We develop a new degree theory for 4‐dimensional, asymptotically conical gradient expanding solitons. Our theory implies the existence of gradient expanding solitons that are asymptotic to any given cone over with non‐negative scalar curvature. We also obtain a similar existence result for cones whose link is diffeomorphic to if we allow the expanding soliton to have orbifold singularities. Our theory reveals the existence of a new topological invariant, called the expander degree , applicable to a particular class of compact, smooth 4‐orbifolds with boundary. This invariant is roughly equal to a signed count of all possible gradient expanding solitons that can be defined on the interior of the orbifold and are asymptotic to any fixed cone metric with non‐negative scalar curvature. If the expander degree of an orbifold is non‐zero, then gradient expanding solitons exist for any such cone metric. We show that the expander degree of the 4‐disk and any orbifold of the form equals 1. Additionally, we demonstrate that the expander degree of certain orbifolds, including exotic 4‐disks, vanishes. Our theory also sheds light on the relation between gradient and non‐gradient expanding solitons with respect to their asymptotic model. More specifically, we show that among the set of asymptotically conical expanding solitons, the subset of those solitons that are gradient forms a union of connected components.
{"title":"Degree theory for 4‐dimensional asymptotically conical gradient expanding solitons","authors":"Richard H. Bamler, Eric Chen","doi":"10.1002/cpa.70024","DOIUrl":"https://doi.org/10.1002/cpa.70024","url":null,"abstract":"We develop a new degree theory for 4‐dimensional, asymptotically conical gradient expanding solitons. Our theory implies the existence of gradient expanding solitons that are asymptotic to any given cone over with non‐negative scalar curvature. We also obtain a similar existence result for cones whose link is diffeomorphic to if we allow the expanding soliton to have orbifold singularities. Our theory reveals the existence of a new topological invariant, called the <jats:italic>expander degree</jats:italic> , applicable to a particular class of compact, smooth 4‐orbifolds with boundary. This invariant is roughly equal to a signed count of all possible gradient expanding solitons that can be defined on the interior of the orbifold and are asymptotic to any fixed cone metric with non‐negative scalar curvature. If the expander degree of an orbifold is non‐zero, then gradient expanding solitons exist for any such cone metric. We show that the expander degree of the 4‐disk and any orbifold of the form equals 1. Additionally, we demonstrate that the expander degree of certain orbifolds, including exotic 4‐disks, vanishes. Our theory also sheds light on the relation between gradient and non‐gradient expanding solitons with respect to their asymptotic model. More specifically, we show that among the set of asymptotically conical expanding solitons, the subset of those solitons that are <jats:italic>gradient</jats:italic> forms a union of connected components.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"66 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145759574","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}
Sébastien Boucksom, Mattias Jonsson, Antonio Trusiani
Generalizing previous results of Arezzo–Pacard–Singer, Seyyedali–Székelyhidi, and Hallam, we prove the invariance under smooth blowups of the class of weighted extremal Kähler manifolds, modulo a log‐concavity assumption on the first weight. Through recent work of Di Nezza–Jubert–Lahdili and Han–Liu, this is obtained as a consequence of a general uniform coercivity estimate for the (relative, weighted) Mabuchi energy on the blowup, which applies more generally to any equivariant resolution of singularities of Fano type of a compact Kähler klt space whose Mabuchi energy is assumed to be coercive.
