Pub Date : 2025-12-11DOI: 10.1016/j.aim.2025.110731
Liam Mazurowski , Tongrui Wang , Xuan Yao
We formulate a conjecture relating the topology of a manifold's universal cover with the existence of metrics with positive m-intermediate curvature. We prove the result for manifolds of dimension and for most choices of m when . As a corollary, we show that a closed, aspherical 6-manifold cannot admit a metric with positive 4-intermediate curvature.
{"title":"On the topology of manifolds with positive intermediate curvature","authors":"Liam Mazurowski , Tongrui Wang , Xuan Yao","doi":"10.1016/j.aim.2025.110731","DOIUrl":"10.1016/j.aim.2025.110731","url":null,"abstract":"<div><div>We formulate a conjecture relating the topology of a manifold's universal cover with the existence of metrics with positive <em>m</em>-intermediate curvature. We prove the result for manifolds of dimension <span><math><mi>n</mi><mo>∈</mo><mo>{</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>}</mo></math></span> and for most choices of <em>m</em> when <span><math><mi>n</mi><mo>=</mo><mn>6</mn></math></span>. As a corollary, we show that a closed, aspherical 6-manifold cannot admit a metric with positive 4-intermediate curvature.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"485 ","pages":"Article 110731"},"PeriodicalIF":1.5,"publicationDate":"2025-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145749190","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}
Pub Date : 2025-12-10DOI: 10.1016/j.aim.2025.110730
Martin A. Guest , Alexander R. Its , Chang-Shou Lin
The purpose of this article is to answer comprehensively, with self-contained proofs, a longstanding question posed by the physicists Cecotti and Vafa in the 1990's concerning solutions of the topological-antitopological fusion equations of Toda type, the tt*-Toda equations. These equations are also of interest in differential geometry and the theory of integrable systems as a certain real form of the radial Toda equations. We describe the complete set of solutions explicitly in terms of their asymptotics, and in terms of their Stokes data. In view of the wide-ranging nature of our methods, we also provide explanatory remarks intended to make the article accessible to researchers in different areas.
{"title":"The tt*-Toda equations of An type","authors":"Martin A. Guest , Alexander R. Its , Chang-Shou Lin","doi":"10.1016/j.aim.2025.110730","DOIUrl":"10.1016/j.aim.2025.110730","url":null,"abstract":"<div><div>The purpose of this article is to answer comprehensively, with self-contained proofs, a longstanding question posed by the physicists Cecotti and Vafa in the 1990's concerning solutions of the topological-antitopological fusion equations of Toda type, the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> tt*-Toda equations. These equations are also of interest in differential geometry and the theory of integrable systems as a certain real form of the radial Toda equations. We describe the complete set of solutions explicitly in terms of their asymptotics, and in terms of their Stokes data. In view of the wide-ranging nature of our methods, we also provide explanatory remarks intended to make the article accessible to researchers in different areas.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"485 ","pages":"Article 110730"},"PeriodicalIF":1.5,"publicationDate":"2025-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145749189","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}
Pub Date : 2025-12-10DOI: 10.1016/j.aim.2025.110729
Arnaud Eteve, Cong Xue
Let G be a generically reductive group over a smooth projective curve X over a finite field. For any finite set I, we show that nearby cycles commute with proper direct image from stacks of shtukas to . This generalizes some results of Salmon and the authors.
{"title":"Nearby cycles commute with proper direct image on stacks of shtukas","authors":"Arnaud Eteve, Cong Xue","doi":"10.1016/j.aim.2025.110729","DOIUrl":"10.1016/j.aim.2025.110729","url":null,"abstract":"<div><div>Let <em>G</em> be a generically reductive group over a smooth projective curve <em>X</em> over a finite field. For any finite set <em>I</em>, we show that nearby cycles commute with proper direct image from stacks of shtukas to <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>I</mi></mrow></msup></math></span>. This generalizes some results of Salmon and the authors.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"485 ","pages":"Article 110729"},"PeriodicalIF":1.5,"publicationDate":"2025-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145748345","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}
Pub Date : 2025-12-10DOI: 10.1016/j.aim.2025.110726
Nicola Di Vittorio
Derivators, introduced independently by Grothendieck and Heller in the 1980s, provide a categorical framework for studying homotopy theory. They are based on the idea that, while the homotopy 1-category of a single model category or -category retains only limited information, the structured collection of homotopy 1-categories of diagram categories often suffices for many homotopical purposes. In this paper, we introduce a set of axioms for a 2-dimensional analog of derivators: a refinement of the homotopy 2-category of an enriched model category or -category into a coherent system of homotopy 2-categories of higher categories of diagrams. We show that these axioms are satisfied in a variety of models, including standard ones related to -category theory. Moreover, we prove that the axioms are preserved under a certain shift operation.
