Pub Date : 2022-04-06DOI: 10.2140/pjm.2022.316.249
Hussein Cheikh Ali
We consider the second best constant in the Hardy-Sobolev inequality on a Riemannian manifold. More precisely, we are interested with the existence of extremal functions for this inequality. This problem was tackled by Djadli-Druet [5] for Sobolev inequalities. Here, we establish the corresponding result for the singular case. In addition, we perform a blow-up analysis of solutions Hardy-Sobolev equations of minimizing type. This yields informations on the value of the second best constant in the related Riemannian functional inequality.
{"title":"The second best constant for the Hardy–Sobolev\u0000inequality on manifolds","authors":"Hussein Cheikh Ali","doi":"10.2140/pjm.2022.316.249","DOIUrl":"https://doi.org/10.2140/pjm.2022.316.249","url":null,"abstract":"We consider the second best constant in the Hardy-Sobolev inequality on a Riemannian manifold. More precisely, we are interested with the existence of extremal functions for this inequality. This problem was tackled by Djadli-Druet [5] for Sobolev inequalities. Here, we establish the corresponding result for the singular case. In addition, we perform a blow-up analysis of solutions Hardy-Sobolev equations of minimizing type. This yields informations on the value of the second best constant in the related Riemannian functional inequality.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45179232","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 : 2022-04-06DOI: 10.2140/pjm.2022.316.453
Xingwang Xu, J. Ye
{"title":"Conformal vector fields and σk-scalar\u0000curvatures","authors":"Xingwang Xu, J. Ye","doi":"10.2140/pjm.2022.316.453","DOIUrl":"https://doi.org/10.2140/pjm.2022.316.453","url":null,"abstract":"","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45745193","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 : 2022-04-06DOI: 10.2140/pjm.2022.316.409
Liat Kessler
{"title":"A symplectic form on the space of embedded symplectic surfaces and its reduction by reparametrizations","authors":"Liat Kessler","doi":"10.2140/pjm.2022.316.409","DOIUrl":"https://doi.org/10.2140/pjm.2022.316.409","url":null,"abstract":"","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48326902","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}
The $J^k$ space of $k$-jets of a real function of one real variable $x$ admits the structure of a sub-Riemannian manifold, which then has an associated Hamiltonian geodesic flow, and it is integrable. As in any Hamiltonian flow, a natural question is the existence of periodic solutions. Does $J^k$ have periodic geodesics? This study will find the action-angle coordinates in $T^*J^k$ for the geodesic flow and demonstrate that geodesics in $J^k$ are never periodic.
{"title":"No periodic geodesics in jet space","authors":"Alejandro Bravo-Doddoli","doi":"10.2140/pjm.2023.322.11","DOIUrl":"https://doi.org/10.2140/pjm.2023.322.11","url":null,"abstract":"The $J^k$ space of $k$-jets of a real function of one real variable $x$ admits the structure of a sub-Riemannian manifold, which then has an associated Hamiltonian geodesic flow, and it is integrable. As in any Hamiltonian flow, a natural question is the existence of periodic solutions. Does $J^k$ have periodic geodesics? This study will find the action-angle coordinates in $T^*J^k$ for the geodesic flow and demonstrate that geodesics in $J^k$ are never periodic.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43587428","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 : 2022-03-29DOI: 10.2140/pjm.2023.323.281
Saki Kanou, K. Sakai
We show that the integration of a 1-cocycle I(X) of the space of long knots in R^3 over the Fox-Hatcher 1-cycles gives rise to a Vassiliev invariant of order exactly three. This result can be seen as a continuation of the previous work of the second named author, proving that the integration of I(X) over the Gramain 1-cycles is the Casson invariant, the unique nontrivial Vassiliev invariant of order two (up to scalar multiplications). The result in the present paper is also analogous to part of Mortier's result. Our result differs from, but is motivated by, Mortier's one in that the 1-cocycle I(X) is given by the configuration space integrals associated with graphs while Mortier's cocycle is obtained in a combinatorial way.
{"title":"The Fox–Hatcher cycle and a Vassiliev invariant\u0000of order three","authors":"Saki Kanou, K. Sakai","doi":"10.2140/pjm.2023.323.281","DOIUrl":"https://doi.org/10.2140/pjm.2023.323.281","url":null,"abstract":"We show that the integration of a 1-cocycle I(X) of the space of long knots in R^3 over the Fox-Hatcher 1-cycles gives rise to a Vassiliev invariant of order exactly three. This result can be seen as a continuation of the previous work of the second named author, proving that the integration of I(X) over the Gramain 1-cycles is the Casson invariant, the unique nontrivial Vassiliev invariant of order two (up to scalar multiplications). The result in the present paper is also analogous to part of Mortier's result. Our result differs from, but is motivated by, Mortier's one in that the 1-cocycle I(X) is given by the configuration space integrals associated with graphs while Mortier's cocycle is obtained in a combinatorial way.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49550335","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}
Vishnu Kakkat, M. Khuri, Jordan Rainone, G. Weinstein
Extending recent work in 5 dimensions, we prove the existence and uniqueness of solutions to the reduced Einstein equations for vacuum black holes in $(n+3)$-dimensional spacetimes admitting the isometry group $mathbb{R}times U(1)^{n}$, with Kaluza-Klein asymptotics for $ngeq3$. This is equivalent to establishing existence and uniqueness for singular harmonic maps $varphi: mathbb{R}^3setminusGammarightarrow SL(n+1,mathbb{R})/SO(n+1)$ with prescribed blow-up along $Gamma$, a subset of the $z$-axis in $mathbb{R}^3$. We also analyze the topology of the domain of outer communication for these spacetimes, by developing an appropriate generalization of the plumbing construction used in the lower dimensional case. Furthermore, we provide a counterexample to a conjecture of Hollands-Ishibashi concerning the topological classification of the domain of outer communication. A refined version of the conjecture is then presented and established in spacetime dimensions less than 8.
