Pub Date : 2024-08-10DOI: 10.4310/cag.2023.v31.n8.a4
Ping Li
Various Alexandrov–Fenchel type inequalities have appeared and played important roles in convex geometry, matrix theory and complex algebraic geometry. It has been noticed for some time that they share some striking analogies and have intimate relationships. The purpose of this article is to shed new light on this by comparatively investigating them in several aspects. The principal result in this article is a complete solution to the equality characterization problem of various Alexandrov–Fenchel type inequalities for intersection numbers of nef and big classes on compact Kähler manifolds, extending some earlier related results. In addition to this central result, we also give a geometric proof of the complex version of the Alexandrov–Fenchel inequality for mixed discriminants and a determinantal generalization of various Alexandrov–Fenchel type inequalities.
{"title":"The Alexandrov–Fenchel type inequalities, revisited","authors":"Ping Li","doi":"10.4310/cag.2023.v31.n8.a4","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n8.a4","url":null,"abstract":"Various Alexandrov–Fenchel type inequalities have appeared and played important roles in convex geometry, matrix theory and complex algebraic geometry. It has been noticed for some time that they share some striking analogies and have intimate relationships. The purpose of this article is to shed new light on this by comparatively investigating them in several aspects. <i>The principal result</i> in this article is a complete solution to the equality characterization problem of various Alexandrov–Fenchel type inequalities for intersection numbers of nef and big classes on compact Kähler manifolds, extending some earlier related results. In addition to this central result, we also give a geometric proof of the complex version of the Alexandrov–Fenchel inequality for mixed discriminants and a determinantal generalization of various Alexandrov–Fenchel type inequalities.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141969618","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 : 2024-08-10DOI: 10.4310/cag.2023.v31.n8.a5
Benjamin Bode
A geometric braid $B$ can be interpreted as a loop in the space of monic complex polynomials with distinct roots. This loop defines a function $g : mathbb{C} times S^1 to C$ that vanishes on $B$. We define the set of P‑fibered braids as those braids that can be represented by loops of polynomials such that the corresponding function g induces a fibration arg $g : (mathbb{C} times S^1) setminus B to S^1$. We show that a certain satellite operation produces new P‑fibered braids from known ones. We also use P‑fibered braids to prove that any braid $B$ with $n$ strands, $k_{-}$ negative and $k_{+}$ positive crossings can be turned into a braid whose closure is fibered by adding at least $frac{k_{-} +1}{n}$ negative or $frac{k_{+} +1}{n}$ positive full twists to it. Using earlier constructions of P‑fibered braids we prove that every link is a sublink of a real algebraic link, i.e., a link of an isolated singularity of a polynomial map $mathbb{R}^4 to mathbb{R}^2$.
几何辫状结构 $B$ 可以解释为具有不同根的单复多项式空间中的一个环。这个循环定义了一个函数 $g ：到times S^1 to C$ 在 $B$ 上消失。我们将 P 纤维辫的集合定义为那些可以用多项式的环来表示的辫，使得相应的函数 g 可以诱导一个纤度 arg $g : (mathbb{C} times S^1) setminus B to S^1$.我们证明了某种卫星操作可以从已知的 P 纤维辫产生新的 P 纤维辫。我们还利用 P 纤维辫证明，任何具有 $n$ 股、$k_{-}$ 负交叉和 $k_{+}$ 正交叉的 $B$ 辫子，都可以通过添加至少 $frac{k_{-}+{n}$ 负交叉或 $k_{+}$ 正交叉，变成闭合是纤维的辫子。+1}{n}$ 负捻或 $frac{k_{+} +1}{n}$ 正捻。利用早先的 P 纤维辫的构造，我们证明了每个链接都是实代数链接的子链接，即多项式映射 $mathbb{R}^4 to mathbb{R}^2$的孤立奇点的链接。
