Pub Date : 2024-07-26DOI: 10.4310/cag.2023.v31.n7.a4
de Hoop,Maarten V., Ilmavirta,Joonas, Lassas,Matti, Saksala,Teemu
We prove that the boundary distance map of a smooth compact Finsler manifold with smooth boundary determines its topological and differentiable structures. We construct the optimal fiberwise open subset of its tangent bundle and show that the boundary distance map determines the Finsler function in this set but not in its exterior. If the Finsler function is fiberwise real analytic, it is determined uniquely. We also discuss the smoothness of the distance function between interior and boundary points. We recall how the fastest $qP$-polarized waves in anisotropic elastic medium are a given as solutions of the second order hyperbolic pseudodifferential equation $(frac{partial ^{2}}{partial t^{2}}-lambda ^{1}(x,D))u(t,x)=h(t,x)$ on ${mathbb R}^{1+3}$, where $sqrt{lambda ^{1}}$ is the Legendre transform of a fiberwise real analytic Finsler function $F$ on ${mathbb R}^{3}$. If $M subset {mathbb R}^{3}$ is a $F$-convex smooth bounded domain we say that a travel time of $u$ to $z in partial M$ is the first time $t>0$ when the wavefront set of $u$ arrives in $(t,z)$. The aforementioned geometric result can then be utilized to determine the isometry class of $(overline M,F)$ if we have measured a large amount of travel times of $qP$-polarized waves, issued from a dense set of unknown interior point sources on $M$.
{"title":"Determination of a compact Finsler manifold from its boundary distance map and an inverse problem in elasticity","authors":"de Hoop,Maarten V., Ilmavirta,Joonas, Lassas,Matti, Saksala,Teemu","doi":"10.4310/cag.2023.v31.n7.a4","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n7.a4","url":null,"abstract":"We prove that the boundary distance map of a smooth compact Finsler manifold with smooth boundary determines its topological and differentiable structures. We construct the optimal fiberwise open subset of its tangent bundle and show that the boundary distance map determines the Finsler function in this set but not in its exterior. If the Finsler function is fiberwise real analytic, it is determined uniquely. We also discuss the smoothness of the distance function between interior and boundary points. We recall how the fastest $qP$-polarized waves in anisotropic elastic medium are a given as solutions of the second order hyperbolic pseudodifferential equation $(frac{partial ^{2}}{partial t^{2}}-lambda ^{1}(x,D))u(t,x)=h(t,x)$ on ${mathbb R}^{1+3}$, where $sqrt{lambda ^{1}}$ is the Legendre transform of a fiberwise real analytic Finsler function $F$ on ${mathbb R}^{3}$. If $M subset {mathbb R}^{3}$ is a $F$-convex smooth bounded domain we say that a travel time of $u$ to $z in partial M$ is the first time $t>0$ when the wavefront set of $u$ arrives in $(t,z)$. The aforementioned geometric result can then be utilized to determine the isometry class of $(overline M,F)$ if we have measured a large amount of travel times of $qP$-polarized waves, issued from a dense set of unknown interior point sources on $M$.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"41 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780479","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.a8
Hirsch,Sven, Lesourd,Martin
Carlotto-Li have generalized Marques' path connectedness result for positive scalar curvature $R>0$ metrics on closed $3$-manifolds to the case of compact $3$-manifolds with $R>0$ and mean convex boundary $H>0$. Using their result, we show that the space of asymptotically flat metrics with nonnegative scalar curvature and mean convex boundary on $mathbb{R}^{3}backslash B^{3}$ is path connected. The argument bypasses Cerf's theorem, which was used in Marques' proof but which becomes inapplicable in the presence of a boundary. We also show path connectedness for a class of maximal initial data sets with marginally outer trapped boundary.
{"title":"On the moduli space of asymptotically flat manifolds with boundary and the constraint equations","authors":"Hirsch,Sven, Lesourd,Martin","doi":"10.4310/cag.2023.v31.n7.a8","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n7.a8","url":null,"abstract":"Carlotto-Li have generalized Marques' path connectedness result for positive scalar curvature $R>0$ metrics on closed $3$-manifolds to the case of compact $3$-manifolds with $R>0$ and mean convex boundary $H>0$. Using their result, we show that the space of asymptotically flat metrics with nonnegative scalar curvature and mean convex boundary on $mathbb{R}^{3}backslash B^{3}$ is path connected. The argument bypasses Cerf's theorem, which was used in Marques' proof but which becomes inapplicable in the presence of a boundary. We also show path connectedness for a class of maximal initial data sets with marginally outer trapped boundary.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"19 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780474","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.a5
Hu,Ying
In this article, we study the Euler class of taut foliations on the Dehn fillings of a $mathbb{Q}$-homology solid torus. We give a necessary and sufficient condition for the Euler class of a foliation transverse to the core of the filling solid torus to vanish. We apply this condition to taut foliations on Dehn fillings of hyperbolic fibered manifolds and obtain many new left-orderable Dehn filling slopes on these manifolds. For instance, we show that when $X$ is the exterior of the pretzel knot $P(-2,3,2r+1)$, for $rgeq 3$, $pi _{1}(X(alpha _{n}))$ is left-orderable for a sequence of positive slopes $alpha _{n}$ with $alpha _{0} =2g-2$ and $alpha _{n}to 2g-1$. Lastly, we prove that given any $mathbb{Q}$-homology solid torus, the set of slopes for which the corresponding Dehn fillings admit a taut foliation transverse to the core with zero Euler class is nowhere dense in $mathbb{R}cup {frac{1}{0}}$.