{"title":"Towards 2-derivators for formal ∞-category theory","authors":"Nicola Di Vittorio","doi":"10.1016/j.aim.2025.110726","DOIUrl":"10.1016/j.aim.2025.110726","url":null,"abstract":"<div><div>Derivators, introduced independently by Grothendieck and Heller in the 1980s, provide a categorical framework for studying homotopy theory. They are based on the idea that, while the homotopy 1-category of a single model category or <span><math><mo>(</mo><mo>∞</mo><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-category retains only limited information, the structured collection of homotopy 1-categories of diagram categories often suffices for many homotopical purposes. In this paper, we introduce a set of axioms for a 2-dimensional analog of derivators: a refinement of the homotopy 2-category of an enriched model category or <span><math><mo>(</mo><mo>∞</mo><mo>,</mo><mn>2</mn><mo>)</mo></math></span>-category into a coherent system of homotopy 2-categories of higher categories of diagrams. We show that these axioms are satisfied in a variety of models, including standard ones related to <span><math><mo>(</mo><mo>∞</mo><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-category theory. Moreover, we prove that the axioms are preserved under a certain shift operation.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"485 ","pages":"Article 110726"},"PeriodicalIF":1.5,"publicationDate":"2025-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145749192","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}
Pub Date : 2025-12-09DOI: 10.1016/j.aim.2025.110727
Peter van Hintum , Peter Keevash
<div><div>Motivated by the Polynomial Freiman-Ruzsa (PFR) Conjecture, we develop a theory of locality in sumsets, with several applications to John-type approximation and stability of sets with small doubling. One highlight shows that if <span><math><mi>A</mi><mo>⊂</mo><mi>Z</mi></math></span> with <span><math><mo>|</mo><mi>A</mi><mo>+</mo><mi>A</mi><mo>|</mo><mo>≤</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>ϵ</mi><mo>)</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msup><mo>|</mo><mi>A</mi><mo>|</mo></math></span> is non-degenerate then <em>A</em> is covered by <span><math><mi>O</mi><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> translates of a <em>d</em>-dimensional generalised arithmetic progression (<em>d</em>-GAP) <em>P</em> with <span><math><mo>|</mo><mi>P</mi><mo>|</mo><mo>≤</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>ϵ</mi></mrow></msub><mo>(</mo><mo>|</mo><mi>A</mi><mo>|</mo><mo>)</mo></math></span>; thus we obtain one of the polynomial bounds required by PFR, under the non-degeneracy assumption that <em>A</em> is not efficiently covered by <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>ϵ</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>)</mo></math></span> translates of a <span><math><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-GAP.</div><div>We also prove a stability result showing for any <span><math><mi>ϵ</mi><mo>,</mo><mi>α</mi><mo>></mo><mn>0</mn></math></span> that if <span><math><mi>A</mi><mo>⊆</mo><mi>Z</mi></math></span> with <span><math><mo>|</mo><mi>A</mi><mo>+</mo><mi>A</mi><mo>|</mo><mo>≤</mo><mo>(</mo><mn>2</mn><mo>−</mo><mi>ϵ</mi><mo>)</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msup><mo>|</mo><mi>A</mi><mo>|</mo></math></span> is non-degenerate then some <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>⊂</mo><mi>A</mi></math></span> with <span><math><mo>|</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>|</mo><mo>></mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo><mo>|</mo><mi>A</mi><mo>|</mo></math></span> is efficiently covered by either a <span><math><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-GAP or <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>)</mo></math></span> translates of a <em>d</em>-GAP. This ‘dimension-free’ bound for approximate covering makes for a surprising contrast with exact covering, where the required number of translates not only grows with <em>d</em>, but does so exponentially. Another highlight shows that if <span><math><mi>A</mi><mo>⊂</mo><mi>Z</mi></math></span> is non-degenerate with <span><math><mo>|</mo><mi>A</mi><mo>+</mo><mi>A</mi><mo>|</mo><mo>≤</mo><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msup><mo>+</mo><mi>ℓ</mi><mo>)</mo><mo>|</mo><mi>A</mi><mo>|</mo></math></span> and <span><math><mi>ℓ</mi><mo>≤</mo><mn>0.1</mn><mo>⋅</m