{"title":"Twisting and satellite operations on P-fibered braids","authors":"Benjamin Bode","doi":"10.4310/cag.2023.v31.n8.a5","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n8.a5","url":null,"abstract":"A geometric braid $B$ can be interpreted as a loop in the space of monic complex polynomials with distinct roots. This loop defines a function $g : mathbb{C} times S^1 to C$ that vanishes on $B$. We define the set of P‑<i>fibered </i>braids as those braids that can be represented by loops of polynomials such that the corresponding function g induces a fibration arg $g : (mathbb{C} times S^1) setminus B to S^1$. We show that a certain satellite operation produces new P‑fibered braids from known ones. We also use P‑fibered braids to prove that any braid $B$ with $n$ strands, $k_{-}$ negative and $k_{+}$ positive crossings can be turned into a braid whose closure is fibered by adding at least $frac{k_{-} +1}{n}$ negative or $frac{k_{+} +1}{n}$ positive full twists to it. Using earlier constructions of P‑fibered braids we prove that every link is a sublink of a real algebraic link, i.e., a link of an isolated singularity of a polynomial map $mathbb{R}^4 to mathbb{R}^2$.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947981","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 : 2024-08-10DOI: 10.4310/cag.2023.v31.n8.a6
Romain Gicquaud
We study the prescribed scalar curvature problem, namely finding which function can be obtained as the scalar curvature of a metric in a given conformal class. We deal with the case of asymptotically hyperbolic manifolds and restrict ourselves to non positive prescribed scalar curvature. Following [$href{https://dx.doi.org/10.4310/CAG.2018.v26.n5.a5}{14}$, $href{https://doi.org/10.1090/S0002-9947-1995-1321588-5}{26}$], we obtain a necessary and sufficient condition on the zero set of the prescribed scalar curvature so that the problem admits a (unique) solution.
{"title":"Prescribed non-positive scalar curvature on asymptotically hyperbolic manifolds with application to the Lichnerowicz equation","authors":"Romain Gicquaud","doi":"10.4310/cag.2023.v31.n8.a6","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n8.a6","url":null,"abstract":"We study the prescribed scalar curvature problem, namely finding which function can be obtained as the scalar curvature of a metric in a given conformal class. We deal with the case of asymptotically hyperbolic manifolds and restrict ourselves to non positive prescribed scalar curvature. Following [$href{https://dx.doi.org/10.4310/CAG.2018.v26.n5.a5}{14}$, $href{https://doi.org/10.1090/S0002-9947-1995-1321588-5}{26}$], we obtain a necessary and sufficient condition on the zero set of the prescribed scalar curvature so that the problem admits a (unique) solution.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947982","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 : 2024-08-10DOI: 10.4310/cag.2023.v31.n8.a7
Jiawei Liu
In this paper, by using smooth approximation, we give a new proof of Donaldson’s existence conjecture that there exist conical Kähler–Einstein metrics with positive Ricci curvatures on Fano manifolds.
{"title":"On the existence of the conical Kähler–Einstein metrics on Fano manifolds","authors":"Jiawei Liu","doi":"10.4310/cag.2023.v31.n8.a7","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n8.a7","url":null,"abstract":"In this paper, by using smooth approximation, we give a new proof of Donaldson’s existence conjecture that there exist conical Kähler–Einstein metrics with positive Ricci curvatures on Fano manifolds.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141969617","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 : 2024-08-10DOI: 10.4310/cag.2023.v31.n8.a1
Lothar Schiemanowski
The regularity of limit spaces of Riemannian manifolds with $L^p$ curvature bounds, $p gt n/2$, is investigated under no apriori noncollapsing assumption. A regular subset, defined by a local volume growth condition for a limit measure, is shown to carry the structure of a Riemannian manifold. One consequence of this is a compactness theorem for Riemannian manifolds with $L^p$ curvature bounds and an a priori volume growth assumption in the pointed Cheeger–Gromov topology. A different notion of convergence is also studied, which replaces the exhaustion by balls in the pointed Cheeger–Gromov topology with an exhaustion by volume non-collapsed regions. Assuming in addition a lower bound on the Ricci curvature, the compactness theorem is extended to this topology. Moreover, we study how a convergent sequence of manifolds disconnects topologically in the limit. In two dimensions, building on results of Shioya, the structure of limit spaces is described in detail: it is seen to be a union of an incomplete Riemannian surface and $1$-dimensional length spaces.
