Pub Date : 2020-09-01DOI: 10.1016/J.DIFGEO.2021.101724
D. Alekseevsky, M. Ganji, G. Schmalz, A. Spiro
{"title":"Lorentzian manifolds with shearfree congruences and Kähler-Sasaki geometry","authors":"D. Alekseevsky, M. Ganji, G. Schmalz, A. Spiro","doi":"10.1016/J.DIFGEO.2021.101724","DOIUrl":"https://doi.org/10.1016/J.DIFGEO.2021.101724","url":null,"abstract":"","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"79 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89017219","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct a family of Hermitian metrics on the Hopf surface $ mathbb{S}^3times mathbb{S}^1$, whose fundamental classes represent distinct cohomology classes in the Aeppli cohomology group. These metrics are locally conformally Kahler. Among the toric fibres of $pi:mathbb{S}^{3} times mathbb{S}^1tomathbb{C} P^1$ two of them are stable minimal surfaces and each of the two has a neighbourhood so that fibres therein are given by stable harmonic maps from 2-torus and outside, far away from the two tori, there are unstable harmonic ones that are also unstable minimal surfaces.
{"title":"On stability of the fibres of Hopf surfaces as harmonic maps and minimal surfaces","authors":"Jingyi Chen, Liding Huang","doi":"10.1090/tran/8520","DOIUrl":"https://doi.org/10.1090/tran/8520","url":null,"abstract":"We construct a family of Hermitian metrics on the Hopf surface $ mathbb{S}^3times mathbb{S}^1$, whose fundamental classes represent distinct cohomology classes in the Aeppli cohomology group. These metrics are locally conformally Kahler. Among the toric fibres of $pi:mathbb{S}^{3} times mathbb{S}^1tomathbb{C} P^1$ two of them are stable minimal surfaces and each of the two has a neighbourhood so that fibres therein are given by stable harmonic maps from 2-torus and outside, far away from the two tori, there are unstable harmonic ones that are also unstable minimal surfaces.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75577930","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a smooth free action of a compact connected Lie group $G$ on a smooth manifold $M$, we show that the space of $G$-invariant Riemannian metrics on $M$ whose automorphism group is precisely $G$ is open dense in the space of all $G$-invariant metrics, provided the dimension of $M$ is "sufficiently large" compared to that of $G$. As a consequence, it follows that every compact connected Lie group can be realized as the automorphism group of some compact connected Riemannian manifold. Along the way we also show, under less restrictive conditions on both dimensions and actions, that the space of $G$-invariant metrics whose automorphism groups preserve the $G$-orbits is dense $G_{delta}$ in the space of all $G$-invariant metrics.
{"title":"Prescribed Riemannian Symmetries","authors":"A. Chirvasitu","doi":"10.3842/SIGMA.2021.030","DOIUrl":"https://doi.org/10.3842/SIGMA.2021.030","url":null,"abstract":"Given a smooth free action of a compact connected Lie group $G$ on a smooth manifold $M$, we show that the space of $G$-invariant Riemannian metrics on $M$ whose automorphism group is precisely $G$ is open dense in the space of all $G$-invariant metrics, provided the dimension of $M$ is \"sufficiently large\" compared to that of $G$. As a consequence, it follows that every compact connected Lie group can be realized as the automorphism group of some compact connected Riemannian manifold. \u0000Along the way we also show, under less restrictive conditions on both dimensions and actions, that the space of $G$-invariant metrics whose automorphism groups preserve the $G$-orbits is dense $G_{delta}$ in the space of all $G$-invariant metrics.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76727052","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give explicit harmonic representatives of Dolbeault cohomology of Oeljeklaus-Toma manifolds and show that they are geometrically Dolbeault formal. We also give explicit harmonic representatives of Bott-Chern cohomology of Oeljeklaus-Toma manifolds of type $(s,1)$ and study the Angella-Tomassini inequality.
{"title":"Remarks on Dolbeault cohomology of Oeljeklaus-Toma manifolds and Hodge theory","authors":"H. Kasuya","doi":"10.1090/proc/15436","DOIUrl":"https://doi.org/10.1090/proc/15436","url":null,"abstract":"We give explicit harmonic representatives of Dolbeault cohomology of Oeljeklaus-Toma manifolds and show that they are geometrically Dolbeault formal. We also give explicit harmonic representatives of Bott-Chern cohomology of Oeljeklaus-Toma manifolds of type $(s,1)$ and study the Angella-Tomassini inequality.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90096985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-08-14DOI: 10.1142/S1793525321500230
Baris Coskunuzer
We study the asymptotic Plateau problem in $mathbb{H}_2times mathbb{R}$. We give the first examples of non-fillable finite curves with no thin tail in the asymptotic cylinder. Furthermore, we study the fillability question for infinite curves in the asymptotic boundary.