{"title":"Euler class of taut foliations and Dehn filling","authors":"Hu,Ying","doi":"10.4310/cag.2023.v31.n7.a5","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n7.a5","url":null,"abstract":"In this article, we study the Euler class of taut foliations on the Dehn fillings of a $mathbb{Q}$-homology solid torus. We give a necessary and sufficient condition for the Euler class of a foliation transverse to the core of the filling solid torus to vanish. We apply this condition to taut\u0000foliations on Dehn fillings of hyperbolic fibered manifolds and obtain many new left-orderable Dehn filling slopes on these manifolds. For instance, we show that when $X$ is the exterior of the pretzel knot $P(-2,3,2r+1)$, for $rgeq 3$, $pi _{1}(X(alpha _{n}))$ is left-orderable for a sequence of positive slopes $alpha _{n}$ with $alpha _{0} =2g-2$ and $alpha _{n}to 2g-1$. Lastly, we prove that given any $mathbb{Q}$-homology solid torus, the set of slopes for which the corresponding Dehn fillings admit a taut foliation transverse to the core with zero Euler class is nowhere dense in $mathbb{R}cup {frac{1}{0}}$.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"66 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780472","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}
We consider the Cauchy problem for the heat diffusion equation in the whole Euclidean space consisting of two media with different constant conductivities, where initially one has temperature 0 and the other has temperature 1. Suppose that the interface is connected and uniformly of class $C^{6}$. We show that if the interface has a time-invariant constant temperature, then it must be a hyperplane.
{"title":"A characterization of a hyperplane in two-phase heat conductorsu","authors":"Cavallina,Lorenzo, Sakaguchi,Shigeru, Udagawa,Seiichi","doi":"10.4310/cag.2023.v31.n7.a9","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n7.a9","url":null,"abstract":"We consider the Cauchy problem for the heat diffusion equation in the whole Euclidean space consisting of two media with different constant conductivities, where initially one has temperature 0 and the other has temperature 1. Suppose that the interface is connected and uniformly of class $C^{6}$. We show that if the interface has a time-invariant constant temperature, then it must be a hyperplane.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"10 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780475","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 : 2024-07-24DOI: 10.4310/cag.2023.v31.n6.a2
An,Zhongshan
We prove that the space of initial data sets which solve the constraint equations and have fixed Bartnik boundary data is a Banach manifold. Moreover if an initial data set on this constraint manifold is a critical point of the ADM total mass, then it must admit a generalised Killing vector field which is asymptotically proportional to the ADM energy-momentum vector.
{"title":"On mass-minimizing extensions of Bartnik boundary data","authors":"An,Zhongshan","doi":"10.4310/cag.2023.v31.n6.a2","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n6.a2","url":null,"abstract":"We prove that the space of initial data sets which solve the constraint equations and have fixed Bartnik boundary data is a Banach manifold. Moreover if an initial data set on this constraint manifold is a critical point of the ADM total mass, then it must admit a generalised Killing vector field which is asymptotically proportional to the ADM energy-momentum vector.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"422 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780477","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 : 2024-07-24DOI: 10.4310/cag.2023.v31.n6.a3
Furutani,Ryoga, Koda,Yuya
A stable map of a closed orientable 3-manifold into the real plane is called a stable map of a link in the manifold if the link is contained in the set of definite fold points. We give a complete characterization of the hyperbolic links in the $3$-sphere that admit stable maps into the real plane with exactly one (connected component of a) fiber having two singular points.
{"title":"Stable maps and hyperbolic links","authors":"Furutani,Ryoga, Koda,Yuya","doi":"10.4310/cag.2023.v31.n6.a3","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n6.a3","url":null,"abstract":"A stable map of a closed orientable 3-manifold into the real plane is called a stable map of a link in the manifold if the link is contained in the set of definite fold points. We give a complete characterization of the hyperbolic links in the $3$-sphere that admit stable maps into the real plane with exactly one (connected component of a) fiber having two singular points.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"5 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780480","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 : 2024-07-24DOI: 10.4310/cag.2023.v31.n6.a7
Jauregui,Jeffrey L.
Based on the isoperimetric inequality, G. Huisken proposed a definition of total mass in general relativity that is equivalent to the ADM mass for smooth asymptotically flat 3-manifolds of nonnegative scalar curvature, but that is well-defined in lower regularity. In a similar vein, we use the isocapacitary inequality (bounding capacity from below in terms of volume) to suggest a new definition of total mass. We prove an inequality between it and the ADM mass, and prove the reverse inequality with harmonically flat asymptotics, or, with general asymptotics, for exhaustions by balls (as opposed to arbitrary compact sets). This approach to mass may have applications to problems involving low regularity metrics and convergence in general relativity, and may have some advantages relative to the isoperimetric mass. Some conjectures, analogs of known results for CMC surfaces and isoperimetric regions, are proposed.
