Pub Date : 2024-07-24DOI: 10.4310/cag.2023.v31.n6.a4
Alves,Claudianor O.
In this work we use variational methods to prove the existence of multiple solutions for the following class of problem $$- epsilon Delta_1 u + V(x)frac{u}{|u|} = f(u) quad mbox{in} quad mathbb{R}^N, quad u in BV(mathbb{R}^N), $$ where $Delta_1$ is the $1-$Laplacian operator and $epsilon$ is a positive parameter. It is proved that the numbers of solutions is at least the numbers of global minimum points of $V$ when $epsilon$ is small enough.
在这项工作中，我们使用变分法证明了以下一类问题的多解存在性 $$- epsilon Delta_1 u + V(x)frac{u}{|u|} = f(u) quad mbox{in}quad u in BV(mathbb{R}^N).quad mathbb{R}^N, quad u in BV(mathbb{R}^N), $$ 其中 $Delta_1$ 是 1-$ 拉普拉斯算子，$epsilon$ 是一个正参数。研究证明，当 $epsilon$ 足够小时，解的数目至少是 $V$ 全局最小点的数目。
{"title":"On existence of multiple solutions to a class of problems involving the 1-Laplace operator in whole $mathbb{R}^N$","authors":"Alves,Claudianor O.","doi":"10.4310/cag.2023.v31.n6.a4","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n6.a4","url":null,"abstract":"In this work we use variational methods to prove the existence of multiple solutions for the following class of problem $$- epsilon Delta_1 u + V(x)frac{u}{|u|} = f(u) quad mbox{in} quad mathbb{R}^N, quad u in BV(mathbb{R}^N), $$ where $Delta_1$ is the $1-$Laplacian operator and $epsilon$ is a positive parameter. It is proved that the numbers of solutions is at least the numbers of global minimum points of $V$ when $epsilon$ is small enough.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780253","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 : 2024-07-16DOI: 10.4310/cag.2023.v31.n5.a7
Ozawa,Ryunosuke, Sakurai,Yohei, Yamada,Taiki
In a previous work, the authors [15] have introduced a Lin-Lu-Yau type Ricci curvature for directed graphs, and obtained a diameter comparison of Bonnet-Myers type. In this paper, we investigate rigidity properties for the equality case, and conclude a maximal diameter theorem of Cheng type.
{"title":"Maximal diameter theorem for directed graphs of positive Ricci curvature","authors":"Ozawa,Ryunosuke, Sakurai,Yohei, Yamada,Taiki","doi":"10.4310/cag.2023.v31.n5.a7","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n5.a7","url":null,"abstract":"In a previous work, the authors [15] have introduced a Lin-Lu-Yau type Ricci curvature for directed graphs, and obtained a diameter comparison of Bonnet-Myers type. In this paper, we investigate rigidity properties for the equality case, and conclude a maximal diameter theorem of Cheng type.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721697","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 : 2024-07-16DOI: 10.4310/cag.2023.v31.n5.a6
Bray,Hubert, Stern,Daniel
For a homotopically energy-minimizing map $u: N^3to S^1$ on a compact, oriented $3$-manifold $N$ with boundary, we establish an identity relating the average Euler characteristic of the level sets $u^{-1}{theta}$ to the scalar curvature of $N$ and the mean curvature of the boundary $partial N$. As an application, we obtain some natural geometric estimates for the Thurston norm on $3$-manifolds with boundary, generalizing results of Kronheimer-Mrowka and the second named author from the closed setting. By combining these techniques with results from minimal surface theory, we obtain moreover a characterization of the Thurston norm via scalar curvature and the harmonic norm for general closed, oriented three-manifolds, extending Kronheimer and Mrowka's characterization for irreducible manifolds to arbitrary topologies.
{"title":"Scalar curvature and harmonic one-forms on three-manifolds with boundary","authors":"Bray,Hubert, Stern,Daniel","doi":"10.4310/cag.2023.v31.n5.a6","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n5.a6","url":null,"abstract":"For a homotopically energy-minimizing map $u: N^3to S^1$ on a compact, oriented $3$-manifold $N$ with boundary, we establish an identity relating the average Euler characteristic of the level sets $u^{-1}{theta}$ to the scalar curvature of $N$ and the mean curvature of the boundary $partial N$. As an application, we obtain some natural geometric estimates for the Thurston norm on $3$-manifolds with boundary, generalizing results of Kronheimer-Mrowka and the second named author from the closed setting. By combining these techniques with results from minimal surface theory, we obtain moreover a characterization of the Thurston norm via scalar curvature and the harmonic norm for general closed, oriented three-manifolds, extending Kronheimer and Mrowka's characterization for irreducible manifolds to arbitrary topologies.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141722507","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 : 2024-07-16DOI: 10.4310/cag.2023.v31.n5.a4
Guang,Qiang, Wang,Zhichao, Zhou,Xin
In this paper, we prove the existence of the free boundary minimal hypersurface of least area in compact manifolds with boundary. Such a hypersurface can be viewed as the ground state of the volume spectrum introduced by Gromov. Moreover, we characterize the orientation and Morse index of them.