{"title":"Locality in sumsets","authors":"Peter van Hintum , Peter Keevash","doi":"10.1016/j.aim.2025.110727","DOIUrl":"10.1016/j.aim.2025.110727","url":null,"abstract":"<div><div>Motivated by the Polynomial Freiman-Ruzsa (PFR) Conjecture, we develop a theory of locality in sumsets, with several applications to John-type approximation and stability of sets with small doubling. One highlight shows that if <span><math><mi>A</mi><mo>⊂</mo><mi>Z</mi></math></span> with <span><math><mo>|</mo><mi>A</mi><mo>+</mo><mi>A</mi><mo>|</mo><mo>≤</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>ϵ</mi><mo>)</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msup><mo>|</mo><mi>A</mi><mo>|</mo></math></span> is non-degenerate then <em>A</em> is covered by <span><math><mi>O</mi><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> translates of a <em>d</em>-dimensional generalised arithmetic progression (<em>d</em>-GAP) <em>P</em> with <span><math><mo>|</mo><mi>P</mi><mo>|</mo><mo>≤</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>ϵ</mi></mrow></msub><mo>(</mo><mo>|</mo><mi>A</mi><mo>|</mo><mo>)</mo></math></span>; thus we obtain one of the polynomial bounds required by PFR, under the non-degeneracy assumption that <em>A</em> is not efficiently covered by <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>ϵ</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>)</mo></math></span> translates of a <span><math><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-GAP.</div><div>We also prove a stability result showing for any <span><math><mi>ϵ</mi><mo>,</mo><mi>α</mi><mo>></mo><mn>0</mn></math></span> that if <span><math><mi>A</mi><mo>⊆</mo><mi>Z</mi></math></span> with <span><math><mo>|</mo><mi>A</mi><mo>+</mo><mi>A</mi><mo>|</mo><mo>≤</mo><mo>(</mo><mn>2</mn><mo>−</mo><mi>ϵ</mi><mo>)</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msup><mo>|</mo><mi>A</mi><mo>|</mo></math></span> is non-degenerate then some <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>⊂</mo><mi>A</mi></math></span> with <span><math><mo>|</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>|</mo><mo>></mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo><mo>|</mo><mi>A</mi><mo>|</mo></math></span> is efficiently covered by either a <span><math><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-GAP or <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>)</mo></math></span> translates of a <em>d</em>-GAP. This ‘dimension-free’ bound for approximate covering makes for a surprising contrast with exact covering, where the required number of translates not only grows with <em>d</em>, but does so exponentially. Another highlight shows that if <span><math><mi>A</mi><mo>⊂</mo><mi>Z</mi></math></span> is non-degenerate with <span><math><mo>|</mo><mi>A</mi><mo>+</mo><mi>A</mi><mo>|</mo><mo>≤</mo><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msup><mo>+</mo><mi>ℓ</mi><mo>)</mo><mo>|</mo><mi>A</mi><mo>|</mo></math></span> and <span><math><mi>ℓ</mi><mo>≤</mo><mn>0.1</mn><mo>⋅</m","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"485 ","pages":"Article 110727"},"PeriodicalIF":1.5,"publicationDate":"2025-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145749187","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}
Pub Date : 2025-12-08DOI: 10.1016/j.aim.2025.110712
Daniel Panazzolo , Maja Resman , Loïc Teyssier
A saddle loop is a germ of a holomorphic foliation near a homoclinic saddle connection. We prove that they are classified by their Poincaré first-return map. We also prove that they are formally rigid when the Poincaré map is multivalued. Finally, we provide a list of all analytic classes of Liouville-integrable saddle loops.
