Pub Date : 2024-01-04DOI: 10.4310/cag.2023.v31.n3.a6
Thomas Raujouan
Using the DPW method, we construct genus zero Alexandrov-embedded constant mean curvature (greater than one) surfaces with any number of Delaunay ends in the hyperbolic space.
{"title":"Constant mean curvature $n$-noids in hyperbolic space","authors":"Thomas Raujouan","doi":"10.4310/cag.2023.v31.n3.a6","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n3.a6","url":null,"abstract":"Using the DPW method, we construct genus zero Alexandrov-embedded constant mean curvature (greater than one) surfaces with any number of Delaunay ends in the hyperbolic space.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"17 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139102751","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 : 2024-01-04DOI: 10.4310/cag.2023.v31.n3.a4
Matthias Wink
This paper studies cohomogeneity one Ricci solitons. If the isotropy representation of the principal orbit $G/K$ consists of two inequivalent $operatorname{Ad}_K$-invariant irreducible summands, the existence of continuous families of non-homothetic complete steady and expanding Ricci solitons on non-trivial bundles is shown. These examples were detected numerically by Buzano–Dancer–Gallaugher–Wang. The analysis of the corresponding Ricci flat trajectories is used to reconstruct Einstein metrics of positive scalar curvature due to Böhm. The techniques also apply to $m$-quasi-Einstein metrics.
{"title":"Cohomogeneity one Ricci solitons from Hopf fibrations","authors":"Matthias Wink","doi":"10.4310/cag.2023.v31.n3.a4","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n3.a4","url":null,"abstract":"This paper studies cohomogeneity one Ricci solitons. If the isotropy representation of the principal orbit $G/K$ consists of two inequivalent $operatorname{Ad}_K$-invariant irreducible summands, the existence of continuous families of non-homothetic complete steady and expanding Ricci solitons on non-trivial bundles is shown. These examples were detected numerically by Buzano–Dancer–Gallaugher–Wang. The analysis of the corresponding Ricci flat trajectories is used to reconstruct Einstein metrics of positive scalar curvature due to Böhm. The techniques also apply to $m$-quasi-Einstein metrics.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"17 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139102613","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 : 2024-01-04DOI: 10.4310/cag.2023.v31.n3.a5
Shubham Dwivedi, Ragini Singhal
$defG{mathrm{G}_2}$We study the deformation theory of nearly $G$ manifolds. These are seven-dimensional manifolds admitting real Killing spinors. We show that the infinitesimal deformations of nearly $G$ structures are obstructed in general. Explicitly, we prove that the infinitesimal deformations of the homogeneous nearly $G$ structure on the Aloff–Wallach space are all obstructed to second order. We also completely describe the cohomology of nearly $G$ manifolds.
{"title":"Deformation theory of nearly $G_2$ manifolds","authors":"Shubham Dwivedi, Ragini Singhal","doi":"10.4310/cag.2023.v31.n3.a5","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n3.a5","url":null,"abstract":"$defG{mathrm{G}_2}$We study the deformation theory of nearly $G$ manifolds. These are seven-dimensional manifolds admitting real Killing spinors. We show that the infinitesimal deformations of nearly $G$ structures are obstructed in general. Explicitly, we prove that the infinitesimal deformations of the homogeneous nearly $G$ structure on the Aloff–Wallach space are all obstructed to second order. We also completely describe the cohomology of nearly $G$ manifolds.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"23 11 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139102706","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 : 2024-01-04DOI: 10.4310/cag.2023.v31.n3.a2
Sierra Knavel, Rolland Trapp
This paper studies embedded totally geodesic surfaces in fully augmented link complements. Not surprisingly, there are no closed embedded totally geodesic surfaces. Non-compact surfaces disjoint from crossing disks are seen to be punctured spheres orthogonal to the standard cell decomposition, while those that intersect crossing disks do so in very restricted ways. Finally we show there is an augmentation of any checkerboard surface in which that surface becomes totally geodesic.
{"title":"Embedded totally geodesic surfaces in fully augmented links","authors":"Sierra Knavel, Rolland Trapp","doi":"10.4310/cag.2023.v31.n3.a2","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n3.a2","url":null,"abstract":"This paper studies embedded totally geodesic surfaces in fully augmented link complements. Not surprisingly, there are no closed embedded totally geodesic surfaces. Non-compact surfaces disjoint from crossing disks are seen to be punctured spheres orthogonal to the standard cell decomposition, while those that intersect crossing disks do so in very restricted ways. Finally we show there is an augmentation of any checkerboard surface in which that surface becomes totally geodesic.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"23 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139102616","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 : 2024-01-04DOI: 10.4310/cag.2023.v31.n3.a1
Theodora Bourni, Mat Langford
We present a simple connection between differential Harnack inequalities for hypersurface flows and natural concavity properties of their time-of-arrival functions. We prove these concavity properties directly for a large class of flows by applying a concavity maximum principle argument to the corresponding level set flow equations. In particular, this yields a short proof of Hamilton’s differential Harnack inequality for mean curvature flow and, more generally, Andrews’ differential Harnack inequalities for certain “$alpha$- inverse-concave” flows.
