Pub Date : 2019-11-02DOI: 10.4310/cag.2022.v30.n5.a4
Q. Cheng, G. Wei
In this paper, We define a $mathcal{F}$-functional and study $mathcal{F}$-stability of $lambda$-hypersurfaces, which extend a result of Colding-Minicozzi. Lower bound growth and upper bound growth of area for complete and non-compact $lambda$-hypersurfaces are studied.
{"title":"Stability and area growth of $lambda$-hypersurfaces","authors":"Q. Cheng, G. Wei","doi":"10.4310/cag.2022.v30.n5.a4","DOIUrl":"https://doi.org/10.4310/cag.2022.v30.n5.a4","url":null,"abstract":"In this paper, We define a $mathcal{F}$-functional and study $mathcal{F}$-stability of $lambda$-hypersurfaces, which extend a result of Colding-Minicozzi. Lower bound growth and upper bound growth of area for complete and non-compact $lambda$-hypersurfaces are studied.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2019-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43504091","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 : 2019-10-24DOI: 10.4310/CAG.2020.V28.N8.A9
Gao Chen, Jeff A. Viaclovsky, Ruobing Zhang
For any elliptic K3 surface $mathfrak{F}: mathcal{K} rightarrow mathbb{P}^1$, we construct a family of collapsing Ricci-flat K"ahler metrics such that curvatures are uniformly bounded away from singular fibers, and which Gromov-Hausdorff limit to $mathbb{P}^1$ equipped with the McLean metric. There are well-known examples of this type of collapsing, but the key point of our construction is that we can additionally give a precise description of the metric degeneration near each type of singular fiber, without any restriction on the types of singular fibers.
对于任意椭圆型K3曲面$mathfrak{F}: mathcal{K} 右列mathbb{P}^1$，我们构造了一个坍缩的Ricci-flat K ahler度量族，使得曲率与奇异纤维有一致的界，并且具有McLean度量，其Gromov-Hausdorff极限为$mathbb{P}^1$。这类坍缩有很多众所周知的例子，但我们构造的关键是我们可以在不限制奇异纤维类型的情况下，对每一类奇异纤维附近的度规退化给出精确的描述。
{"title":"Collapsing Ricci-flat metrics on elliptic K3 surfaces","authors":"Gao Chen, Jeff A. Viaclovsky, Ruobing Zhang","doi":"10.4310/CAG.2020.V28.N8.A9","DOIUrl":"https://doi.org/10.4310/CAG.2020.V28.N8.A9","url":null,"abstract":"For any elliptic K3 surface $mathfrak{F}: mathcal{K} rightarrow mathbb{P}^1$, we construct a family of collapsing Ricci-flat K\"ahler metrics such that curvatures are uniformly bounded away from singular fibers, and which Gromov-Hausdorff limit to $mathbb{P}^1$ equipped with the McLean metric. There are well-known examples of this type of collapsing, but the key point of our construction is that we can additionally give a precise description of the metric degeneration near each type of singular fiber, without any restriction on the types of singular fibers.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2019-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46209691","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 : 2019-10-16DOI: 10.4310/cag.2022.v30.n9.a8
Guofang Wang, C. Xia
In this paper we introduce a Guan-Li type volume preserving mean curvature flow for free boundary hypersurfaces in a ball. We give a concept of star-shaped free boundary hypersurfaces in a ball and show that the Guan-Li type mean curvature flow has long time existence and converges to a free boundary spherical cap, provided the initial data is star-shaped.
{"title":"Guan–Li type mean curvature flow for free boundary hypersurfaces in a ball","authors":"Guofang Wang, C. Xia","doi":"10.4310/cag.2022.v30.n9.a8","DOIUrl":"https://doi.org/10.4310/cag.2022.v30.n9.a8","url":null,"abstract":"In this paper we introduce a Guan-Li type volume preserving mean curvature flow for free boundary hypersurfaces in a ball. We give a concept of star-shaped free boundary hypersurfaces in a ball and show that the Guan-Li type mean curvature flow has long time existence and converges to a free boundary spherical cap, provided the initial data is star-shaped.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2019-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46153027","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 : 2019-09-09DOI: 10.4310/CAG.2020.v28.n8.a6
S. Bradlow, Lucas C. Branco, L. Schaposnik
We examine Higgs bundles for non-compact real forms of SO(4,C) and the isogenous complex group SL(2,C)XSL(2,C). This involves a study of non-regular fibers in the corresponding Hitchin fibrations and provides interesting examples of non-abelian spectral data.