{"title":"ADM mass and the capacity-volume deficit at infinity","authors":"Jauregui,Jeffrey L.","doi":"10.4310/cag.2023.v31.n6.a7","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n6.a7","url":null,"abstract":"Based on the isoperimetric inequality, G. Huisken proposed a definition of total mass in general relativity that is equivalent to the ADM mass for smooth asymptotically flat 3-manifolds of nonnegative scalar curvature, but that is well-defined in lower regularity. In a similar vein, we use the isocapacitary inequality (bounding capacity from below in terms of volume) to suggest a new definition of total mass. We prove an inequality between it and the ADM mass, and prove the reverse inequality with harmonically flat asymptotics, or, with general asymptotics, for exhaustions by balls (as opposed to arbitrary compact sets). This approach to mass may have applications to problems involving low regularity metrics and convergence in general relativity, and may have some advantages relative to the isoperimetric mass. Some conjectures, analogs of known results for CMC surfaces and isoperimetric regions, are proposed.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"245 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780257","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}
We study generalizations of Lorentzian warped products with one-dimensional base of the form $Itimes_f X$, where $I$ is an interval, $X$ is a length space and $f$ is a positive continuous function. These generalized cones furnish an important class of Lorentzian length spaces in the sense of [39], displaying optimal causality properties that allow for explicit descriptions of all underlying notions. In addition, synthetic sectional curvature bounds of generalized cones are directly related to metric curvature bounds of the fiber $X$. The interest in such spaces comes both from metric geometry and from General Relativity, where warped products underlie important cosmological models (FLRW spacetimes). Moreover, we prove singularity theorems for these spaces, showing that non-positive lower timelike curvature bounds imply the existence of incomplete timelike geodesics.
{"title":"Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems","authors":"Alexander,Stephanie B., Graf,Melanie, Kunzinger,Michael, Sämann,Clemens","doi":"10.4310/cag.2023.v31.n6.a5","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n6.a5","url":null,"abstract":"We study generalizations of Lorentzian warped products with one-dimensional base of the form $Itimes_f X$, where $I$ is an interval, $X$ is a length space and $f$ is a positive continuous function. These generalized cones furnish an important class of Lorentzian length spaces in the sense of [39], displaying optimal causality properties that allow for explicit descriptions of all underlying notions. In addition, synthetic sectional curvature bounds of generalized cones are directly related to metric curvature bounds of the fiber $X$. The interest in such spaces comes both from metric geometry and from General Relativity, where warped products underlie important cosmological models (FLRW spacetimes). Moreover, we prove singularity theorems for these spaces, showing that non-positive lower timelike curvature bounds imply the existence of incomplete timelike geodesics.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"46 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780254","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 : 2024-07-24DOI: 10.4310/cag.2023.v31.n6.a1
Lee,Tsung-Ju, Lian,Bong H., Zhang,Dingxin
We verify a formula on the solution rank of the tautological system arising from ample complete intersections in a projective homogeneous space of a semisimple group conjectured by Huang-Lian-Yau-Yu [1]. As an application, we prove the existence of the rank one point for such a system, where mirror symmetry is expected.
{"title":"On a conjecture of Huang-Lian-Yau-Yu","authors":"Lee,Tsung-Ju, Lian,Bong H., Zhang,Dingxin","doi":"10.4310/cag.2023.v31.n6.a1","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n6.a1","url":null,"abstract":"We verify a formula on the solution rank of the tautological system arising from ample complete intersections in a projective homogeneous space of a semisimple group conjectured by Huang-Lian-Yau-Yu [1]. As an application, we prove the existence of the rank one point for such a system, where mirror symmetry is expected.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"134 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780478","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}
In this paper we prove that in the class of metric measure space with Alexandrov curvature bounded from below the Riemannian curvature-dimension condition $RCD^*(K,N)$ with $Kin mathbb{R}$ & $Nin [1,infty)$ is preserved under doubling and gluing constructions provided the weight in the measure is semiconcave.
在本文中，我们证明了在一类具有亚历山德罗夫曲率的度量空间中，只要度量中的权重是半凹的，那么在具有亚历山德罗夫曲率的度量空间中，黎曼曲率维度条件 $RCD^*(K,N)$ with $Kin mathbb{R}$ & $Nin [1,infty)$ 在加倍和粘合构造下是保留的。
{"title":"On gluing Alexandrov spaces with lower Ricci curvature bounds","authors":"Kapovitch,Vitali, Ketterer,Christian, Sturm,Karl-Theodor","doi":"10.4310/cag.2023.v31.n6.a6","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n6.a6","url":null,"abstract":"In this paper we prove that in the class of metric measure space with Alexandrov curvature bounded from below the Riemannian curvature-dimension condition $RCD^*(K,N)$ with $Kin mathbb{R}$ & $Nin [1,infty)$ is preserved under doubling and gluing constructions provided the weight in the measure is semiconcave.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":"56 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780255","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：