{"title":"Free boundary minimal hypersurfaces with least area","authors":"Guang,Qiang, Wang,Zhichao, Zhou,Xin","doi":"10.4310/cag.2023.v31.n5.a4","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n5.a4","url":null,"abstract":"In this paper, we prove the existence of the free boundary minimal hypersurface of least area in compact manifolds with boundary. Such a hypersurface can be viewed as the ground state of the volume spectrum introduced by Gromov. Moreover, we characterize the orientation and Morse index of them.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141722506","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 : 2024-07-16DOI: 10.4310/cag.2023.v31.n5.a3
Ball,Gavin, Madnick,Jesse
We study associative submanifolds of the Berger space $mathrm{SO}(5)/mathrm{SO}(3)$ endowed with its homogeneous nearly-parallel $mathrm{G}_2$-structure. We focus on two geometrically interesting classes: the ruled associatives, and the associatives with special Gauss map. We show that the associative submanifolds ruled by a certain special type of geodesic are in correspondence with pseudo-holomorphic curves in $mathrm{Gr}^+_2 !left( T S^4 right)$. Using this correspondence, together with a theorem of Bryant on superminimal surfaces in $S^4,$ we prove the existence of infinitely many topological types of compact immersed associative 3-folds in $mathrm{SO}(5)/mathrm{SO}(3)$. An associative submanifold of the Berger space is said to have special Gauss map if its tangent spaces have non-trivial $mathrm{SO}(3)$-stabiliser. We classify the associative submanifolds with special Gauss map in the cases where the stabiliser contains an element of order greater than 2. In particular, we find several homogeneous examples of this type.
{"title":"Associative submanifolds of the Berger space","authors":"Ball,Gavin, Madnick,Jesse","doi":"10.4310/cag.2023.v31.n5.a3","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n5.a3","url":null,"abstract":"We study associative submanifolds of the Berger space $mathrm{SO}(5)/mathrm{SO}(3)$ endowed with its homogeneous nearly-parallel $mathrm{G}_2$-structure. We focus on two geometrically interesting classes: the ruled associatives, and the associatives with special Gauss map. We show that the associative submanifolds ruled by a certain special type of geodesic are in correspondence with pseudo-holomorphic curves in $mathrm{Gr}^+_2 !left( T S^4 right)$. Using this correspondence, together with a theorem of Bryant on superminimal surfaces in $S^4,$ we prove the existence of infinitely many topological types of compact immersed associative 3-folds in $mathrm{SO}(5)/mathrm{SO}(3)$. An associative submanifold of the Berger space is said to have special Gauss map if its tangent spaces have non-trivial $mathrm{SO}(3)$-stabiliser. We classify the associative submanifolds with special Gauss map in the cases where the stabiliser contains an element of order greater than 2. In particular, we find several homogeneous examples of this type.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141722503","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 : 2024-07-16DOI: 10.4310/cag.2023.v31.n5.a1
Maximo,Davi
For any closed Riemannian three-manifold, we prove that for any sequence of closed embedded minimal surfaces with uniformly bounded index, the genus can only grow at most linearly with respect to the area.
{"title":"A note on minimal surfaces with bounded index","authors":"Maximo,Davi","doi":"10.4310/cag.2023.v31.n5.a1","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n5.a1","url":null,"abstract":"For any closed Riemannian three-manifold, we prove that for any sequence of closed embedded minimal surfaces with uniformly bounded index, the genus can only grow at most linearly with respect to the area.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721693","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 : 2024-07-16DOI: 10.4310/cag.2023.v31.n5.a5
Fowdar,Udhav
We investigate the $mathbb{T}^2$-quotient of a torsion free $Spin(7)$-structure on an $8$-manifold under the assumption that the quotient $6$-manifold is Kähler. We show that there exists either a Hamiltonian $S^1$ or $mathbb{T}^2$ action on the quotient preserving the complex structure. Performing a Kähler reduction in each case reduces the problem of finding $Spin(7)$ metrics to studying a system of PDEs on either a $4$- or $2$-manifold with trivial canonical bundle, which in the compact case corresponds to either $mathbb{T}^4$, a $K3$ surface or an elliptic curve. By reversing this construction we give infinitely many new explicit examples of $Spin(7)$ holonomy metrics. In the simplest case, our result can be viewed as an extension of the Gibbons-Hawking ansatz.
{"title":"Spin(7) metrics from Kähler geometry","authors":"Fowdar,Udhav","doi":"10.4310/cag.2023.v31.n5.a5","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n5.a5","url":null,"abstract":"We investigate the $mathbb{T}^2$-quotient of a torsion free $Spin(7)$-structure on an $8$-manifold under the assumption that the quotient $6$-manifold is Kähler. We show that there exists either a Hamiltonian $S^1$ or $mathbb{T}^2$ action on the quotient preserving the complex structure. Performing a Kähler reduction in each case reduces the problem of finding $Spin(7)$ metrics to studying a system of PDEs on either a $4$- or $2$-manifold with trivial canonical bundle, which in the compact case corresponds to either $mathbb{T}^4$, a $K3$ surface or an elliptic curve. By reversing this construction we give infinitely many new explicit examples of $Spin(7)$ holonomy metrics. In the simplest case, our result can be viewed as an extension of the Gibbons-Hawking ansatz.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721698","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 : 2024-07-16DOI: 10.4310/cag.2023.v31.n5.a8
Delay,Erwann
We adapt the Bartnik method to provide a Hilbert manifold structure for the space of solutions, without KID's, to the vacuum constraint equations on compact manifold of any dimension $geq 3$. In the course, we prove that some fibers of the scalar curvature or the constraint operator are Hilbert submanifolds. We also study some operators and inequalities related to the KID's operator. Finally we comment the adaptation to some non-compact manifolds.
