Pub Date : 2024-07-29DOI: 10.4310/cag.2023.v31.n10.a2
Li,Tian-Jun, Mak,Cheuk Yu, Min,Jie
This paper investigates the symplectic and contact topology associated to circular spherical divisors. We classify, up to toric equivalence, all concave circular spherical divisors $ D $ that can be embedded symplectically into a closed symplectic 4-manifold and show they are all realized as symplectic log Calabi-Yau pairs if their complements are minimal. We then determine the Stein fillability and rational homology type of all minimal symplectic fillings for the boundary torus bundles of such $D$. When $ D $ is anticanonical and convex, we give explicit Betti number bounds for Stein fillings of its boundary contact torus bundle.
本文研究了与圆球面骰子相关的交映拓扑学和接触拓扑学。我们对所有可以交映嵌入封闭交映 4-manifold的凹圆球卜元 $ D $ 进行了分类（直到环等价），并证明如果它们的补集是最小的，它们都可以实现为交映 log Calabi-Yau 对。然后，我们确定了这种 $D$ 的边界环束的所有最小交映填充的 Steinability 和有理同调类型。当 $ D $ 是反谐和凸时，我们给出了其边界接触环束的斯坦因填充的明确贝蒂数边界。
{"title":"Circular spherical divisors and their contact topology","authors":"Li,Tian-Jun, Mak,Cheuk Yu, Min,Jie","doi":"10.4310/cag.2023.v31.n10.a2","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n10.a2","url":null,"abstract":"This paper investigates the symplectic and contact topology associated to circular spherical divisors. We classify, up to toric equivalence, all concave circular spherical divisors $ D $ that can be embedded symplectically into a closed symplectic 4-manifold and show they are all realized as symplectic log Calabi-Yau pairs if their complements are minimal. We then determine the Stein fillability and rational homology type of all minimal symplectic fillings for the boundary torus bundles of such $D$. When $ D $ is anticanonical and convex, we give explicit Betti number bounds for Stein fillings of its boundary contact torus bundle.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"45 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-29DOI: 10.4310/cag.2023.v31.n10.a4
Albanese,Michael, LeBrun,Claude
For compact complex surfaces $(M^{4}, J)$ of Kähler type, it was previously shown [30] that the sign of the Yamabe invariant $mathscr{Y}(M)$ only depends on the Kodaira dimension $text{Kod} (M, J)$. In this paper, we prove that this pattern in fact extends to all compact complex surfaces except those of class VII. In the process, we also reprove a result from [2] that explains why the exclusion of class VII is essential here.
对于凯勒类型的紧凑复曲面$(M^{4}, J)$，之前已经证明[30]山边不变量$mmathscr{Y}(M)$的符号只取决于柯达伊拉维度$text{Kod} (M, J)$。在本文中，我们证明了这一模式事实上扩展到了除第 VII 类之外的所有紧凑复曲面。在此过程中，我们还重新证明了[2]中的一个结果，它解释了为什么这里必须排除第 VII 类。
{"title":"Kodaira dimension & the Yamabe problem, II","authors":"Albanese,Michael, LeBrun,Claude","doi":"10.4310/cag.2023.v31.n10.a4","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n10.a4","url":null,"abstract":"For compact complex surfaces $(M^{4}, J)$ of Kähler type, it was previously shown [30] that the sign of the Yamabe invariant $mathscr{Y}(M)$ only depends on the Kodaira dimension $text{Kod} (M, J)$. In this paper, we prove that this pattern in fact extends to all compact complex surfaces except those of class VII. In the process, we also reprove a result from [2] that explains why the exclusion of class VII is essential here.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"45 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-29DOI: 10.4310/cag.2023.v31.n10.a1
Freedman,Michael, Krushkal,Vyacheslav, Leininger,Christopher J., Reid,Alan W.
