For the nongradient exclusion process, we prove the quantitative homogenization of the diffusion matrix and the conductivity by local functions. The proof relies on the renormalization approach developed by Armstrong, Kuusi, Mourrat, and Smart, while the new challenge here is the hard core constraint of particle number on every site. Therefore, a coarse‐grained method is proposed to lift the configuration to a larger space without exclusion, and a gradient coupling between two systems is applied to capture the spatial cancellation. We then strengthen the convergence rate to be uniform concerning the density, and integrate it into the work by Funaki, Uchiyama, and Yau [ IMA Vol. Math. Appl ., 77 (1996), pp. 1–40.] to yield a quantitative hydrodynamic limit. Our new approach avoids showing the characterization of closed forms and provides stronger results. The extension is discussed for the model in the presence of disorder on the bonds.
{"title":"Quantitative Homogenization and Hydrodynamic Limit of Nongradient Exclusion Process","authors":"Tadahisa Funaki, Chenlin Gu, Han Wang","doi":"10.1002/cpa.70034","DOIUrl":"https://doi.org/10.1002/cpa.70034","url":null,"abstract":"For the nongradient exclusion process, we prove the quantitative homogenization of the diffusion matrix and the conductivity by local functions. The proof relies on the renormalization approach developed by Armstrong, Kuusi, Mourrat, and Smart, while the new challenge here is the hard core constraint of particle number on every site. Therefore, a coarse‐grained method is proposed to lift the configuration to a larger space without exclusion, and a gradient coupling between two systems is applied to capture the spatial cancellation. We then strengthen the convergence rate to be uniform concerning the density, and integrate it into the work by Funaki, Uchiyama, and Yau [ <jats:italic>IMA Vol. Math. Appl</jats:italic> ., 77 (1996), pp. 1–40.] to yield a quantitative hydrodynamic limit. Our new approach avoids showing the characterization of closed forms and provides stronger results. The extension is discussed for the model in the presence of disorder on the bonds.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"38 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146095737","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}
The Guillemin boundary condition naturally appears in the study of Kähler geometry of toric manifolds. In the present paper, the following Guillemin boundary value problem is investigated: , where and is a simple convex polytope in . The solvability of this problem is given under the necessary and sufficient condition. The key issue in the proof is to obtain the boundary regularity of . Due to the difficulty caused by the structure of the equation itself and the singularity of , special attention is required to understand the influence of different singularity types at various positions on and how these impact the behavior of in its vicinity.
{"title":"Monge–Ampère Equation With Guillemin Boundary Condition in High Dimension","authors":"Genggeng Huang, Weiming Shen","doi":"10.1002/cpa.70033","DOIUrl":"https://doi.org/10.1002/cpa.70033","url":null,"abstract":"The Guillemin boundary condition naturally appears in the study of Kähler geometry of toric manifolds. In the present paper, the following Guillemin boundary value problem is investigated: , where and is a simple convex polytope in . The solvability of this problem is given under the necessary and sufficient condition. The key issue in the proof is to obtain the boundary regularity of . Due to the difficulty caused by the structure of the equation itself and the singularity of , special attention is required to understand the influence of different singularity types at various positions on and how these impact the behavior of in its vicinity.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"55 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146089381","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}
Yifan Chen, Xiaoou Cheng, Jonathan Niles‐Weed, Jonathan Weare
The unadjusted Langevin algorithm is commonly used to sample probability distributions in extremely high‐dimensional settings. However, existing analyses of the algorithm for strongly log‐concave distributions suggest that, as the dimension of the problem increases, the number of iterations required to ensure convergence within a desired error in the metric scales in proportion to or . In this paper, we argue that, despite this poor scaling of the error for the full set of variables, the behavior for a small number of variables can be significantly better: A number of iterations proportional to , up to logarithmic terms in , often suffices for the algorithm to converge to within a desired error for all ‐marginals. We refer to this effect as delocalization of bias . We show that the delocalization effect does not hold universally and prove its validity for Gaussian distributions and strongly log‐concave distributions with certain sparse interactions. Our analysis relies on a novel metric to measure convergence. A key technical challenge we address is the lack of a one‐step contraction property in this metric. Finally, we use asymptotic arguments to explore potential generalizations of the delocalization effect beyond the Gaussian and sparse interactions setting.