推广了Arezzo-Pacard-Singer， seyyedali - sz kelyhidi， and Hallam之前的结果，证明了一类加权极值Kähler流形在光滑膨胀下的不变性，在第一权值上取对数凹性的模。通过Di Nezza-Jubert-Lahdili和Han-Liu最近的工作，这是作为（相对的，加权的）Mabuchi能量在blowup上的一般均匀矫顽力估计的结果得到的，它更普遍地适用于紧化Kähler klt空间的Fano型奇点的任何等变分辨率，其中Mabuchi能量被假设为矫顽力。
{"title":"Weighted extremal kähler metrics on resolutions of singularities","authors":"Sébastien Boucksom, Mattias Jonsson, Antonio Trusiani","doi":"10.1002/cpa.70026","DOIUrl":"https://doi.org/10.1002/cpa.70026","url":null,"abstract":"Generalizing previous results of Arezzo–Pacard–Singer, Seyyedali–Székelyhidi, and Hallam, we prove the invariance under smooth blowups of the class of weighted extremal Kähler manifolds, modulo a log‐concavity assumption on the first weight. Through recent work of Di Nezza–Jubert–Lahdili and Han–Liu, this is obtained as a consequence of a general uniform coercivity estimate for the (relative, weighted) Mabuchi energy on the blowup, which applies more generally to any equivariant resolution of singularities of Fano type of a compact Kähler klt space whose Mabuchi energy is assumed to be coercive.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"32 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145673663","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}
We study isoperimetric inequalities on “slabs”, namely weighted Riemannian manifolds obtained as the product of the uniform measure on a finite length interval with a codimension‐one base. As our two main applications, we consider the case when the base is the flat torus and the standard Gaussian measure on . The isoperimetric conjecture on the three‐dimensional cube predicts that minimizers are enclosed by spheres about a corner, cylinders about an edge and coordinate planes. This has only been established for relative volumes close to 0, and 1 by compactness arguments. Our analysis confirms the isoperimetric conjecture on the three‐dimensional cube with side lengths in a new range of relative volumes . In particular, we confirm the conjecture for the standard cube () for all , when for the entire range where spheres are conjectured to be minimizing, and also for all . When we reduce the validity of the full conjecture to establishing that the half‐plane is an isoperimetric minimizer. We also show that the analogous conjecture on a high‐dimensional cube is false for . In the case of a slab with a Gaussian base of width , we identify a phase transition when and when . In particular, while products of half‐planes with are always minimizing when , when they are never minimizing, being beaten by Gaussian unduloids. In the range , a potential trichotomy occurs.
{"title":"Isoperimetric inequalities on slabs with applications to cubes and Gaussian slabs","authors":"Emanuel Milman","doi":"10.1002/cpa.70020","DOIUrl":"https://doi.org/10.1002/cpa.70020","url":null,"abstract":"We study isoperimetric inequalities on “slabs”, namely weighted Riemannian manifolds obtained as the product of the uniform measure on a finite length interval with a codimension‐one base. As our two main applications, we consider the case when the base is the flat torus and the standard Gaussian measure on . The isoperimetric conjecture on the three‐dimensional cube predicts that minimizers are enclosed by spheres about a corner, cylinders about an edge and coordinate planes. This has only been established for relative volumes close to 0, and 1 by compactness arguments. Our analysis confirms the isoperimetric conjecture on the three‐dimensional cube with side lengths in a new range of relative volumes . In particular, we confirm the conjecture for the standard cube () for all , when for the entire range where spheres are conjectured to be minimizing, and also for all . When we reduce the validity of the full conjecture to establishing that the half‐plane is an isoperimetric minimizer. We also show that the analogous conjecture on a high‐dimensional cube is false for . In the case of a slab with a Gaussian base of width , we identify a phase transition when and when . In particular, while products of half‐planes with are always minimizing when , when they are never minimizing, being beaten by Gaussian unduloids. In the range , a potential trichotomy occurs.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"206 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145608974","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}
Understanding the transition mechanism of boundary layer flows is of great significance in physics and engineering, especially due to the current development of supersonic and hypersonic aircraft. In this paper, we construct multiple unstable acoustic modes so‐called, Mack modes , which play a crucial role during the early stage of transition in the supersonic boundary layer. To this end, we develop an inner‐outer gluing iteration to solve a hyperbolic‐elliptic mixed type and singular system.
{"title":"Mack modes in supersonic boundary layer","authors":"Nader Masmoudi, Yuxi Wang, Di Wu, Zhifei Zhang","doi":"10.1002/cpa.70022","DOIUrl":"https://doi.org/10.1002/cpa.70022","url":null,"abstract":"Understanding the transition mechanism of boundary layer flows is of great significance in physics and engineering, especially due to the current development of supersonic and hypersonic aircraft. In this paper, we construct multiple unstable acoustic modes so‐called, Mack modes , which play a crucial role during the early stage of transition in the supersonic boundary layer. To this end, we develop an inner‐outer gluing iteration to solve a hyperbolic‐elliptic mixed type and singular system.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"23 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145554293","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}
Dorin Bucur, Jimmy Lamboley, Mickaël Nahon, Raphaël Prunier
Let be an open set with the same volume as the unit ball and let be the ‐th eigenvalue of the Laplace operator of with Dirichlet boundary conditions on . In this work, we answer the following question: Ifis small, how large canbe? We establish quantitative bounds of the form with sharp exponents depending on the multiplicity of . We first show that such an inequality is valid with for any , improving previous known results and providing the sharpest possible exponent. Then, through the study of a vectorial free boundary problem, we show that one can achieve the better exponent if is simple. We also obtain a similar result for the whole cluster of eigenvalues when is multiple, thus providing a complete answer to the question above. As a consequence of these results, we obtain the persistence of the ball as the minimizer for a large class of spectral functionals which are small perturbations of the fundamental eigenvalue on the one hand, and a full reverse Kohler–Jobin inequality on the other hand, solving an open problem formulated by M. Van Den Berg, G. Buttazzo and A. Pratelli.
