{"title":"Prescribed mean curvature min-max theory in some non-compact manifolds","authors":"Liam Mazurowski","doi":"10.1016/j.aim.2025.110133","DOIUrl":"10.1016/j.aim.2025.110133","url":null,"abstract":"<div><div>This paper develops a technique for applying one-parameter prescribed mean curvature min-max theory in certain non-compact manifolds. We give two main applications. First, fix a dimension <span><math><mn>3</mn><mo>≤</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>≤</mo><mn>7</mn></math></span> and consider a smooth function <span><math><mi>h</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>→</mo><mi>R</mi></math></span> which is asymptotic to a positive constant near infinity. We show that, under certain additional assumptions on <em>h</em>, there exists a closed hypersurface Σ in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> with mean curvature prescribed by <em>h</em>. Second, let <span><math><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>,</mo><mi>g</mi><mo>)</mo></math></span> be an asymptotically flat 3-manifold with no boundary and fix a constant <span><math><mi>c</mi><mo>></mo><mn>0</mn></math></span>. We show that, under an additional assumption on <em>M</em>, it is possible to find a closed surface Σ of constant mean curvature <em>c</em> in <em>M</em>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"464 ","pages":"Article 110133"},"PeriodicalIF":1.5,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143277976","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":"Curvature operators on Kähler manifolds","authors":"Barry Minemyer","doi":"10.1016/j.aim.2025.110142","DOIUrl":"10.1016/j.aim.2025.110142","url":null,"abstract":"<div><div>We prove that there exist Kähler manifolds that are not homotopy equivalent to a quotient of complex hyperbolic space but which admit a Riemannian metric with nonpositive curvature operator. This shows that Kähler manifolds do not satisfy the same type of rigidity with respect to the curvature operator as quaternionic hyperbolic and Cayley hyperbolic manifolds and are thus more similar to real hyperbolic manifolds in this setting. Along the way we also calculate explicit values for the eigenvalues of the curvature operator with respect to the standard complex hyperbolic metric.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"464 ","pages":"Article 110142"},"PeriodicalIF":1.5,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143274451","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":"Symmetric homoclinic tangles in reversible dynamical systems have positive topological entropy","authors":"A.J. Homburg , J.S.W. Lamb , D.V. Turaev","doi":"10.1016/j.aim.2025.110131","DOIUrl":"10.1016/j.aim.2025.110131","url":null,"abstract":"<div><div>We consider reversible vector fields in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup></math></span> such that the set of fixed points of the involutory reversing symmetry is <em>n</em>-dimensional. Let such system have a smooth one-parameter family of symmetric periodic orbits which is of saddle type in normal directions. We establish that the topological entropy is positive when the stable and unstable manifolds of this family of periodic orbits have a strongly-transverse intersection.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"464 ","pages":"Article 110131"},"PeriodicalIF":1.5,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165800","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":"A connected sum formula for embedded contact homology","authors":"Luya Wang","doi":"10.1016/j.aim.2025.110130","DOIUrl":"10.1016/j.aim.2025.110130","url":null,"abstract":"<div><div>Given two closed contact three-manifolds, one can form their contact connected sum via the Weinstein one-handle attachment. We study how pseudo-holomorphic curves in the symplectization behave under this operation. As a result, we give a connected sum formula for embedded contact homology.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"464 ","pages":"Article 110130"},"PeriodicalIF":1.5,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165801","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":"Unital operads, monoids, monads, and bar constructions","authors":"J.P. May, Ruoqi Zhang, Foling Zou","doi":"10.1016/j.aim.2024.110065","DOIUrl":"10.1016/j.aim.2024.110065","url":null,"abstract":"<div><div>We give a description of unital operads in a symmetric monoidal category as monoids in a monoidal category of unital <figure><img></figure>-sequences. This is a new variant of Kelly's old description of operads as monoids in the monoidal category of symmetric sequences. The monads associated to unital operads are the ones of interest in iterated loop space theory and factorization homology, among many other applications. Our new description of unital operads allows an illuminating comparison between the two-sided monadic bar constructions used in such applications and “classical” monoidal two-sided bar constructions. It also allows a more conceptual understanding of the scanning map central to non-abelian Poincaré duality in factorization homology.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110065"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164287","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}
Jean-Baptiste Campesato , Goulwen Fichou , Adam Parusiński