{"title":"Bartnik Hilbert manifold structure on fibers of the scalar curvature and the constraint operator","authors":"Delay,Erwann","doi":"10.4310/cag.2023.v31.n5.a8","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n5.a8","url":null,"abstract":"We adapt the Bartnik method to provide a Hilbert manifold structure for the space of solutions, without KID's, to the vacuum constraint equations on compact manifold of any dimension $geq 3$. In the course, we prove that some fibers of the scalar curvature or the constraint operator are Hilbert submanifolds. We also study some operators and inequalities related to the KID's operator. Finally we comment the adaptation to some non-compact manifolds.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721699","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 : 2024-07-16DOI: 10.4310/cag.2023.v31.n5.a2
Berndt,Jürgen, Sanmartı́n-López,Vı́ctor
We study submanifolds whose principal curvatures, counted with multiplicities, do not depend on the normal direction. Such submanifolds, which we briefly call CPC submanifolds, are always austere, hence minimal, and have constant principal curvatures. Well-known classes of examples include totally geodesic submanifolds, homogeneous austere hypersurfaces, and singular orbits of cohomogeneity one actions. The main purpose of this article is to present a systematic approach to the construction and classification of homogeneous submanifolds whose principal curvatures are independent of the normal direction in irreducible Riemannian symmetric spaces of non-compact type and rank $geq 2$. In particular, we provide a large number of new examples of non-totally geodesic CPC submanifolds not coming from cohomogeneity one actions (note that only one example was known previously, namely a particular 11-dimensional submanifold of the Cayley hyperbolic plane).
{"title":"Submanifolds with constant principal curvatures in symmetric spaces","authors":"Berndt,Jürgen, Sanmartı́n-López,Vı́ctor","doi":"10.4310/cag.2023.v31.n5.a2","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n5.a2","url":null,"abstract":"We study submanifolds whose principal curvatures, counted with multiplicities, do not depend on the normal direction. Such submanifolds, which we briefly call CPC submanifolds, are always austere, hence minimal, and have constant principal curvatures. Well-known classes of examples include totally geodesic submanifolds, homogeneous austere hypersurfaces, and singular orbits of cohomogeneity one actions. The main purpose of this article is to present a systematic approach to the construction and classification of homogeneous submanifolds whose principal curvatures are independent of the normal direction in irreducible Riemannian symmetric spaces of non-compact type and rank $geq 2$. In particular, we provide a large number of new examples of non-totally geodesic CPC submanifolds not coming from cohomogeneity one actions (note that only one example was known previously, namely a particular 11-dimensional submanifold of the Cayley hyperbolic plane).","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721694","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.a3
Julius Baldauf, Ao Sun
We prove that a closed immersed plane curve with total curvature $2 pi m$ has entropy at least $m$ times the entropy of the embedded circle, as long as it generates a type I singularity under the curve shortening flow (CSF). We construct closed immersed plane curves of total curvature $2 pi m$ whose entropy is less than $m$ times the entropy of the embedded circle. As an application, we extend Colding–Minicozzi’s notion of a generic mean curvature flow to closed immersed plane curves by constructing a piecewise CSF whose only singularities are embedded circles and type II singularities.
我们证明总曲率为 2 pi m$ 的封闭沉浸平面曲线的熵至少是内嵌圆的熵的 $m$ 倍，只要它在曲线缩短流（CSF）下产生 I 型奇点。我们构造了总曲率为 $2 pi m$ 的封闭沉浸平面曲线，其熵小于嵌入圆熵的 $m$ 倍。作为应用，我们通过构造片断 CSF，将 Colding-Minicozzi 的一般平均曲率流概念扩展到闭合沉浸平面曲线，其唯一奇点是嵌入圆和第二类奇点。
{"title":"Sharp entropy bounds for plane curves and dynamics of the curve shortening flow","authors":"Julius Baldauf, Ao Sun","doi":"10.4310/cag.2023.v31.n3.a3","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n3.a3","url":null,"abstract":"We prove that a closed immersed plane curve with total curvature $2 pi m$ has entropy at least $m$ times the entropy of the embedded circle, as long as it generates a type I singularity under the curve shortening flow (CSF). We construct closed immersed plane curves of total curvature $2 pi m$ whose entropy is less than $m$ times the entropy of the embedded circle. As an application, we extend Colding–Minicozzi’s notion of a generic mean curvature flow to closed immersed plane curves by constructing a piecewise CSF whose only singularities are embedded circles and type II singularities.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139102750","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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