{"title":"Rigidity of saddle loops","authors":"Daniel Panazzolo , Maja Resman , Loïc Teyssier","doi":"10.1016/j.aim.2025.110712","DOIUrl":"10.1016/j.aim.2025.110712","url":null,"abstract":"<div><div>A saddle loop is a germ of a holomorphic foliation near a homoclinic saddle connection. We prove that they are classified by their Poincaré first-return map. We also prove that they are formally rigid when the Poincaré map is multivalued. Finally, we provide a list of all analytic classes of Liouville-integrable saddle loops.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"485 ","pages":"Article 110712"},"PeriodicalIF":1.5,"publicationDate":"2025-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145749191","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}
Pub Date : 2025-12-03DOI: 10.1016/j.aim.2025.110552
Jin-Jie Yang, Shou-Fu Tian , Zhi-Qiang Li
<div><div>We consider the long-time asymptotic behavior of the modified Camassa-Holm (mCH) equation with finite density initial data<span><span><span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><msub><mrow><mo>(</mo><mo>(</mo><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo><mi>m</mi><mo>)</mo></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mi>m</mi><mo>=</mo><mi>u</mi><mo>−</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>,</mo><mspace></mspace><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>∈</mo><mi>R</mi><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>,</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><mspace></mspace><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mo>±</mo><mo>∞</mo></mrow></munder><mo></mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>,</mo><mspace></mspace><mspace></mspace><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mn>1</mn><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>4</mn><mo>,</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>m</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>=</mo><mn>0</mn><mo>)</mo></math></span> and <span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mn>1</mn><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span>. By deforming the Lax pair of the mCH equation, we successfully establish a well-defined mapping from initial values to reflection coefficients. Then by developing the <span><math><mover><mrow><mo>∂</mo></mrow><mo>‾</mo></mover></math></span>-nonlinear steepest descent method, we reveal that the mCH equation has four different asymptotic regions depending on <span><math><mi>τ</mi><mo>:</mo><mo>=</mo><mi>x</mi><mo>/</mo><mi>t</mi></math></span>. For the region <span><math><mi>τ</mi><mo>∈</mo><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mn>3</mn><mo>/</mo><mn>4</mn><mo>)</mo><mo>∪</mo><mo>(</mo><mn>3</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> without steady-state phase points, we strictly prove that the solution of the mCH equation can be characterized by the soliton solution and an error term, and further prove the asymptotic stability of the <em>N</em>-soliton solution. In the regions <span><math><mi>τ</mi><mo>∈</mo><mo>(</mo><mn>3</mn><mo>/</mo><mn>4</mn><mo>,</mo><mn>1</mn
{"title":"The modified Camassa-Holm equation with nonzero background: Soliton resolution conjecture and asymptotic stability of N-soliton solutions","authors":"Jin-Jie Yang, Shou-Fu Tian , Zhi-Qiang Li","doi":"10.1016/j.aim.2025.110552","DOIUrl":"10.1016/j.aim.2025.110552","url":null,"abstract":"<div><div>We consider the long-time asymptotic behavior of the modified Camassa-Holm (mCH) equation with finite density initial data<span><span><span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><msub><mrow><mo>(</mo><mo>(</mo><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><msubsup><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>)</mo><mi>m</mi><mo>)</mo></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mi>m</mi><mo>=</mo><mi>u</mi><mo>−</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>,</mo><mspace></mspace><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>∈</mo><mi>R</mi><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>,</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><mspace></mspace><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mo>±</mo><mo>∞</mo></mrow></munder><mo></mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>,</mo><mspace></mspace><mspace></mspace><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mn>1</mn><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>4</mn><mo>,</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>m</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>=</mo><mn>0</mn><mo>)</mo></math></span> and <span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mn>1</mn><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span>. By deforming the Lax pair of the mCH equation, we successfully establish a well-defined mapping from initial values to reflection coefficients. Then by developing the <span><math><mover><mrow><mo>∂</mo></mrow><mo>‾</mo></mover></math></span>-nonlinear steepest descent method, we reveal that the mCH equation has four different asymptotic regions depending on <span><math><mi>τ</mi><mo>:</mo><mo>=</mo><mi>x</mi><mo>/</mo><mi>t</mi></math></span>. For the region <span><math><mi>τ</mi><mo>∈</mo><mo>(</mo><mo>−</mo><mo>∞</mo><mo>,</mo><mn>3</mn><mo>/</mo><mn>4</mn><mo>)</mo><mo>∪</mo><mo>(</mo><mn>3</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> without steady-state phase points, we strictly prove that the solution of the mCH equation can be characterized by the soliton solution and an error term, and further prove the asymptotic stability of the <em>N</em>-soliton solution. In the regions <span><math><mi>τ</mi><mo>∈</mo><mo>(</mo><mn>3</mn><mo>/</mo><mn>4</mn><mo>,</mo><mn>1</mn","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"484 ","pages":"Article 110552"},"PeriodicalIF":1.5,"publicationDate":"2025-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145693585","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}
Pub Date : 2025-12-01DOI: 10.1016/j.aim.2025.110699
Mario Garcia-Fernandez , Raul Gonzalez Molina , Jeffrey Streets
We define a natural extension of pluriclosed flow aiming at constructing solutions of the Hull-Strominger system. We give several geometric formulations of this flow, which yield a series of a priori estimates for the flow and also for the Hull-Strominger system. The evolution equations are derived using the theory of string algebroids, a class of Courant algebroids which occur naturally in higher gauge theory. Using this, we interpret the flow as generalized Ricci flow and also as a higher/coupled version of Hermitian-Yang-Mills flow, proving furthermore that it is compatible with symmetry reduction. Regarding analytical results, we prove a priori estimates for uniformly parabolic solutions. This in particular settles the question of smooth regularity of uniformly elliptic solutions of the Hull-Strominger system, generalizing Yau's estimate for the complex Monge-Ampère equation. We prove global existence and convergence results for the flow on special backgrounds, and discuss a conjectural relationship of the flow to the geometrization of Reid's fantasy.