We introduce and study the notion of filling links in $3$-manifolds: a link $L$ is filling in $M$ if for any $1$-spine $G$ of $M$ which is disjoint from $L$, $pi _{1}(G)$ injects into $pi _{1}(Msmallsetminus L)$. A weaker "$k$-filling" version concerns injectivity modulo $k$-th term of the lower central series. For each $kgeq 2$ we construct a $k$-filling link in the $3$-torus. The proof relies on an extension of the Stallings theorem which may be of independent interest. We discuss notions related to "filling" links in $3$-manifolds, and formulate several open problems. The appendix by C. Leininger and A. Reid establishes the existence of a filling hyperbolic link in any closed orientable $3$-manifold with $pi _{1}(M)$ of rank $2$.
{"title":"Filling links and spines in 3-manifolds","authors":"Freedman,Michael, Krushkal,Vyacheslav, Leininger,Christopher J., Reid,Alan W.","doi":"10.4310/cag.2023.v31.n10.a1","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n10.a1","url":null,"abstract":"We introduce and study the notion of filling links in $3$-manifolds: a link $L$ is filling in $M$ if for any $1$-spine $G$ of $M$ which is disjoint from $L$, $pi _{1}(G)$ injects into $pi _{1}(Msmallsetminus L)$. A weaker \"$k$-filling\" version concerns injectivity modulo $k$-th term of the lower central series. For each $kgeq 2$ we construct a $k$-filling link in the $3$-torus. The proof relies on an extension of the Stallings theorem which may be of independent interest. We discuss notions related to \"filling\" links in $3$-manifolds, and formulate several open problems. The appendix by C. Leininger and A. Reid establishes the existence of a filling hyperbolic link in any closed orientable $3$-manifold with $pi _{1}(M)$ of rank $2$.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"169 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863780","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-29DOI: 10.4310/cag.2023.v31.n10.a7
Wang,Yi, Xu,Hang
We study the variational structure of the complex $k$-Hessian equation on bounded domain $Xsubset mathbb C^{n}$ with boundary $M=partial X$. We prove that the Dirichlet problem $sigma _{k} (partial bar{partial} u) =0$ in $X$, and $u=f$ on $M$ is variational and we give an explicit construction of the associated functional $ mathcal{E}_{k}(u)$. Moreover we prove $ mathcal{E}_{k}(u)$ satisfies the Dirichlet principle. In a special case when $k=2$, our constructed functional $ mathcal{E}_{2}(u)$ involves the Hermitian mean curvature of the boundary, the notion first introduced and studied by X. Wang [37]. Earlier work of J. Case and and the first author of this article [9] introduced a boundary operator for the (real) $k$-Hessian functional which satisfies the Dirichlet principle. The present paper shows that there is a parallel picture in the complex setting.
我们研究了边界为 $M=partial X$ 的有界域 $Xsubset mathbb C^{n}$ 上复 $k$-Hessian 方程的变分结构。我们证明了德里赫特问题 $sigma _{k}(partial bar{partial} u) =0$ in $X$, and $u=f$ on $M$ is variational and we give an explicit construction of the associated functional $ mathcal{E}_{k}(u)$.此外，我们还证明 $ mathcal{E}_{k}(u)$ 满足德里赫特原理。在 $k=2$ 的特殊情况下，我们构造的函数 $ mathcal{E}_{2}(u)$ 涉及边界的赫尔墨斯平均曲率，这一概念由王旭东首次提出并研究[37]。J. Case 和本文第一作者的早期研究[9]为（实）$k$-Hessian 函数引入了一个满足狄利克特原理的边界算子。本文表明，在复数环境中也有类似的情况。
{"title":"The Dirichlet principle for the complex $k$-Hessian functional","authors":"Wang,Yi, Xu,Hang","doi":"10.4310/cag.2023.v31.n10.a7","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n10.a7","url":null,"abstract":"We study the variational structure of the complex $k$-Hessian equation on bounded domain $Xsubset mathbb C^{n}$ with boundary $M=partial X$. We prove that the Dirichlet problem $sigma _{k} (partial bar{partial} u) =0$ in $X$, and $u=f$ on $M$ is variational and we give an explicit construction of the associated functional $ mathcal{E}_{k}(u)$. Moreover we prove $ mathcal{E}_{k}(u)$ satisfies the Dirichlet principle. In a special case when $k=2$, our constructed functional $ mathcal{E}_{2}(u)$ involves the Hermitian mean curvature of the boundary, the notion first introduced and studied by X. Wang [37]. Earlier work of J. Case and and the first author of this article [9] introduced a boundary operator for the (real) $k$-Hessian functional which satisfies the Dirichlet principle. The present paper shows that there is a parallel picture in the complex setting.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"74 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-29DOI: 10.4310/cag.2023.v31.n10.a3
Li,Song-Ying
In the paper, we apply Hörmander's weighted $L^{2}$ estimate for $overline{partial }$ to study the Corona problem on the unit ball $B_{n}$ in ${mathbf{C}}^{n}$. We introduce a new holomorphic function space ${mathcal S}(B_{n})$ which is slightly small than $H^{infty}(B_{n})$. We can solve the Corona problems on ${mathcal S}(B_{n})$ instead of $H^{infty}(B_{n})$. We also provide a new proof of $H^{infty }cdot BMOA$ solution for the Corona problem which was first obtained by Varopoulos [41].