{"title":"Orthogonal Higgs bundles with singular spectral curves","authors":"S. Bradlow, Lucas C. Branco, L. Schaposnik","doi":"10.4310/CAG.2020.v28.n8.a6","DOIUrl":"https://doi.org/10.4310/CAG.2020.v28.n8.a6","url":null,"abstract":"We examine Higgs bundles for non-compact real forms of SO(4,C) and the isogenous complex group SL(2,C)XSL(2,C). This involves a study of non-regular fibers in the corresponding Hitchin fibrations and provides interesting examples of non-abelian spectral data.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2019-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45784839","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 : 2019-08-27DOI: 10.4310/cag.2022.v30.n2.a2
Pak-Yeung Chan
We study the non Ricci flat gradient steady Kahler Ricci soliton with non-negative Ricci curvature and weak integrability condition of the scalar curvature $S$, namely $underline{lim}_{rto infty} r^{-1}int_{B_r} S=0$, and show that it is a quotient of $Sigmatimes mathbb{C}^{n-1-k}times N^k$, where $Sigma$ and $N$ denote the Hamilton's cigar soliton and some compact Kahler Ricci flat manifold respectively. As an application, we prove that any non Ricci flat gradient steady Kahler Ricci soliton with $Ricgeq 0$, together with subquadratic volume growth or $limsup_{rto infty} rS<1$ must have universal covering space isometric to $Sigmatimes mathbb{C}^{n-1-k}times N^k$.
{"title":"Gradient steady Kähler–Ricci solitons with non-negative Ricci curvature and integrable scalar curvature","authors":"Pak-Yeung Chan","doi":"10.4310/cag.2022.v30.n2.a2","DOIUrl":"https://doi.org/10.4310/cag.2022.v30.n2.a2","url":null,"abstract":"We study the non Ricci flat gradient steady Kahler Ricci soliton with non-negative Ricci curvature and weak integrability condition of the scalar curvature $S$, namely $underline{lim}_{rto infty} r^{-1}int_{B_r} S=0$, and show that it is a quotient of $Sigmatimes mathbb{C}^{n-1-k}times N^k$, where $Sigma$ and $N$ denote the Hamilton's cigar soliton and some compact Kahler Ricci flat manifold respectively. As an application, we prove that any non Ricci flat gradient steady Kahler Ricci soliton with $Ricgeq 0$, together with subquadratic volume growth or $limsup_{rto infty} rS<1$ must have universal covering space isometric to $Sigmatimes mathbb{C}^{n-1-k}times N^k$.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2019-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48868634","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 : 2019-06-20DOI: 10.4310/cag.2022.v30.n7.a1
Aghil Alaee, S. Yau
We prove positive mass theorem with angular momentum and charges for axially symmetric, simply connected, maximal, complete initial data sets with two ends, one designated asymptotically flat and the other either (Kaluza-Klein) asymptotically flat or asymptotically cylindrical, for 4-dimensional Einstein-Maxwell theory and $5$-dimensional minimal supergravity theory which metrics fail to be $C^1$ and second fundamental forms and electromagnetic fields fail to be $C^0$ across an axially symmetric hypersurface $Sigma$. Furthermore, we remove the completeness and simple connectivity assumptions in this result and prove it for manifold with boundary such that the mean curvature of the boundary is non-positive.
{"title":"Positive mass theorem for initial data sets with corners along a hypersurface","authors":"Aghil Alaee, S. Yau","doi":"10.4310/cag.2022.v30.n7.a1","DOIUrl":"https://doi.org/10.4310/cag.2022.v30.n7.a1","url":null,"abstract":"We prove positive mass theorem with angular momentum and charges for axially symmetric, simply connected, maximal, complete initial data sets with two ends, one designated asymptotically flat and the other either (Kaluza-Klein) asymptotically flat or asymptotically cylindrical, for 4-dimensional Einstein-Maxwell theory and $5$-dimensional minimal supergravity theory which metrics fail to be $C^1$ and second fundamental forms and electromagnetic fields fail to be $C^0$ across an axially symmetric hypersurface $Sigma$. Furthermore, we remove the completeness and simple connectivity assumptions in this result and prove it for manifold with boundary such that the mean curvature of the boundary is non-positive.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2019-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42822969","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 : 2019-05-12DOI: 10.4310/cag.2022.v30.n7.a3
Weimin Sheng, Caihong Yi
We consider a shrinking flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_n^{beta}, where u is the support function of the hypersurface, alpha, beta are two constants, and beta>0, sigma_n is the n-th symmetric polynomial of the principle curvature radii of the hypersurface. We prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a soliton which is a solution of an elliptic equation, when the constants alpha, beta belong to a suitable range, provided the initial hypersuface is origin-symmetric and f is a smooth positive even function on S^n. For the case alpha>= 1+n*beta, beta>0, we prove that the flow converges smoothly after normalisation to a unique smooth solution of an elliptic equation without any constraint on the initial hypersuface and smooth positive function f. When beta=1, our argument provides a uniform proof to the existence of the solutions to the equation of L_p Minkowski problem for p belongs to (-n-1,+infty).
