Pub Date : 2021-10-09DOI: 10.1142/s1793525321500540
Zhi Li, G. Wei, Gangyi Chen
In this paper, we obtain the classification theorems for 3-dimensional complete [Formula: see text]-translators [Formula: see text] with constant squared norm [Formula: see text] of the second fundamental form and constant [Formula: see text] in the Euclidean space [Formula: see text].
{"title":"Complete 3-dimensional λ-translators in the Euclidean space ℝ4","authors":"Zhi Li, G. Wei, Gangyi Chen","doi":"10.1142/s1793525321500540","DOIUrl":"https://doi.org/10.1142/s1793525321500540","url":null,"abstract":"In this paper, we obtain the classification theorems for 3-dimensional complete [Formula: see text]-translators [Formula: see text] with constant squared norm [Formula: see text] of the second fundamental form and constant [Formula: see text] in the Euclidean space [Formula: see text].","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"39 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80518363","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-08DOI: 10.1142/s179352532350036x
S. Auyeung
We define a broad class of local Lagrangian intersections which we call quasi-minimally degenerate (QMD) before developing techniques for studying their local Floer homology. In some cases, one may think of such intersections as modeled on minimally degenerate functions as defined by Kirwan. One major result of this paper is: if $L_0,L_1$ are two Lagrangian submanifolds whose intersection decomposes into QMD sets, there is a spectral sequence converging to their Floer homology $HF_*(L_0,L_1)$ whose $E^1$ page is obtained from local data given by the QMD pieces. The $E^1$ terms are the singular homologies of submanifolds with boundary that come from perturbations of the QMD sets. We then give some applications of these techniques towards studying affine varieties, reproducing some prior results using our more general framework.
{"title":"Local Lagrangian Floer Homology of Quasi-Minimally Degenerate Intersections","authors":"S. Auyeung","doi":"10.1142/s179352532350036x","DOIUrl":"https://doi.org/10.1142/s179352532350036x","url":null,"abstract":"We define a broad class of local Lagrangian intersections which we call quasi-minimally degenerate (QMD) before developing techniques for studying their local Floer homology. In some cases, one may think of such intersections as modeled on minimally degenerate functions as defined by Kirwan. One major result of this paper is: if $L_0,L_1$ are two Lagrangian submanifolds whose intersection decomposes into QMD sets, there is a spectral sequence converging to their Floer homology $HF_*(L_0,L_1)$ whose $E^1$ page is obtained from local data given by the QMD pieces. The $E^1$ terms are the singular homologies of submanifolds with boundary that come from perturbations of the QMD sets. We then give some applications of these techniques towards studying affine varieties, reproducing some prior results using our more general framework.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87799277","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-04DOI: 10.1142/s1793525323500127
E. Kerman, Yuanpu Liang
In this paper we settle three basic questions concerning the Gutt-Hutchings capacities. Our primary result settles a version of the recognition question in the negative. We prove that the Gutt-Hutchings capacities together with the volume, do not constitute a complete set of symplectic invariants for star-shaped domains with smooth boundary. We also establish two independence properties. We prove that, even for star-shaped domains with smooth boundaries, these capacities are independent from the volume. We also prove that the capacities are mutually independent by constructing, for any $j in mathbb{N}$, a family of star-shaped domains, with smooth boundary and the same volume, whose capacities are all equal but the $j^{th}$. The constructions underlying these results are not exotic. They are convex and concave toric domains. A key to the progress made here is a significant simplification of the formulae of Gutt and Hutchings for the capacities of such domains which holds under an additional symmetry assumption. This simplification allows us to identify new blind spots of the capacities which are used to construct the desired examples.
{"title":"On symplectic capacities and their blind spots","authors":"E. Kerman, Yuanpu Liang","doi":"10.1142/s1793525323500127","DOIUrl":"https://doi.org/10.1142/s1793525323500127","url":null,"abstract":"In this paper we settle three basic questions concerning the Gutt-Hutchings capacities. Our primary result settles a version of the recognition question in the negative. We prove that the Gutt-Hutchings capacities together with the volume, do not constitute a complete set of symplectic invariants for star-shaped domains with smooth boundary. We also establish two independence properties. We prove that, even for star-shaped domains with smooth boundaries, these capacities are independent from the volume. We also prove that the capacities are mutually independent by constructing, for any $j in mathbb{N}$, a family of star-shaped domains, with smooth boundary and the same volume, whose capacities are all equal but the $j^{th}$. The constructions underlying these results are not exotic. They are convex and concave toric domains. A key to the progress made here is a significant simplification of the formulae of Gutt and Hutchings for the capacities of such domains which holds under an additional symmetry assumption. This simplification allows us to identify new blind spots of the capacities which are used to construct the desired examples.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89198540","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-02DOI: 10.1142/s1793525321500473
Junchao Shentu, Chen Zhao
The existence of Kähler Einstein metrics with mixed cone and cusp singularity has received considerable attentions in recent years. It is believed that such kind of metric would give rise to important geometric invariants. We computed their [Formula: see text]-Hodge–Frölicher spectral sequence under the Dirichlet and Neumann boundary conditions and examine the pure Hodge structures on them. It turns out that these cohomologies agree well with the de Rham cohomology of a good compactification.