{"title":"Differential Harnack inequalities via Concavity of the arrival time","authors":"Theodora Bourni, Mat Langford","doi":"10.4310/cag.2023.v31.n3.a1","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n3.a1","url":null,"abstract":"We present a simple connection between differential Harnack inequalities for hypersurface flows and natural concavity properties of their time-of-arrival functions. We prove these concavity properties directly for a large class of flows by applying a concavity maximum principle argument to the corresponding level set flow equations. In particular, this yields a short proof of Hamilton’s differential Harnack inequality for mean curvature flow and, more generally, Andrews’ differential Harnack inequalities for certain “$alpha$- inverse-concave” flows.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"21 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139102754","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 : 2023-12-06DOI: 10.4310/cag.2023.v31.n2.a5
Livio Liechti
We construct divide knots with arbitrary smooth four-genus but topological four-genus equal to one. In particular, for strongly quasipositive fibred knots, the ratio between the topological and the smooth four-genus can be arbitrarily close to zero.
{"title":"Divide knots of maximal genus defect","authors":"Livio Liechti","doi":"10.4310/cag.2023.v31.n2.a5","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n2.a5","url":null,"abstract":"We construct divide knots with arbitrary smooth four-genus but topological four-genus equal to one. In particular, for strongly quasipositive fibred knots, the ratio between the topological and the smooth four-genus can be arbitrarily close to zero.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"19 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138578983","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 : 2023-12-06DOI: 10.4310/cag.2023.v31.n2.a4
Wen Han, Bobo Hua
We study the eigenvalues of the Dirichlet-to-Neumann operator on a finite subgraph of the integer lattice Zn. We estimate the first n+1 eigenvalues using the number of vertices of the subgraph. As a corollary, we prove that the first non-trivial eigenvalue of the Dirichlet-to-Neumann operator tends to zero as the number of vertices of the subgraph tends to infinity.
{"title":"Steklov eigenvalue problem on subgraphs of integer lattices","authors":"Wen Han, Bobo Hua","doi":"10.4310/cag.2023.v31.n2.a4","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n2.a4","url":null,"abstract":"We study the eigenvalues of the Dirichlet-to-Neumann operator on a finite subgraph of the integer lattice Zn. We estimate the first n+1 eigenvalues using the number of vertices of the subgraph. As a corollary, we prove that the first non-trivial eigenvalue of the Dirichlet-to-Neumann operator tends to zero as the number of vertices of the subgraph tends to infinity.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"71 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138579092","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 : 2023-12-06DOI: 10.4310/cag.2023.v31.n2.a7
Marco Pozzetta
We consider minimization problems of functionals given by the difference between the Willmore functional of a closed surface and its area, when the latter is multiplied by a positive constant weight $Lambda$ and when the surfaces are confined in the closure of a bounded open set $Omega subset mathbb{R}^3$. We explicitly solve the minimization problem in the case $Omega = B_1$. We give a description of the value of the infima and of the convergence of minimizing sequences to integer rectifiable varifolds, depending on the parameter $Lambda$. We also analyze some properties of these functionals and we provide some examples. Finally we prove the existence of a $C^{1,alpha} cap W^{2,2}$ embedded surface that is also $C^infty$ inside $Omega$ and such that it achieves the infimum of the problem when the weight $Lambda$ is sufficiently small.
{"title":"Confined Willmore energy and the area functional","authors":"Marco Pozzetta","doi":"10.4310/cag.2023.v31.n2.a7","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n2.a7","url":null,"abstract":"We consider minimization problems of functionals given by the difference between the Willmore functional of a closed surface and its area, when the latter is multiplied by a positive constant weight $Lambda$ and when the surfaces are confined in the closure of a bounded open set $Omega subset mathbb{R}^3$. We explicitly solve the minimization problem in the case $Omega = B_1$. We give a description of the value of the infima and of the convergence of minimizing sequences to integer rectifiable varifolds, depending on the parameter $Lambda$. We also analyze some properties of these functionals and we provide some examples. Finally we prove the existence of a $C^{1,alpha} cap W^{2,2}$ embedded surface that is also $C^infty$ inside $Omega$ and such that it achieves the infimum of the problem when the weight $Lambda$ is sufficiently small.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"8 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138581766","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 : 2023-12-06DOI: 10.4310/cag.2023.v31.n2.a2
S. Montaldo, A. Pámpano
Biconservative surfaces of Riemannian $3$-space forms $N^3(rho)$, are either constant mean curvature (CMC) surfaces or rotational linear Weingarten surfaces verifying the relation $3 kappa_1 + kappa_2 = 0$ between their principal curvatures $kappa_1$ and $kappa_2$. We characterise the profile curves of the non-CMC biconservative surfaces as the critical curves for a suitable curvature energy. Moreover, using this characterisation, we prove the existence of a discrete biparametric family of closed, i.e. compact without boundary, non-CMC biconservative surfaces in the round $3$-sphere, $mathbb{S}^3(rho)$. However, none of these closed surfaces is embedded in $mathbb{S}^ (rho)$.