{"title":"An anisotropic shrinking flow and $L_p$ Minkowski problem","authors":"Weimin Sheng, Caihong Yi","doi":"10.4310/cag.2022.v30.n7.a3","DOIUrl":"https://doi.org/10.4310/cag.2022.v30.n7.a3","url":null,"abstract":"We consider a shrinking flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_n^{beta}, where u is the support function of the hypersurface, alpha, beta are two constants, and beta>0, sigma_n is the n-th symmetric polynomial of the principle curvature radii of the hypersurface. We prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a soliton which is a solution of an elliptic equation, when the constants alpha, beta belong to a suitable range, provided the initial hypersuface is origin-symmetric and f is a smooth positive even function on S^n. For the case alpha>= 1+n*beta, beta>0, we prove that the flow converges smoothly after normalisation to a unique smooth solution of an elliptic equation without any constraint on the initial hypersuface and smooth positive function f. When beta=1, our argument provides a uniform proof to the existence of the solutions to the equation of L_p Minkowski problem for p belongs to (-n-1,+infty).","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2019-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48185474","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 : 2019-04-29DOI: 10.4310/cag.2022.v30.n8.a3
Paul Creutz
We give a solution of Plateau's problem for singular curves possibly having self-intersections. The proof is based on the solution of Plateau's problem for Jordan curves in very general metric spaces by Alexander Lytchak and Stefan Wenger and hence works also in a quite general setting. However the main result of this paper seems to be new even in $mathbb{R}^n$.
{"title":"Plateau’s problem for singular curves","authors":"Paul Creutz","doi":"10.4310/cag.2022.v30.n8.a3","DOIUrl":"https://doi.org/10.4310/cag.2022.v30.n8.a3","url":null,"abstract":"We give a solution of Plateau's problem for singular curves possibly having self-intersections. The proof is based on the solution of Plateau's problem for Jordan curves in very general metric spaces by Alexander Lytchak and Stefan Wenger and hence works also in a quite general setting. However the main result of this paper seems to be new even in $mathbb{R}^n$.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2019-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45949483","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 : 2019-02-19DOI: 10.4310/CAG.2022.v30.n8.a2
Matteo Capoferri, M. Levitin, D. Vassiliev
We study the propagator of the wave equation on a closed Riemannian manifold $M$. We propose a geometric approach to the construction of the propagator as a single oscillatory integral global both in space and in time with a distinguished complex-valued phase function. This enables us to provide a global invariant definition of the full symbol of the propagator - a scalar function on the cotangent bundle - and an algorithm for the explicit calculation of its homogeneous components. The central part of the paper is devoted to the detailed analysis of the subprincipal symbol; in particular, we derive its explicit small time asymptotic expansion. We present a general geometric construction that allows one to visualise topological obstructions and describe their circumvention with the use of a complex-valued phase function. We illustrate the general framework with explicit examples in dimension two.
{"title":"Geometric wave propagator on Riemannian manifolds","authors":"Matteo Capoferri, M. Levitin, D. Vassiliev","doi":"10.4310/CAG.2022.v30.n8.a2","DOIUrl":"https://doi.org/10.4310/CAG.2022.v30.n8.a2","url":null,"abstract":"We study the propagator of the wave equation on a closed Riemannian manifold $M$. We propose a geometric approach to the construction of the propagator as a single oscillatory integral global both in space and in time with a distinguished complex-valued phase function. This enables us to provide a global invariant definition of the full symbol of the propagator - a scalar function on the cotangent bundle - and an algorithm for the explicit calculation of its homogeneous components. The central part of the paper is devoted to the detailed analysis of the subprincipal symbol; in particular, we derive its explicit small time asymptotic expansion. We present a general geometric construction that allows one to visualise topological obstructions and describe their circumvention with the use of a complex-valued phase function. We illustrate the general framework with explicit examples in dimension two.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43867466","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 : 2019-02-18DOI: 10.4310/cag.2022.v30.n6.a5
J. Lauret, Marina Nicolini
Only two examples of extremally Ricci pinched G2-structures can be found in the literature and they are both homogeneous. We study in this paper the existence and structure of such very special closed G2-structures on Lie groups. Strong structural conditions on the Lie algebra are proved to hold. As an application, we obtain three new examples of extremally Ricci pinched G2-structures and that they are all necessarily steady Laplacian solitons. The deformation and rigidity of such structures are also studied.
{"title":"Extremally Ricci pinched $G_2$-structures on Lie groups","authors":"J. Lauret, Marina Nicolini","doi":"10.4310/cag.2022.v30.n6.a5","DOIUrl":"https://doi.org/10.4310/cag.2022.v30.n6.a5","url":null,"abstract":"Only two examples of extremally Ricci pinched G2-structures can be found in the literature and they are both homogeneous. We study in this paper the existence and structure of such very special closed G2-structures on Lie groups. Strong structural conditions on the Lie algebra are proved to hold. As an application, we obtain three new examples of extremally Ricci pinched G2-structures and that they are all necessarily steady Laplacian solitons. The deformation and rigidity of such structures are also studied.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2019-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48481032","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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