{"title":"Convergence of Unadjusted Langevin in High Dimensions: Delocalization of Bias","authors":"Yifan Chen, Xiaoou Cheng, Jonathan Niles‐Weed, Jonathan Weare","doi":"10.1002/cpa.70032","DOIUrl":"https://doi.org/10.1002/cpa.70032","url":null,"abstract":"The unadjusted Langevin algorithm is commonly used to sample probability distributions in extremely high‐dimensional settings. However, existing analyses of the algorithm for strongly log‐concave distributions suggest that, as the dimension of the problem increases, the number of iterations required to ensure convergence within a desired error in the metric scales in proportion to or . In this paper, we argue that, despite this poor scaling of the error for the full set of variables, the behavior for a <jats:italic>small number</jats:italic> of variables can be significantly better: A number of iterations proportional to , up to logarithmic terms in , often suffices for the algorithm to converge to within a desired error for all ‐marginals. We refer to this effect as <jats:italic>delocalization of bias</jats:italic> . We show that the delocalization effect does not hold universally and prove its validity for Gaussian distributions and strongly log‐concave distributions with certain sparse interactions. Our analysis relies on a novel metric to measure convergence. A key technical challenge we address is the lack of a one‐step contraction property in this metric. Finally, we use asymptotic arguments to explore potential generalizations of the delocalization effect beyond the Gaussian and sparse interactions setting.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"260 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146089380","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}
Batchelor predicted that a passive scalar with diffusivity , advected by a smooth fluid velocity, should typically have Fourier mass distributed as for . For a broad class of velocity fields, we give a quantitative lower bound for a version of this prediction summed over constant width annuli in Fourier space. This improves on previously known results, which require the prediction to be summed over the whole ball.
{"title":"Fourier Mass Lower Bounds for Batchelor‐Regime Passive Scalars","authors":"William Cooperman, Keefer Rowan","doi":"10.1002/cpa.70030","DOIUrl":"https://doi.org/10.1002/cpa.70030","url":null,"abstract":"Batchelor predicted that a passive scalar with diffusivity , advected by a smooth fluid velocity, should typically have Fourier mass distributed as for . For a broad class of velocity fields, we give a quantitative lower bound for a version of this prediction summed over constant width annuli in Fourier space. This improves on previously known results, which require the prediction to be summed over the whole ball.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"79 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145993108","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}
Ajay Chandra, Guilherme de Lima Feltes, Hendrik Weber
We show a priori bounds for solutions to in finite volume in the framework of Hairer's Regularity Structures [Invent Math 198:269–504, 2014]. We assume and that is of negative Hölder regularity of order where for an explicit , and that it can be lifted to a model in the sense of Regularity Structures. Our main results guarantee non‐explosion of the solution in finite time and a growth, which is at most polynomial in . Our estimates imply global well‐posedness for the 2‐d generalised parabolic Anderson model on the torus, as well as for the parabolic quantisation of the Sine–Gordon Euclidean quantum fieldtheory (EQFT) on the torus in the regime . We also consider the parabolic quantisation of a massive Sine–Gordon EQFT and derive estimates that imply the existence of the measure for the same range of . Finally, our estimates apply to Itô SPDEs in the sense of Da Prato‐Zabczyk [ Stochastic Equations in Infinite Dimensions , Enc. Math. App., Cambridge Univ. Press, 1992] and imply existence of a stochastic flow beyond the trace‐class regime.
{"title":"A priori bounds for the generalised parabolic Anderson model","authors":"Ajay Chandra, Guilherme de Lima Feltes, Hendrik Weber","doi":"10.1002/cpa.70025","DOIUrl":"https://doi.org/10.1002/cpa.70025","url":null,"abstract":"We show a priori bounds for solutions to in finite volume in the framework of Hairer's Regularity Structures [Invent Math 198:269–504, 2014]. We assume and that is of negative Hölder regularity of order where for an explicit , and that it can be lifted to a model in the sense of Regularity Structures. Our main results guarantee non‐explosion of the solution in finite time and a growth, which is at most polynomial in . Our estimates imply global well‐posedness for the 2‐d generalised parabolic Anderson model on the torus, as well as for the parabolic quantisation of the Sine–Gordon Euclidean quantum fieldtheory (EQFT) on the torus in the regime . We also consider the parabolic quantisation of a massive Sine–Gordon EQFT and derive estimates that imply the existence of the measure for the same range of . Finally, our estimates apply to Itô SPDEs in the sense of Da Prato‐Zabczyk [ <jats:italic>Stochastic Equations in Infinite Dimensions</jats:italic> , Enc. Math. App., Cambridge Univ. Press, 1992] and imply existence of a stochastic flow beyond the trace‐class regime.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"120 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145955151","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 show that any open set that is a finite distance away from a Lipschitz subgraph will become a Lipschitz subgraph after flowing under fractional mean curvature flow for a finite, universal time. Our proof is quantitative and inherently nonlocal, as the corresponding statement is false for classical mean curvature flow. This is the first regularizing effect proven for weak solutions to nonlocal curvature flow.