{"title":"On limit spaces of Riemannian manifolds with volume and integral curvature bounds","authors":"Lothar Schiemanowski","doi":"10.4310/cag.2023.v31.n8.a1","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n8.a1","url":null,"abstract":"The regularity of limit spaces of Riemannian manifolds with $L^p$ curvature bounds, $p gt n/2$, is investigated under no apriori noncollapsing assumption. A regular subset, defined by a local volume growth condition for a limit measure, is shown to carry the structure of a Riemannian manifold. One consequence of this is a compactness theorem for Riemannian manifolds with $L^p$ curvature bounds and an <i>a priori</i> volume growth assumption in the pointed Cheeger–Gromov topology. A different notion of convergence is also studied, which replaces the exhaustion by balls in the pointed Cheeger–Gromov topology with an exhaustion by volume non-collapsed regions. Assuming in addition a lower bound on the Ricci curvature, the compactness theorem is extended to this topology. Moreover, we study how a convergent sequence of manifolds disconnects topologically in the limit. In two dimensions, building on results of Shioya, the structure of limit spaces is described in detail: it is seen to be a union of an incomplete Riemannian surface and $1$-dimensional length spaces.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947977","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 : 2024-08-10DOI: 10.4310/cag.2023.v31.n8.a2
Rayssa Caju, Pedro Gaspar
We prove that given a minimal hypersurface $Gamma$ in a compact Riemannian manifold without boundary, if all the Jacobi fields of $Gamma$ are generated by ambient isometries, then we can find solutions of the Allen–Cahn equation $-varepsilon^2 Delta u + W^prime (u) = 0$ on $M$, for sufficiently small $varepsilon gt 0$, whose nodal sets converge to $Gamma$. This extends the results of Pacard–Ritoré $href{https://doi.org/10.4310/jdg/1090426999}{[41]}$ (in the case of closed manifolds and zero mean curvature).
{"title":"Solutions of the Allen–Cahn equation on closed manifolds in the presence of symmetry","authors":"Rayssa Caju, Pedro Gaspar","doi":"10.4310/cag.2023.v31.n8.a2","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n8.a2","url":null,"abstract":"We prove that given a minimal hypersurface $Gamma$ in a compact Riemannian manifold without boundary, if all the Jacobi fields of $Gamma$ are generated by ambient isometries, then we can find solutions of the Allen–Cahn equation $-varepsilon^2 Delta u + W^prime (u) = 0$ on $M$, for sufficiently small $varepsilon gt 0$, whose nodal sets converge to $Gamma$. This extends the results of Pacard–Ritoré $href{https://doi.org/10.4310/jdg/1090426999}{[41]}$ (in the case of closed manifolds and zero mean curvature).","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141969771","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 : 2024-08-10DOI: 10.4310/cag.2023.v31.n8.a8
Matti Lassas, Tony Liimatainen
We study conformal harmonic coordinates on Riemannian and Lorentzian manifolds, which are coordinates constructed as quotients of solutions to the conformal Laplace equation. We show existence of conformal harmonic coordinates under general conditions and find that the coordinates are a conformal analogue of harmonic coordinates. We prove up to boundary regularity results for conformal mappings. We show that Weyl, Cotton, Bach, and Fefferman–Graham obstruction tensors are elliptic operators in conformal harmonic coordinates if one also normalizes the determinant of the metric. We give a corresponding elliptic regularity results, including the analytic case. We prove a unique continuation result for Bach and obstruction flat manifolds, which are conformally flat near a point. We prove unique continuation results for conformal mappings both on Riemannian and Lorentzian manifolds.
{"title":"Conformal harmonic coordinates","authors":"Matti Lassas, Tony Liimatainen","doi":"10.4310/cag.2023.v31.n8.a8","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n8.a8","url":null,"abstract":"We study conformal harmonic coordinates on Riemannian and Lorentzian manifolds, which are coordinates constructed as quotients of solutions to the conformal Laplace equation. We show existence of conformal harmonic coordinates under general conditions and find that the coordinates are a conformal analogue of harmonic coordinates. We prove up to boundary regularity results for conformal mappings. We show that Weyl, Cotton, Bach, and Fefferman–Graham obstruction tensors are elliptic operators in conformal harmonic coordinates if one also normalizes the determinant of the metric. We give a corresponding elliptic regularity results, including the analytic case. We prove a unique continuation result for Bach and obstruction flat manifolds, which are conformally flat near a point. We prove unique continuation results for conformal mappings both on Riemannian and Lorentzian manifolds.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947983","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 : 2024-08-10DOI: 10.4310/cag.2023.v31.n8.a3
Jaehoon Lee
In this paper, we prove that the closed Lagrangian self-shrinkers in $mathbb{R}^4$ which are symmetric with respect to a hyperplane are given by the products of Abresch–Langer curves. As a corollary, we obtain a new geometric characterization of the Clifford torus as the unique embedded closed Lagrangian self-shrinker symmetric with respect to a hyperplane in $mathbb{R}^4$.