{"title":"Motivic, logarithmic, and topological Milnor fibrations","authors":"Jean-Baptiste Campesato , Goulwen Fichou , Adam Parusiński","doi":"10.1016/j.aim.2024.110075","DOIUrl":"10.1016/j.aim.2024.110075","url":null,"abstract":"<div><div>We compare the topological Milnor fibration and the motivic Milnor fibre of a regular complex function with only normal crossing singularities by introducing their common extension: the complete Milnor fibration. We give two equivalent constructions: the first one extending the classical Kato–Nakayama log-space, and the second one, more geometric, based on the real oriented multigraph construction, a version of the real oriented deformation to the normal cone. As an application, we recover A'Campo's model of the topological Milnor fibration, by quotienting the motivic Milnor fibration with suitable powers of <span><math><msub><mrow><mi>R</mi></mrow><mrow><mo>></mo><mn>0</mn></mrow></msub></math></span>, and show that it determines the classical motivic Milnor fibre.</div><div>We also give precise formulae expressing how the introduced objects change under blowings-up. As an application, we show that the motivic Milnor fibre is well-defined as an element of a suitable Grothendieck ring without requiring that the Lefschetz motive be invertible.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110075"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164291","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":"Stone-Gelfand duality for metrically complete lattice-ordered groups","authors":"Marco Abbadini , Vincenzo Marra , Luca Spada","doi":"10.1016/j.aim.2024.110067","DOIUrl":"10.1016/j.aim.2024.110067","url":null,"abstract":"<div><div>We extend Yosida's 1941 version of Stone-Gelfand duality to metrically complete unital lattice-ordered groups that are no longer required to be real vector spaces. This calls for a generalised notion of compact Hausdorff space whose points carry an arithmetic character to be preserved by continuous maps. The arithmetic character of a point is (the complete isomorphism invariant of) a metrically complete additive subgroup of the real numbers containing 1—namely, either <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mi>Z</mi></math></span> for an integer <span><math><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo></math></span>, or the whole of <span><math><mi>R</mi></math></span>. The main result needed to establish the extended duality theorem is a substantial generalisation of Urysohn's Lemma to such “arithmetic” compact Hausdorff spaces. The original duality is obtained by considering the full subcategory of spaces every point of which is assigned the entire group of real numbers. In the Introduction we indicate motivations from and connections with the theory of dimension groups.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110067"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164297","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":"The space of traces of the free group and free products of matrix algebras","authors":"Joav Orovitz, Raz Slutsky, Itamar Vigdorovich","doi":"10.1016/j.aim.2024.110053","DOIUrl":"10.1016/j.aim.2024.110053","url":null,"abstract":"<div><div>We show that the space of traces of free products of the form <span><math><mi>C</mi><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>⁎</mo><mi>C</mi><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are compact metrizable spaces without isolated points, is a Poulsen simplex, i.e., every trace is a pointwise limit of extreme traces. In particular, the space of traces of the free group <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> on <span><math><mn>2</mn><mo>≤</mo><mi>d</mi><mo>≤</mo><mo>∞</mo></math></span> generators is a Poulsen simplex, and we demonstrate that this is no longer true for many virtually free groups. Using a similar strategy, we show that the space of traces of the free product of matrix algebras <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo><mo>⁎</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> is a Poulsen simplex as well, answering a question of Musat and Rørdam for <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>. Similar results are shown for certain faces of the simplices above, such as the face of finite-dimensional traces or amenable traces.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110053"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164298","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":"Stable local dynamics: Expansion, quasi-conformality and ergodicity","authors":"Abbas Fakhari , Meysam Nassiri , Hesam Rajabzadeh","doi":"10.1016/j.aim.2024.110088","DOIUrl":"10.1016/j.aim.2024.110088","url":null,"abstract":"<div><div>In this paper, we study stable ergodicity of the action of groups of diffeomorphisms on smooth manifolds. Such actions are known to exist only on one-dimensional manifolds. The aim of this paper is to introduce a geometric method to overcome this restriction and to construct higher dimensional examples. In particular, we show that every closed manifold admits stably ergodic finitely generated group actions by diffeomorphisms of class <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>α</mi></mrow></msup></math></span>. We also prove the stable ergodicity of certain algebraic actions, including the natural action of a generic pair of matrices near the identity on a sphere of arbitrary dimension. These are consequences of the <em>quasi-conformal blender</em>, a local and stable mechanism/phenomenon introduced in this paper, which encapsulates our method for proving stable local ergodicity by providing quasi-conformal orbits with fine controlled geometry. The quasi-conformal blender is developed in the context of pseudo-semigroup actions of locally defined smooth diffeomorphisms, which allows for applications in diverse settings.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110088"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143149612","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":"Joyce structures and their twistor spaces","authors":"Tom Bridgeland","doi":"10.1016/j.aim.2024.110089","DOIUrl":"10.1016/j.aim.2024.110089","url":null,"abstract":"<div><div>Joyce structures are a class of geometric structures which first arose in relation to holomorphic generating functions for Donaldson-Thomas invariants. They can be thought of as non-linear analogues of Frobenius structures, or as special classes of complex hyperkähler manifolds. We give a detailed introduction to Joyce structures, with particular focus on the geometry of the associated twistor space. We also prove several new results.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110089"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150239","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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