{"title":"Weighted $L^{2}$ estimates for $overline{partial }$ and the Corona problem of several complex variables","authors":"Li,Song-Ying","doi":"10.4310/cag.2023.v31.n10.a3","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n10.a3","url":null,"abstract":"In the paper, we apply Hörmander's weighted $L^{2}$ estimate for $overline{partial }$ to study the Corona problem on the unit ball $B_{n}$ in ${mathbf{C}}^{n}$. We introduce a new holomorphic function space ${mathcal S}(B_{n})$ which is slightly small than $H^{infty}(B_{n})$. We can solve the Corona problems on ${mathcal S}(B_{n})$ instead of $H^{infty}(B_{n})$. We also provide a new proof of $H^{infty }cdot BMOA$ solution for the Corona problem which was first obtained by Varopoulos [41].","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"28 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-26DOI: 10.4310/cag.2023.v31.n7.a3
Huang,Xiaojun, Li,Xiaoshan
Let $Omega $ be a Stein space with a compact smooth strongly pseudoconvex boundary. We prove that the boundary is spherical if its Bergman metric over $text{Reg}(Omega )$ is Kähler-Einstein.
{"title":"Bergman-Einstein metric on a Stein space with a strongly pseudoconvex boundary","authors":"Huang,Xiaojun, Li,Xiaoshan","doi":"10.4310/cag.2023.v31.n7.a3","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n7.a3","url":null,"abstract":"Let $Omega $ be a Stein space with a compact smooth strongly pseudoconvex boundary. We prove that the boundary is spherical if its Bergman metric over $text{Reg}(Omega )$ is Kähler-Einstein.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"46 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780471","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-26DOI: 10.4310/cag.2023.v31.n7.a6
Di Matteo,Gianmichele
In [8], Enders, Müller and Topping showed that any blow up sequence of a Type I Ricci flow near a singular point converges to a non-trivial gradient Ricci soliton, leading them to conclude that for such flows all reasonable definitions of singular points agree with each other. We prove the analogous result for the harmonic Ricci flow, generalizing in particular results of Guo, Huang and Phong [11] and Shi [25]. In order to obtain our result, we develop refined compactness theorems, a new pseudolocality theorem, and a notion of reduced length and volume based at the singular time for the harmonic Ricci flow.
