </mfrac>\u0000 </msqrt>\u0000 </mrow>\u0000 <annotation>$E_{infty }(p)=2sqrt {frac{p-1}{p}}$</annotation>\u0000 </semantics></math> 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 <i>approximate local maximum</i>. In fact the latter behavior holds for any Hamiltonian obeying natural smoothness estimates, even with disorder-dependent initialization and on exponential time-scales.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 11","pages":"4065-4099"},"PeriodicalIF":3.1,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140621595","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Every finite graph arises as the singular set of a compact 3-D calibrated area minimizing surface","authors":"Zhenhua Liu","doi":"10.1002/cpa.22194","DOIUrl":"10.1002/cpa.22194","url":null,"abstract":"<p>Given any (not necessarily connected) combinatorial finite graph and any compact smooth 6-manifold <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>M</mi>\u0000 <mn>6</mn>\u0000 </msup>\u0000 <annotation>$M^6$</annotation>\u0000 </semantics></math> with the third Betti number <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>b</mi>\u0000 <mn>3</mn>\u0000 </msub>\u0000 <mo>≠</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$b_3not=0$</annotation>\u0000 </semantics></math>, we construct a calibrated 3-dimensional homologically area minimizing surface on <span></span><math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$M$</annotation>\u0000 </semantics></math> equipped in a smooth metric <span></span><math>\u0000 <semantics>\u0000 <mi>g</mi>\u0000 <annotation>$g$</annotation>\u0000 </semantics></math>, 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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mi>L</mi>\u0000 <mo>(</mo>\u0000 <mn>6</mn>\u0000 <mo>,</mo>\u0000 <mi>R</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$GL(6,mathbb {R})$</annotation>\u0000 </semantics></math> twisted special Lagrangian form. The constructions are based on some unpublished ideas of Professor Camillo De Lellis and Professor Robert Bryant.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 9","pages":"3670-3707"},"PeriodicalIF":3.1,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22194","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140069780","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Delta-convex structure of the singular set of distance functions","authors":"Tatsuya Miura,&nbsp;Minoru Tanaka","doi":"10.1002/cpa.22195","DOIUrl":"10.1002/cpa.22195","url":null,"abstract":"<p>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.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 9","pages":"3631-3669"},"PeriodicalIF":3.1,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140063984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Non-degenerate minimal submanifolds as energy concentration sets: A variational approach","authors":"Guido De Philippis,&nbsp;Alessandro Pigati","doi":"10.1002/cpa.22193","DOIUrl":"10.1002/cpa.22193","url":null,"abstract":"<p>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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>U</mi>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$U(1)$</annotation>\u0000 </semantics></math>-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.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 8","pages":"3581-3627"},"PeriodicalIF":3.0,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139987545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Liouville-type theorem for cylindrical cones","authors":"Nick Edelen,&nbsp;Gábor Székelyhidi","doi":"10.1002/cpa.22192","DOIUrl":"10.1002/cpa.22192","url":null,"abstract":"&lt;p&gt;Suppose that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;mo&gt;⊂&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathbf {C}_0^n subset mathbb {R}^{n+1}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a smooth strictly minimizing and strictly stable minimal hypercone (such as the Simons cone), &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;l&lt;/mi&gt;\u0000 &lt;mo&gt;≥&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$l ge 0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;annotation&gt;$M$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; a complete embedded minimal hypersurface of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mi&gt;l&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathbb {R}^{n+1+l}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; lying to one side of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mi&gt;l&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathbf {C}= mathbf {C}_0 times mathbb {R}^l$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. If the density at infinity of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;annotation&gt;$M$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is less than twice the density of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathbf {C}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, then we show that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mi&gt;l&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 ","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 8","pages":"3557-3580"},"PeriodicalIF":3.0,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139938269","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Diameter estimates in Kähler geometry","authors":"Bin Guo,&nbsp;Duong H. Phong,&nbsp;Jian Song,&nbsp;Jacob Sturm","doi":"10.1002/cpa.22196","DOIUrl":"10.1002/cpa.22196","url":null,"abstract":"<p>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 <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mi>∞</mi>\u0000 </msup>\u0000 <annotation>$L^infty$</annotation>\u0000 </semantics></math> 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.