{"title":"Eventual regularization of fractional mean curvature flow","authors":"Stephen Cameron","doi":"10.1002/cpa.70028","DOIUrl":"https://doi.org/10.1002/cpa.70028","url":null,"abstract":"We show that any open set that is a finite distance away from a Lipschitz subgraph will become a Lipschitz subgraph after flowing under fractional mean curvature flow for a finite, universal time. Our proof is quantitative and inherently nonlocal, as the corresponding statement is false for classical mean curvature flow. This is the first regularizing effect proven for weak solutions to nonlocal curvature flow.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"38 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145955149","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 show that a complete contractible 3‐manifold with positive scalar curvature and bounded geometry must be . We also show that an open handlebody of genus larger than 1 does not admit complete metrics with positive scalar curvature and bounded geometry. Our results rely on the maximal weak solution to inverse mean curvature flow due to the third‐named author.
{"title":"3‐Manifolds With Positive Scalar Curvature and Bounded Geometry","authors":"Otis Chodosh, Yi Lai, Kai Xu","doi":"10.1002/cpa.70029","DOIUrl":"https://doi.org/10.1002/cpa.70029","url":null,"abstract":"We show that a complete contractible 3‐manifold with positive scalar curvature and bounded geometry must be . We also show that an open handlebody of genus larger than 1 does not admit complete metrics with positive scalar curvature and bounded geometry. Our results rely on the maximal weak solution to inverse mean curvature flow due to the third‐named author.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"4 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145937959","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 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":"10.1002/cpa.70026","url":null,"abstract":"<p>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.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"79 4","pages":"1073-1148"},"PeriodicalIF":2.7,"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}
<p>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 <span></span><math> <semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>/</mo> <mn>2</mn> <msup> <mi>Z</mi> <mn>2</mn> </msup> </mrow> <annotation>$mathbb {R}^2 / 2 mathbb {Z}^2$</annotation> </semantics></math> and the standard Gaussian measure on <span></span><math> <semantics> <msup> <mi>R</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <annotation>$mathbb {R}^{n-1}$</annotation> </semantics></math>. 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, <span></span><math> <semantics> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> <annotation>$1/2$</annotation> </semantics></math> and 1 by compactness arguments. Our analysis confirms the isoperimetric conjecture on the three-dimensional cube with side lengths <span></span><math> <semantics> <mrow> <mo>(</mo> <mi>β</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> <annotation>$(beta,1,1)$</annotation> </semantics></math> in a new range of relative volumes <span></span><math> <semantics> <mrow> <mover> <mi>v</mi> <mo>¯</mo> </mover> <mo>∈</mo> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> <mo>]</mo> </mrow> </mrow> <annotation>$bar{v} in [0,1/2]$</annotation> </semantics></math>. In particular, we co
{"title":"Isoperimetric inequalities on slabs with applications to cubes and Gaussian slabs","authors":"Emanuel Milman","doi":"10.1002/cpa.70020","DOIUrl":"10.1002/cpa.70020","url":null,"abstract":"<p>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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 <msup>\u0000 <mi>Z</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathbb {R}^2 / 2 mathbb {Z}^2$</annotation>\u0000 </semantics></math> and the standard Gaussian measure on <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msup>\u0000 <annotation>$mathbb {R}^{n-1}$</annotation>\u0000 </semantics></math>. 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, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$1/2$</annotation>\u0000 </semantics></math> and 1 by compactness arguments. Our analysis confirms the isoperimetric conjecture on the three-dimensional cube with side lengths <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>β</mi>\u0000 <mo>,</mo>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(beta,1,1)$</annotation>\u0000 </semantics></math> in a new range of relative volumes <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mover>\u0000 <mi>v</mi>\u0000 <mo>¯</mo>\u0000 </mover>\u0000 <mo>∈</mo>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$bar{v} in [0,1/2]$</annotation>\u0000 </semantics></math>. In particular, we co","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"79 4","pages":"1012-1072"},"PeriodicalIF":2.7,"publicationDate":"2025-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.70020","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":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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