{"title":"Pluriclosed flow and the Hull-Strominger system","authors":"Mario Garcia-Fernandez , Raul Gonzalez Molina , Jeffrey Streets","doi":"10.1016/j.aim.2025.110699","DOIUrl":"10.1016/j.aim.2025.110699","url":null,"abstract":"<div><div>We define a natural extension of pluriclosed flow aiming at constructing solutions of the Hull-Strominger system. We give several geometric formulations of this flow, which yield a series of a priori estimates for the flow and also for the Hull-Strominger system. The evolution equations are derived using the theory of string algebroids, a class of Courant algebroids which occur naturally in higher gauge theory. Using this, we interpret the flow as generalized Ricci flow and also as a higher/coupled version of Hermitian-Yang-Mills flow, proving furthermore that it is compatible with symmetry reduction. Regarding analytical results, we prove a priori <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> estimates for uniformly parabolic solutions. This in particular settles the question of smooth regularity of uniformly elliptic solutions of the Hull-Strominger system, generalizing Yau's <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> estimate for the complex Monge-Ampère equation. We prove global existence and convergence results for the flow on special backgrounds, and discuss a conjectural relationship of the flow to the geometrization of Reid's fantasy.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"485 ","pages":"Article 110699"},"PeriodicalIF":1.5,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145694482","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}
Pub Date : 2025-12-01DOI: 10.1016/j.aim.2025.110682
Ezra Miller
The geometric and algebraic theory of monomial ideals and multigraded modules is initiated over real-exponent polynomial rings and, more generally, monoid algebras for real polyhedral cones. The main results include the generalization of Nakayama's lemma; complete theories of minimal and dense primary, secondary, and irreducible decomposition, including associated and attached faces; socles and tops; minimality and density for downset hulls, upset covers, and fringe presentations; Matlis duality; and geometric analysis of staircases. Modules that are semialgebraic or piecewise-linear (PL) have the relevant property preserved by functorial constructions as well as by minimal primary and secondary decompositions. And when the modules in question are subquotients of the group itself, such as monomial ideals and quotients modulo them, minimal primary and secondary decompositions are canonical, as are irreducible decompositions up to the new real-exponent notion of density.
{"title":"Essential graded algebra over polynomial rings with real exponents","authors":"Ezra Miller","doi":"10.1016/j.aim.2025.110682","DOIUrl":"10.1016/j.aim.2025.110682","url":null,"abstract":"<div><div>The geometric and algebraic theory of monomial ideals and multigraded modules is initiated over real-exponent polynomial rings and, more generally, monoid algebras for real polyhedral cones. The main results include the generalization of Nakayama's lemma; complete theories of minimal and dense primary, secondary, and irreducible decomposition, including associated and attached faces; socles and tops; minimality and density for downset hulls, upset covers, and fringe presentations; Matlis duality; and geometric analysis of staircases. Modules that are semialgebraic or piecewise-linear (PL) have the relevant property preserved by functorial constructions as well as by minimal primary and secondary decompositions. And when the modules in question are subquotients of the group itself, such as monomial ideals and quotients modulo them, minimal primary and secondary decompositions are canonical, as are irreducible decompositions up to the new real-exponent notion of density.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"485 ","pages":"Article 110682"},"PeriodicalIF":1.5,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145694480","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}
Pub Date : 2025-12-01DOI: 10.1016/j.aim.2025.110689
Fabrizio Bianchi , Tien-Cuong Dinh
We consider the unique measure of maximal entropy of an automorphism of a compact Kähler manifold with simple action on cohomology. We show that it is exponentially mixing of all orders with respect to Hölder observables. It follows that the Central Limit Theorem (CLT) holds for these observables. In particular, our result applies to all automorphisms of compact Kähler surfaces with positive entropy.
{"title":"Exponential mixing of all orders and CLT for automorphisms of compact Kähler manifolds","authors":"Fabrizio Bianchi , Tien-Cuong Dinh","doi":"10.1016/j.aim.2025.110689","DOIUrl":"10.1016/j.aim.2025.110689","url":null,"abstract":"<div><div>We consider the unique measure of maximal entropy of an automorphism of a compact Kähler manifold with simple action on cohomology. We show that it is exponentially mixing of all orders with respect to Hölder observables. It follows that the Central Limit Theorem (CLT) holds for these observables. In particular, our result applies to all automorphisms of compact Kähler surfaces with positive entropy.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"485 ","pages":"Article 110689"},"PeriodicalIF":1.5,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145694481","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}
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