{"title":"Analysis of Type I singularities in the harmonic Ricci flow","authors":"Di Matteo,Gianmichele","doi":"10.4310/cag.2023.v31.n7.a6","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n7.a6","url":null,"abstract":"In [8], Enders, Müller and Topping showed that any blow up sequence of a Type I Ricci flow near a singular point converges to a non-trivial gradient Ricci soliton, leading them to conclude that for such flows all reasonable definitions of singular points agree with each other. We prove the analogous result for the harmonic Ricci flow, generalizing in particular results of Guo, Huang and Phong [11] and Shi [25]. In order to obtain our result, we develop refined compactness theorems, a new pseudolocality theorem, and a notion of reduced length and volume based at the singular time for the harmonic Ricci flow.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"40 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-26DOI: 10.4310/cag.2023.v31.n7.a1
Hirsch,Sven, Miao,Pengzi, Tsang,Tin-Yau
We study the mass of asymptotically flat $3$-manifolds with boundary using the method of Bray-Kazaras-Khuri-Stern cite{BKKS}. More precisely, we derive a mass formula on the union of an asymptotically flat manifold and fill-ins of its boundary, and give new sufficient conditions guaranteeing the positivity of the mass. Motivation to such consideration comes from studying the quasi-local mass of the boundary surface. If the boundary isometrically embeds in the Euclidean space, we apply the formula to obtain convergence of the Brown-York mass along large surfaces tending to $infty$ which include the scaling of any fixed coordinate-convex surface.
{"title":"Mass of asymptotically flat 3-manifolds with boundary","authors":"Hirsch,Sven, Miao,Pengzi, Tsang,Tin-Yau","doi":"10.4310/cag.2023.v31.n7.a1","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n7.a1","url":null,"abstract":"We study the mass of asymptotically flat $3$-manifolds with boundary using the method of Bray-Kazaras-Khuri-Stern cite{BKKS}. More precisely, we derive a mass formula on the union of an asymptotically flat manifold and fill-ins of its boundary, and give new sufficient conditions guaranteeing the positivity of the mass. Motivation to such consideration comes from studying the quasi-local mass of the boundary surface. If the boundary isometrically embeds in the Euclidean space, we apply the formula to obtain convergence of the Brown-York mass along large surfaces tending to $infty$ which include the scaling of any fixed coordinate-convex surface.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"54 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780469","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-26DOI: 10.4310/cag.2023.v31.n7.a2
Kunikawa,Keita, Sakurai,Yohei
In this article, we establish a monotonicity formula of Hamilton type entropy along Ricci flow on compact surfaces with boundary. We also study the relation between our entropy functional and the $mathcal{W}$-functional of Perelman type.
{"title":"Hamilton type entropy formula along the Ricci flow on surfaces with boundary","authors":"Kunikawa,Keita, Sakurai,Yohei","doi":"10.4310/cag.2023.v31.n7.a2","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n7.a2","url":null,"abstract":"In this article, we establish a monotonicity formula of Hamilton type entropy along Ricci flow on compact surfaces with boundary. We also study the relation between our entropy functional and the $mathcal{W}$-functional of Perelman type.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"63 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780470","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-26DOI: 10.4310/cag.2023.v31.n7.a7
Feng,Yu, Shi,Yiqian, Song,Jijian, Xu,Bin
We propose a conjecture that the monodromy group of a singular hyperbolic metric on a non-hyperbolic Riemann surface is Zariski dense in $text{PSL}(2,,{mathbb R})$. By using meromorphic differentials and affine connections, we obtain evidence of the conjecture that the monodromy group of the singular hyperbolic metric cannot be contained in four classes of one-dimensional Lie subgroups of $text{PSL}(2,,{mathbb R})$. Moreover, we confirm the conjecture if the Riemann surface is the once punctured Riemann sphere, the twice punctured Riemann sphere, a once punctured torus or a compact Riemann surface.
{"title":"Singular hyperbolic metrics and negative subharmonic functions","authors":"Feng,Yu, Shi,Yiqian, Song,Jijian, Xu,Bin","doi":"10.4310/cag.2023.v31.n7.a7","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n7.a7","url":null,"abstract":"We propose a conjecture that the monodromy group of a singular hyperbolic metric on a non-hyperbolic Riemann surface is Zariski dense in $text{PSL}(2,,{mathbb R})$. By using meromorphic differentials and affine connections, we obtain evidence of the conjecture that the monodromy group of the singular hyperbolic metric cannot be contained in four classes of one-dimensional Lie subgroups of $text{PSL}(2,,{mathbb R})$. Moreover, we confirm the conjecture if the Riemann surface is the once punctured Riemann sphere, the twice punctured Riemann sphere, a once punctured torus or a compact Riemann surface.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"26 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780476","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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