设一个与单位球体积相同的开集，设有狄利克雷边界条件的拉普拉斯算子的第一个特征值。在这项工作中，我们回答了以下问题：如果是小的，可以有多大？我们根据的多重性，建立了具有尖锐指数形式的定量界限。我们首先证明了这样的不等式对于任何都是有效的，改进了以前已知的结果并提供了可能的最尖锐的指数。然后，通过对无向量边界问题的研究，我们证明了如果简单，可以得到较好的指数。当为倍数时，我们对整个特征值簇也得到了类似的结果，从而对上面的问题提供了一个完整的答案。作为这些结果的结果，我们得到了球作为一大类谱泛函的最小值的持续性，这些泛函一方面是基本特征值的小扰动，另一方面是完全逆的Kohler-Jobin不等式，解决了M. Van Den Berg， G. Buttazzo和a . Pratelli提出的开放问题。
{"title":"Sharp quantitative stability of the Dirichlet spectrum near the ball","authors":"Dorin Bucur, Jimmy Lamboley, Mickaël Nahon, Raphaël Prunier","doi":"10.1002/cpa.70021","DOIUrl":"https://doi.org/10.1002/cpa.70021","url":null,"abstract":"Let be an open set with the same volume as the unit ball and let be the ‐th eigenvalue of the Laplace operator of with Dirichlet boundary conditions on . In this work, we answer the following question: <jats:disp-quote content-type=\"quotation\"> <jats:italic>If</jats:italic> <jats:italic>is small, how large can</jats:italic> <jats:italic>be?</jats:italic> </jats:disp-quote> We establish quantitative bounds of the form with sharp exponents depending on the multiplicity of . We first show that such an inequality is valid with for any , improving previous known results and providing the sharpest possible exponent. Then, through the study of a vectorial free boundary problem, we show that one can achieve the better exponent if is simple. We also obtain a similar result for the whole cluster of eigenvalues when is multiple, thus providing a complete answer to the question above. As a consequence of these results, we obtain the persistence of the ball as the minimizer for a large class of spectral functionals which are small perturbations of the fundamental eigenvalue on the one hand, and a full reverse Kohler–Jobin inequality on the other hand, solving an open problem formulated by M. Van Den Berg, G. Buttazzo and A. Pratelli.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"24 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145509281","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}
We are interested in inequalities that bound the Riesz means of the eigenvalues of the Dirichlet and Neumann Laplacians in terms of their semiclassical counterpart. We show that the classical inequalities of Berezin–Li–Yau and Kröger, valid for Riesz exponents , extend to certain values , provided the underlying domain is convex. We also study the corresponding optimization problems and describe the implications of a possible failure of Pólya's conjecture for convex sets in terms of Riesz means. These findings allow us to describe the asymptotic behavior of solutions of a spectral shape optimization problem for convex sets.
{"title":"Semiclassical inequalities for Dirichlet and Neumann Laplacians on convex domains","authors":"Rupert L. Frank, Simon Larson","doi":"10.1002/cpa.70019","DOIUrl":"https://doi.org/10.1002/cpa.70019","url":null,"abstract":"We are interested in inequalities that bound the Riesz means of the eigenvalues of the Dirichlet and Neumann Laplacians in terms of their semiclassical counterpart. We show that the classical inequalities of Berezin–Li–Yau and Kröger, valid for Riesz exponents , extend to certain values , provided the underlying domain is convex. We also study the corresponding optimization problems and describe the implications of a possible failure of Pólya's conjecture for convex sets in terms of Riesz means. These findings allow us to describe the asymptotic behavior of solutions of a spectral shape optimization problem for convex sets.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"27 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145381744","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}
In this paper we give a characterization of Avila's quantized acceleration of the Lyapunov exponent via the number of zeros of the Dirichlet determinants in finite volume. As applications, we prove ‐Hölder continuity of the integrated density of states for supercritical quasi‐periodic Schrödinger operators restricted to the th stratum, for any and . We establish Anderson localization for all Diophantine frequencies for the operator with even analytic potential function on the first supercritical stratum, which has positive measure if it is nonempty.