{"title":"Minimal Surfaces in $mathbb{H}_2times mathbb{R}$: Nonfillable Curves","authors":"Baris Coskunuzer","doi":"10.1142/S1793525321500230","DOIUrl":"https://doi.org/10.1142/S1793525321500230","url":null,"abstract":"We study the asymptotic Plateau problem in $mathbb{H}_2times mathbb{R}$. We give the first examples of non-fillable finite curves with no thin tail in the asymptotic cylinder. Furthermore, we study the fillability question for infinite curves in the asymptotic boundary.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88655911","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-08-13DOI: 10.31801/CFSUASMAS.785489
Mahmut Mak
In this study, we introduce the natural mate and conjugate mate of a Frenet curve in a three dimensional Lie group $ mathbb{G} $ with bi-invariant metric. Also, we give some relationships between a Frenet curve and its natural mate or its conjugate mate in $ mathbb{G} $. Especially, we obtain some results for the natural mate and the conjugate mate of a Frenet curve in $ mathbb{G} $ when the Frenet curve is a general helix, a slant helix, a spherical curve, a rectifying curve, a Salkowski (constant curvature and non-constant torsion), anti-Salkowski (non-constant curvature and constant torsion), Bertrand curve. Finally, we give nice graphics with numeric solution in Euclidean 3-space as a commutative Lie group.
{"title":"Natural and conjugate mates of Frenet curves in three-dimensional Lie group","authors":"Mahmut Mak","doi":"10.31801/CFSUASMAS.785489","DOIUrl":"https://doi.org/10.31801/CFSUASMAS.785489","url":null,"abstract":"In this study, we introduce the natural mate and conjugate mate of a Frenet curve in a three dimensional Lie group $ mathbb{G} $ with bi-invariant metric. Also, we give some relationships between a Frenet curve and its natural mate or its conjugate mate in $ mathbb{G} $. Especially, we obtain some results for the natural mate and the conjugate mate of a Frenet curve in $ mathbb{G} $ when the Frenet curve is a general helix, a slant helix, a spherical curve, a rectifying curve, a Salkowski (constant curvature and non-constant torsion), anti-Salkowski (non-constant curvature and constant torsion), Bertrand curve. Finally, we give nice graphics with numeric solution in Euclidean 3-space as a commutative Lie group.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"52 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81333797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-07-31DOI: 10.1515/9783110700763-011
A. Grigor’yan, Satoshi Ishiwata, L. Saloff‐Coste
In this survey article, we discuss some recent progress on geometric analysis on manifold with ends. In the final section, we construct manifolds with ends with oscillating volume functions which may turn out to have a different heat kernel estimates from those provided by known results.
{"title":"Geometric analysis on manifolds with ends","authors":"A. Grigor’yan, Satoshi Ishiwata, L. Saloff‐Coste","doi":"10.1515/9783110700763-011","DOIUrl":"https://doi.org/10.1515/9783110700763-011","url":null,"abstract":"In this survey article, we discuss some recent progress on geometric analysis on manifold with ends. In the final section, we construct manifolds with ends with oscillating volume functions which may turn out to have a different heat kernel estimates from those provided by known results.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80714569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We survey some recent work using Ricci flow to create a class of local definitions of weak lower scalar curvature bounds that is well defined for $C^0$ metrics. We discuss several properties of these definitions and explain some applications of this approach to questions regarding uniform convergence of metrics with scalar curvature bounded below. Finally, we consider the relationship between this approach and some other generalized notions of lower scalar curvature bounds.
{"title":"Defining Pointwise Lower Scalar Curvature Bounds for C0 Metrics with Regularization by Ricci Flow","authors":"Paula Burkhardt-Guim","doi":"10.3842/sigma.2020.128","DOIUrl":"https://doi.org/10.3842/sigma.2020.128","url":null,"abstract":"We survey some recent work using Ricci flow to create a class of local definitions of weak lower scalar curvature bounds that is well defined for $C^0$ metrics. We discuss several properties of these definitions and explain some applications of this approach to questions regarding uniform convergence of metrics with scalar curvature bounded below. Finally, we consider the relationship between this approach and some other generalized notions of lower scalar curvature bounds.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"71 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87706546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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