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 8","pages":"3520-3556"},"PeriodicalIF":3.0,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139928843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Approximate Gibbsian structure in strongly correlated point fields and generalized Gaussian zero ensembles","authors":"Ujan Gangopadhyay,&nbsp;Subhroshekhar Ghosh,&nbsp;Kin Aun Tan","doi":"10.1002/cpa.22187","DOIUrl":"10.1002/cpa.22187","url":null,"abstract":"&lt;p&gt;Gibbsian structure in random point fields has been a classical tool for studying their spatial properties. However, exact Gibbs property is available only in a relatively limited class of models, and it does not adequately address many random fields with a strongly dependent spatial structure. In this work, we provide a very general framework for approximate Gibbsian structure for strongly correlated random point fields, including those with a highly singular spatial structure. These include processes that exhibit strong spatial &lt;i&gt;rigidity&lt;/i&gt;, in particular, a certain one-parameter family of analytic Gaussian zero point fields, namely the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;α&lt;/mi&gt;\u0000 &lt;annotation&gt;$alpha$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-GAFs, that are known to demonstrate a wide range of such spatial behavior. Our framework entails conditions that may be verified via finite particle approximations to the process, a phenomenon that we call an approximate Gibbs property. We show that these enable one to compare the spatial conditional measures in the infinite volume limit with Gibbs-type densities supported on appropriate singular manifolds, a phenomenon we refer to as a generalized Gibbs property. Our work provides a general mechanism to rigorously understand the limiting behavior of spatial conditioning in strongly correlated point processes with growing system size. We demonstrate the scope and versatility of our approach by showing that a generalized Gibbs property holds with a logarithmic pair potential for the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;α&lt;/mi&gt;\u0000 &lt;annotation&gt;$alpha$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-GAFs for any value of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;α&lt;/mi&gt;\u0000 &lt;annotation&gt;$alpha$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. In this vein, we settle in the affirmative an open question regarding the existence of point processes with any specified level of rigidity. In particular, for the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;α&lt;/mi&gt;\u0000 &lt;annotation&gt;$alpha$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-GAF zero process, we establish the &lt;i&gt;level of rigidity&lt;/i&gt; to be exactly &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;⌊&lt;/mo&gt;\u0000 &lt;mfrac&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mi&gt;α&lt;/mi&gt;\u0000 &lt;/mfrac&gt;\u0000 &lt;mo&gt;⌋&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$lfloor frac{1}{alpha} rfloor$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, a fortiori demonstrating the phenomenon of spatial &lt;i&gt;tolerance&lt;/i&gt; subject to the local conservation of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;⌊&lt;/mo&gt;\u0000 &lt;mfrac&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mi&gt;α&lt;/mi&gt;\u0000 &lt;/mfrac&gt;\u0000 ","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 8","pages":"3427-3519"},"PeriodicalIF":3.0,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138840301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Arnold diffusion in Hamiltonian systems on infinite lattices","authors":"Filippo Giuliani,&nbsp;Marcel Guardia","doi":"10.1002/cpa.22191","DOIUrl":"10.1002/cpa.22191","url":null,"abstract":"<p>We consider a system of infinitely many penduli on an <span></span><math>\u0000 <semantics>\u0000 <mi>m</mi>\u0000 <annotation>$m$</annotation>\u0000 </semantics></math>-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 <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>Z</mi>\u0000 <mi>m</mi>\u0000 </msup>\u0000 <annotation>$mathbb {Z}^m$</annotation>\u0000 </semantics></math>-sequences and in spaces of decaying <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>Z</mi>\u0000 <mi>m</mi>\u0000 </msup>\u0000 <annotation>$mathbb {Z}^m$</annotation>\u0000 </semantics></math>-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.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 8","pages":"3333-3426"},"PeriodicalIF":3.0,"publicationDate":"2023-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138770934","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Chord measures in integral geometry and their Minkowski problems","authors":"Erwin Lutwak,&nbsp;Dongmeng Xi,&nbsp;Deane Yang,&nbsp;Gaoyong Zhang","doi":"10.1002/cpa.22190","DOIUrl":"10.1002/cpa.22190","url":null,"abstract":"<p>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.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 7","pages":"3277-3330"},"PeriodicalIF":3.0,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138565628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}