{"title":"An L2-Poincaré–Dolbeault lemma of spaces with mixed cone-cusp singular metrics","authors":"Junchao Shentu, Chen Zhao","doi":"10.1142/s1793525321500473","DOIUrl":"https://doi.org/10.1142/s1793525321500473","url":null,"abstract":"The existence of Kähler Einstein metrics with mixed cone and cusp singularity has received considerable attentions in recent years. It is believed that such kind of metric would give rise to important geometric invariants. We computed their [Formula: see text]-Hodge–Frölicher spectral sequence under the Dirichlet and Neumann boundary conditions and examine the pure Hodge structures on them. It turns out that these cohomologies agree well with the de Rham cohomology of a good compactification.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"193 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81067458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-13DOI: 10.1142/s1793525323500164
Jiahao Hu
We show the minimal total Betti number of a closed almost complex manifold of dimension $2nge 8$ is four, thus confirming a conjecture of Sullivan except for dimension $6$. Along the way, we prove the only simply connected closed complex manifold having total Betti number three is the complex projective plane.
{"title":"Almost complex manifolds with total betti number three","authors":"Jiahao Hu","doi":"10.1142/s1793525323500164","DOIUrl":"https://doi.org/10.1142/s1793525323500164","url":null,"abstract":"We show the minimal total Betti number of a closed almost complex manifold of dimension $2nge 8$ is four, thus confirming a conjecture of Sullivan except for dimension $6$. Along the way, we prove the only simply connected closed complex manifold having total Betti number three is the complex projective plane.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"55 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89190472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-29DOI: 10.1142/s179352532150045x
Kazuto Takao
We give some local moves of the Stein factorization of the product map of two Morse functions on a closed orientable smooth [Formula: see text]-manifold which can be realized by isotopies of the functions.
{"title":"Local moves of the Stein factorization of the product map of two functions on a 3-manifold","authors":"Kazuto Takao","doi":"10.1142/s179352532150045x","DOIUrl":"https://doi.org/10.1142/s179352532150045x","url":null,"abstract":"We give some local moves of the Stein factorization of the product map of two Morse functions on a closed orientable smooth [Formula: see text]-manifold which can be realized by isotopies of the functions.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"25 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80013742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-20DOI: 10.1142/s1793525323500036
Matt Clay
We introduce a condition on the monodromy of a free-by-cyclic group, Gφ, called the chain flare condition, that implies that the L–torsion, ρ(Gφ), is non-zero. We conjecture that this condition holds whenever the monodromy is exponentially growing.
{"title":"Chain flaring and L2-torsion of free-by-cyclic groups","authors":"Matt Clay","doi":"10.1142/s1793525323500036","DOIUrl":"https://doi.org/10.1142/s1793525323500036","url":null,"abstract":"We introduce a condition on the monodromy of a free-by-cyclic group, Gφ, called the chain flare condition, that implies that the L–torsion, ρ(Gφ), is non-zero. We conjecture that this condition holds whenever the monodromy is exponentially growing.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"11 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88931088","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-19DOI: 10.1142/s179352532350005x
S. Avvakumov, Alexey Balitskiy, Alfredo Hubard, R. Karasev
Inspired by the classical Riemannian systolic inequality of Gromov we present a combinatorial analogue providing a lower bound on the number of vertices of a simplicial complex in terms of its edge-path systole. Similarly to the Riemannian case, where the inequality holds under a topological assumption of"essentiality", our proofs rely on a combinatorial analogue of that assumption. Under a stronger assumption, expressed in terms of cohomology cup-length, we improve our results quantitatively. We also illustrate our methods in the continuous setting, generalizing and improving quantitatively the Minkowski principle of Balacheff and Karam; a corollary of this result is the extension of the Guth--Nakamura cup-length systolic bound from manifolds to complexes.