{"title":"The geometry and topology of stationary multiaxisymmetric vacuum black holes in higher dimensions","authors":"Vishnu Kakkat, M. Khuri, Jordan Rainone, G. Weinstein","doi":"10.2140/pjm.2023.322.59","DOIUrl":"https://doi.org/10.2140/pjm.2023.322.59","url":null,"abstract":"Extending recent work in 5 dimensions, we prove the existence and uniqueness of solutions to the reduced Einstein equations for vacuum black holes in $(n+3)$-dimensional spacetimes admitting the isometry group $mathbb{R}times U(1)^{n}$, with Kaluza-Klein asymptotics for $ngeq3$. This is equivalent to establishing existence and uniqueness for singular harmonic maps $varphi: mathbb{R}^3setminusGammarightarrow SL(n+1,mathbb{R})/SO(n+1)$ with prescribed blow-up along $Gamma$, a subset of the $z$-axis in $mathbb{R}^3$. We also analyze the topology of the domain of outer communication for these spacetimes, by developing an appropriate generalization of the plumbing construction used in the lower dimensional case. Furthermore, we provide a counterexample to a conjecture of Hollands-Ishibashi concerning the topological classification of the domain of outer communication. A refined version of the conjecture is then presented and established in spacetime dimensions less than 8.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49397659","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 : 2022-03-13DOI: 10.2140/pjm.2023.324.143
Fagui Li, Niang-Shin Chen
Let $ M^n$ be a closed immersed minimal hypersurface in the unit sphere $mathbb{S}^{n+1}$. We establish a special isoperimetric inequality of $M^n$. As an application, if the scalar curvature of $ M^n$ is constant, then we get a uniform lower bound independent of $M^n$ for the isoperimetric inequality. In addition, we obtain an inequality between Cheeger's isoperimetric constant and the volume of the nodal set of the height function.
{"title":"An isoperimetric inequality of minimal hypersurfaces in spheres","authors":"Fagui Li, Niang-Shin Chen","doi":"10.2140/pjm.2023.324.143","DOIUrl":"https://doi.org/10.2140/pjm.2023.324.143","url":null,"abstract":"Let $ M^n$ be a closed immersed minimal hypersurface in the unit sphere $mathbb{S}^{n+1}$. We establish a special isoperimetric inequality of $M^n$. As an application, if the scalar curvature of $ M^n$ is constant, then we get a uniform lower bound independent of $M^n$ for the isoperimetric inequality. In addition, we obtain an inequality between Cheeger's isoperimetric constant and the volume of the nodal set of the height function.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44275738","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}
{"title":"Compactness of conformal metrics with integral bounds on Ricci curvature","authors":"Conghan Dong, Yuxiang Li","doi":"10.2140/pjm.2022.316.65","DOIUrl":"https://doi.org/10.2140/pjm.2022.316.65","url":null,"abstract":"","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46114660","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 : 2022-02-26DOI: 10.2140/pjm.2022.316.207
Xiankui Meng, S. Yau
{"title":"Rigidity of CR morphisms","authors":"Xiankui Meng, S. Yau","doi":"10.2140/pjm.2022.316.207","DOIUrl":"https://doi.org/10.2140/pjm.2022.316.207","url":null,"abstract":"","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49586173","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 : 2022-02-20DOI: 10.2140/pjm.2022.321.345
K. Kozioł
Suppose $p geq 5$ is a prime number, and let $G = textrm{SL}_2(mathbb{Q}_p)$. We calculate the derived functors $textrm{R}^nmathcal{R}_B^G(pi)$, where $B$ is a Borel subgroup of $G$, $mathcal{R}_B^G$ is the right adjoint of smooth parabolic induction constructed by Vign'eras, and $pi$ is any smooth, absolutely irreducible, mod $p$ representation of $G$.
{"title":"Derived right adjoints of parabolic induction: an example","authors":"K. Kozioł","doi":"10.2140/pjm.2022.321.345","DOIUrl":"https://doi.org/10.2140/pjm.2022.321.345","url":null,"abstract":"Suppose $p geq 5$ is a prime number, and let $G = textrm{SL}_2(mathbb{Q}_p)$. We calculate the derived functors $textrm{R}^nmathcal{R}_B^G(pi)$, where $B$ is a Borel subgroup of $G$, $mathcal{R}_B^G$ is the right adjoint of smooth parabolic induction constructed by Vign'eras, and $pi$ is any smooth, absolutely irreducible, mod $p$ representation of $G$.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45389075","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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