{"title":"Avila's acceleration via zeros of determinants and applications to Schrödinger cocycles","authors":"Rui Han, Wilhelm Schlag","doi":"10.1002/cpa.70018","DOIUrl":"https://doi.org/10.1002/cpa.70018","url":null,"abstract":"In this paper we give a characterization of Avila's quantized acceleration of the Lyapunov exponent via the number of zeros of the Dirichlet determinants in finite volume. As applications, we prove ‐Hölder continuity of the integrated density of states for supercritical quasi‐periodic Schrödinger operators restricted to the th stratum, for any and . We establish Anderson localization for all Diophantine frequencies for the operator with even analytic potential function on the first supercritical stratum, which has positive measure if it is nonempty.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"125 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145396447","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}
This work addresses the question of uniqueness and regularity of the minimizers of a convex but not strictly convex integral functional with linear growth in a two‐dimensional setting. The integrand – whose precise form derives directly from the theory of perfect plasticity – behaves quadratically close to the origin and grows linearly once a specific threshold is reached. Thus, in contrast with the only existing literature on uniqueness for functionals with linear growth, that is that which pertains to the generalized least gradient, the integrand is not a norm. We make use of hyperbolic conservation laws hidden in the structure of the problem to tackle uniqueness. Our argument strongly relies on the regularity of a vector field – the Cauchy stress in the terminology of perfect plasticity – which allows us to define characteristic lines and then to employ the method of characteristics. Using the detailed structure of the characteristic landscape evidenced in our preliminary study [5], we show that this vector field is actually continuous, save for possibly two points. The different behaviors of the energy density at zero and at infinity imply an inequality constraint on the Cauchy stress. Under a barrier type convexity assumption on the set where the inequality constraint is saturated, we show that uniqueness holds for pure Dirichlet boundary data devoid of any regularity properties, a stronger result than that of uniqueness for a given trace on the whole boundary since our minimizers can fail to attain the boundary data. We also show a partial regularity result for the minimizer.
{"title":"Uniqueness, regularity, and characteristic flow for a non strictly convex singular variational problem","authors":"Jean‐François Babadjian, Gilles A. Francfort","doi":"10.1002/cpa.70015","DOIUrl":"https://doi.org/10.1002/cpa.70015","url":null,"abstract":"This work addresses the question of uniqueness and regularity of the minimizers of a convex but not strictly convex integral functional with linear growth in a two‐dimensional setting. The integrand – whose precise form derives directly from the theory of perfect plasticity – behaves quadratically close to the origin and grows linearly once a specific threshold is reached. Thus, in contrast with the only existing literature on uniqueness for functionals with linear growth, that is that which pertains to the generalized least gradient, the integrand is not a norm. We make use of hyperbolic conservation laws hidden in the structure of the problem to tackle uniqueness. Our argument strongly relies on the regularity of a vector field – the Cauchy stress in the terminology of perfect plasticity – which allows us to define characteristic lines and then to employ the method of characteristics. Using the detailed structure of the characteristic landscape evidenced in our preliminary study [5], we show that this vector field is actually continuous, save for possibly two points. The different behaviors of the energy density at zero and at infinity imply an inequality constraint on the Cauchy stress. Under a barrier type convexity assumption on the set where the inequality constraint is saturated, we show that uniqueness holds for pure Dirichlet boundary data devoid of any regularity properties, a stronger result than that of uniqueness for a given trace on the whole boundary since our minimizers can fail to attain the boundary data. We also show a partial regularity result for the minimizer.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"25 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145306023","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}
Taking place naturally in a gas subject to a given wall temperature distribution, the “ghost effect” exhibits a rare kinetic effect beyond the prediction of classical fluid theory and Fourier law in such a classical problem in physics. As the Knudsen number goes to zero, the finite variation of temperature in the bulk is determined by an infinitesimal, ghost‐like velocity field, created by a given finite variation of the tangential wall temperature as predicted by Maxwell's slip boundary condition. Mathematically, such a finite variation leads to the presence of a severe singularity and a Knudsen layer approximation in the fundamental energy estimate. Neither difficulty is within the reach of any existing PDE theory on the steady Boltzmann equation in a general 3D bounded domain. Consequently, in spite of the discovery of such a ghost effect from temperature variation in as early as 1960s, its mathematical validity has been a challenging and intriguing open question, causing confusion and suspicion. We settle this open question in affirmative if the temperature variation is small but finite, by developing a new framework with four major innovations as follows: (1) a key ‐Hodge decomposition and its corresponding local ‐conservation law eliminate the severe bulk singularity, leading to a reduced energy estimate; (2) a surprising gain in via momentum conservation and a dual Stokes solution; (3) the ‐conservation, energy conservation, and a coupled dual Stokes–Poisson solution reduces to an boundary singularity; (4) a crucial construction of ‐cutoff boundary layer eliminates such boundary singularity via new Hardy's and BV estimates.