{"title":"On the existence of closed biconservative surfaces in space forms","authors":"S. Montaldo, A. Pámpano","doi":"10.4310/cag.2023.v31.n2.a2","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n2.a2","url":null,"abstract":"Biconservative surfaces of Riemannian $3$-space forms $N^3(rho)$, are either constant mean curvature (CMC) surfaces or rotational linear Weingarten surfaces verifying the relation $3 kappa_1 + kappa_2 = 0$ between their principal curvatures $kappa_1$ and $kappa_2$. We characterise the profile curves of the non-CMC biconservative surfaces as the critical curves for a suitable curvature energy. Moreover, using this characterisation, we prove the existence of a discrete biparametric family of closed, i.e. compact without boundary, non-CMC biconservative surfaces in the round $3$-sphere, $mathbb{S}^3(rho)$. However, none of these closed surfaces is embedded in $mathbb{S}^ (rho)$.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"26 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138579407","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 : 2023-12-06DOI: 10.4310/cag.2023.v31.n2.a9
Indranil Biswas, Luis Ángel Calvo, Oscar García-Prada
We introduce real structures on $L$-twisted Higgs pairs over a compact connected Riemann surface $X$ equipped with an antiholomorphic involution, where $L$ is a holomorphic line bundle on $X$ with a real structure, and prove a Hitchin–Kobayashi correspondence for the $L$-twisted Higgs pairs. Real $G^mathbb{R}$-Higgs bundles, where $G^mathbb{R}$ is a real form of a connected semisimple complex affine algebraic group $G$, constitute a particular class of examples of these pairs. In this case, the real structure of the moduli space of $G$-Higgs pairs is defined using a conjugation of $G$ that commutes with the one defining the real form $G^mathbb{R}$ and a compact conjugation of $G$ preserving $G^mathbb{R}$. We establish a homeomorphism between the moduli space of real $G^mathbb{R}$-Higgs bundles and the moduli space of representations of the fundamental group of $X$ in $G^mathbb{R}$ that can be extended to a representation of the orbifold fundamental group of $X$ into a certain enlargement of $G^mathbb{R}$ with quotient $mathbb{Z}/2 mathbb{Z}$. Finally, we show how real $G^mathbb{R}$-Higgs bundles appear naturally as fixed points of certain anti-holomorphic involutions of the moduli space of $G^mathbb{R}$-Higgs bundles, constructed using the real structures on $G$ and $X$. A similar result is proved for the representations of the orbifold fundamental group.
{"title":"Real Higgs pairs and non-abelian Hodge correspondence on a Klein surface","authors":"Indranil Biswas, Luis Ángel Calvo, Oscar García-Prada","doi":"10.4310/cag.2023.v31.n2.a9","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n2.a9","url":null,"abstract":"We introduce real structures on $L$-twisted Higgs pairs over a compact connected Riemann surface $X$ equipped with an antiholomorphic involution, where $L$ is a holomorphic line bundle on $X$ with a real structure, and prove a Hitchin–Kobayashi correspondence for the $L$-twisted Higgs pairs. Real $G^mathbb{R}$-Higgs bundles, where $G^mathbb{R}$ is a real form of a connected semisimple complex affine algebraic group $G$, constitute a particular class of examples of these pairs. In this case, the real structure of the moduli space of $G$-Higgs pairs is defined using a conjugation of $G$ that commutes with the one defining the real form $G^mathbb{R}$ and a compact conjugation of $G$ preserving $G^mathbb{R}$. We establish a homeomorphism between the moduli space of real $G^mathbb{R}$-Higgs bundles and the moduli space of representations of the fundamental group of $X$ in $G^mathbb{R}$ that can be extended to a representation of the orbifold fundamental group of $X$ into a certain enlargement of $G^mathbb{R}$ with quotient $mathbb{Z}/2 mathbb{Z}$. Finally, we show how real $G^mathbb{R}$-Higgs bundles appear naturally as fixed points of certain anti-holomorphic involutions of the moduli space of $G^mathbb{R}$-Higgs bundles, constructed using the real structures on $G$ and $X$. A similar result is proved for the representations of the orbifold fundamental group.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"12 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138578973","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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