{"title":"Systolic inequalities for the number of vertices","authors":"S. Avvakumov, Alexey Balitskiy, Alfredo Hubard, R. Karasev","doi":"10.1142/s179352532350005x","DOIUrl":"https://doi.org/10.1142/s179352532350005x","url":null,"abstract":"Inspired by the classical Riemannian systolic inequality of Gromov we present a combinatorial analogue providing a lower bound on the number of vertices of a simplicial complex in terms of its edge-path systole. Similarly to the Riemannian case, where the inequality holds under a topological assumption of\"essentiality\", our proofs rely on a combinatorial analogue of that assumption. Under a stronger assumption, expressed in terms of cohomology cup-length, we improve our results quantitatively. We also illustrate our methods in the continuous setting, generalizing and improving quantitatively the Minkowski principle of Balacheff and Karam; a corollary of this result is the extension of the Guth--Nakamura cup-length systolic bound from manifolds to complexes.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"14 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81502379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-31DOI: 10.1142/S1793525322500030
Valentin Bosshard
Floer theory for Lagrangian cobordisms was developed by Biran and Cornea in a series of papers [BC13, BC14, BC17] to study the triangulated structure of the derived Fukaya category of monotone symplectic manifolds. This paper explains how to use the language of stops to study Lagrangian cobordisms in Liouville manifolds and the associated exact triangles in the derived wrapped Fukaya category. Furthermore, we compute the cobordism groups of non-compact Riemann surfaces of finite type.
{"title":"Lagrangian Cobordisms in Liouville manifolds","authors":"Valentin Bosshard","doi":"10.1142/S1793525322500030","DOIUrl":"https://doi.org/10.1142/S1793525322500030","url":null,"abstract":"Floer theory for Lagrangian cobordisms was developed by Biran and Cornea in a series of papers [BC13, BC14, BC17] to study the triangulated structure of the derived Fukaya category of monotone symplectic manifolds. This paper explains how to use the language of stops to study Lagrangian cobordisms in Liouville manifolds and the associated exact triangles in the derived wrapped Fukaya category. Furthermore, we compute the cobordism groups of non-compact Riemann surfaces of finite type.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"22 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86365922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-06DOI: 10.1142/s1793525321500680
Tsuyoshi Kato, D. Kishimoto, Mitsunobu Tsutaya
Given a countable metric space, we can consider its end. Then a basis of a Hilbert space indexed by the metric space defines an end of the Hilbert space, which is a new notion and different from an end as a metric space. Such an indexed basis also defines unitary operators of finite propagation, and these operators preserve an end of a Hilbert space. Then, we can define a Hilbert bundle with end, which lightens up new structures of Hilbert bundles. In a special case, we can define characteristic classes of Hilbert bundles with ends, which are new invariants of Hilbert bundles. We show Hilbert bundles with ends appear in natural contexts. First, we generalize the pushforward of a vector bundle along a finite covering to an infinite covering, which is a Hilbert bundle with end under a mild condition. Then we compute characteristic classes of some pushforwards along infinite coverings. Next, we will show the spectral decompositions of nice differential operators give rise to Hilbert bundles with ends, which elucidate new features of spectral decompositions. The spectral decompositions we will consider are the Fourier transform and the harmonic oscillators.
{"title":"Hilbert bundles with ends","authors":"Tsuyoshi Kato, D. Kishimoto, Mitsunobu Tsutaya","doi":"10.1142/s1793525321500680","DOIUrl":"https://doi.org/10.1142/s1793525321500680","url":null,"abstract":"Given a countable metric space, we can consider its end. Then a basis of a Hilbert space indexed by the metric space defines an end of the Hilbert space, which is a new notion and different from an end as a metric space. Such an indexed basis also defines unitary operators of finite propagation, and these operators preserve an end of a Hilbert space. Then, we can define a Hilbert bundle with end, which lightens up new structures of Hilbert bundles. In a special case, we can define characteristic classes of Hilbert bundles with ends, which are new invariants of Hilbert bundles. We show Hilbert bundles with ends appear in natural contexts. First, we generalize the pushforward of a vector bundle along a finite covering to an infinite covering, which is a Hilbert bundle with end under a mild condition. Then we compute characteristic classes of some pushforwards along infinite coverings. Next, we will show the spectral decompositions of nice differential operators give rise to Hilbert bundles with ends, which elucidate new features of spectral decompositions. The spectral decompositions we will consider are the Fourier transform and the harmonic oscillators.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"2 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87587114","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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