{"title":"Ghost effect from Boltzmann theory","authors":"Raffaele Esposito, Yan Guo, Rossana Marra, Lei Wu","doi":"10.1002/cpa.70017","DOIUrl":"https://doi.org/10.1002/cpa.70017","url":null,"abstract":"Taking place naturally in a gas subject to a given wall temperature distribution, the “ghost effect” exhibits a rare kinetic effect beyond the prediction of classical fluid theory and Fourier law in such a classical problem in physics. As the Knudsen number goes to zero, the finite variation of temperature in the bulk is determined by an infinitesimal, ghost‐like velocity field, created by a given <jats:italic>finite</jats:italic> variation of the tangential wall temperature as predicted by Maxwell's slip boundary condition. Mathematically, such a finite variation leads to the presence of a severe singularity and a Knudsen layer approximation in the fundamental energy estimate. Neither difficulty is within the reach of any existing PDE theory on the steady Boltzmann equation in a general 3D bounded domain. Consequently, in spite of the discovery of such a ghost effect from temperature variation in as early as 1960s, its mathematical validity has been a challenging and intriguing open question, causing confusion and suspicion. We settle this open question in affirmative if the temperature variation is small but finite, by developing a new framework with four major innovations as follows: (1) a key ‐Hodge decomposition and its corresponding local ‐conservation law eliminate the severe bulk singularity, leading to a reduced energy estimate; (2) a surprising gain in via momentum conservation and a dual Stokes solution; (3) the ‐conservation, energy conservation, and a coupled dual Stokes–Poisson solution reduces to an boundary singularity; (4) a crucial construction of ‐cutoff boundary layer eliminates such boundary singularity via new Hardy's and BV estimates.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"54 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2025-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145295266","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}
Sylvain E. Cappell, Laurenţiu Maxim, Jörg Schürmann, Julius L. Shaneson
We first investigate torus‐equivariant motivic characteristic classes of toric varieties, and then apply them via the equivariant Riemann–Roch formalism to prove very general Euler–Maclaurin‐type formulae for full‐dimensional simple lattice polytopes.We consider ‐equivariant versions and of the motivic Chern and, resp., Hirzebruch characteristic classes of a toric variety (with corresponding torus ), and extend many known results from the non‐equivariant context to the equivariant setting. For example, the equivariant motivic Chern class is computed as the sum of the equivariant Grothendieck classes of the ‐equivariant sheaves of Zariski‐forms weighted by . Using the motivic, as well as the characteristic class nature of , the corresponding generalized equivariant Hirzebruch‐genus of a ‐invariant Cartier divisor on is also calculated.Further global formulae for are obtained in the simplicial context based on the Cox construction and the equivariant Lefschetz–Riemann–Roch theorem of Edidin–Graham. Alternative proofs of all these results are given via localization techniques at the torus fixed points in ‐equivariant ‐ and, resp., homology theories of toric varieties, due to Brion–Vergne and, resp., Brylinski–Zhang. These localization results apply to any toric variety with a torus fixed point. In localized ‐equivariant ‐theory, we extend a classical formula of Brion for a full‐dimensional lattice polytope to a weighted version. We also generalize the Molien formula of Brion–Vergne for the localized class of the structure sheaf of a simplicial toric variety to the context of . Similarly, we calculate the localized Hirzebruch class in localized ‐equivariant homology, extending the corresponding results of Brylinski–Zhang for the localized Todd class (fitting with the equivariant Hirzebruch class for ).As main applications of our equivariant characteristic class formulae, we provide a geometric perspective on several weighted Euler–Maclaurin‐type formulae for full‐dimensional simple lattice polytopes (corresponding to simplicial toric varieties), coming from the equivariant toric geometry via the equivariant Hirzebruch–Riemann–Roch (for an ample torus invariant Cartier divisor). Our main results even provide generalizations to arbitrary equivariant coherent sheaf coefficients, including algebraic geometric proofs of (weighted versions of) the Euler–Maclaurin formulae of Cappell–Shaneson, Brion–Vergne, Guillemin, and so forth (all of which correspond to the choice of the structure sheaf), via the equivariant Hirzebruch–Riemann–Roch formalism. In particu
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