{"title":"Closed Lagrangian self-shrinkers in $mathbb{R}^4$ symmetric with respect to a hyperplane","authors":"Jaehoon Lee","doi":"10.4310/cag.2023.v31.n8.a3","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n8.a3","url":null,"abstract":"In this paper, we prove that the closed Lagrangian self-shrinkers in $mathbb{R}^4$ which are symmetric with respect to a hyperplane are given by the products of Abresch–Langer curves. As a corollary, we obtain a new geometric characterization of the Clifford torus as the unique embedded closed Lagrangian self-shrinker symmetric with respect to a hyperplane in $mathbb{R}^4$.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947979","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 : 2024-07-29DOI: 10.4310/cag.2023.v31.n10.a5
Allen,Brian
We study the stability of the Positive Mass Theorem (PMT) in the case where a sequence of regions of manifolds with positive scalar curvature $U_{T}^{i}subset M_{i}^{3}$ are foliated by a smooth solution to Inverse Mean Curvature Flow (IMCF) which may not be uniformly controlled near the boundary. Then if $partial U_{T}^{i} = Sigma _{0}^{i} cup Sigma _{T}^{i}$, $m_{H}(Sigma _{T}^{i}) rightarrow 0$ and extra technical conditions are satisfied we show that $U_{T}^{i}$ converges to a flat annulus with respect to Sormani-Wenger Intrinsic Flat (SWIF) convergence.
{"title":"Inverse nean curvature flow and the stability of the positive mass theorem","authors":"Allen,Brian","doi":"10.4310/cag.2023.v31.n10.a5","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n10.a5","url":null,"abstract":"We study the stability of the Positive Mass Theorem (PMT) in the case where a sequence of regions of manifolds with positive scalar curvature $U_{T}^{i}subset M_{i}^{3}$ are foliated by a smooth solution to Inverse Mean Curvature Flow (IMCF) which may not be uniformly controlled near the boundary. Then if $partial U_{T}^{i} = Sigma _{0}^{i} cup Sigma _{T}^{i}$, $m_{H}(Sigma _{T}^{i}) rightarrow 0$ and extra technical conditions are satisfied we show that $U_{T}^{i}$ converges to a flat annulus with respect to Sormani-Wenger Intrinsic Flat (SWIF) convergence.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863783","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 : 2024-07-29DOI: 10.4310/cag.2023.v31.n10.a6
Raffaelli,Matteo
Given a smooth distribution $mathscr{D}$ of $m$-dimensional planes along a smooth regular curve $gamma $ in $mathbb{R}^{m+n}$, we consider the following problem: to find an $m$-dimensional rank-one submanifold of $mathbb{R}^{m+n}$, that is, an $(m-1)$-ruled submanifold with constant tangent space along the rulings, such that its tangent bundle along $gamma $ coincides with $mathscr{D}$. In particular, we give sufficient conditions for the local well-posedness of the problem, together with a parametric description of the solution.
{"title":"The geometric Cauchy problem for rank-one submanifolds","authors":"Raffaelli,Matteo","doi":"10.4310/cag.2023.v31.n10.a6","DOIUrl":"https://doi.org/10.4310/cag.2023.v31.n10.a6","url":null,"abstract":"Given a smooth distribution $mathscr{D}$ of $m$-dimensional planes along a smooth regular curve $gamma $ in $mathbb{R}^{m+n}$, we consider the following problem: to find an $m$-dimensional rank-one submanifold of $mathbb{R}^{m+n}$, that is, an $(m-1)$-ruled submanifold with constant tangent space along the rulings, such that its tangent bundle along $gamma $ coincides with $mathscr{D}$. In particular, we give sufficient conditions for the local well-posedness of the problem, together with a parametric description of the solution.","PeriodicalId":50662,"journal":{"name":"Communications in Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863784","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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