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Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball 球中具有自由边界的凸超曲面的Alexandrov-Fenchel不等式
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-11-14 DOI: 10.4310/jdg/1645207496
Julian Scheuer, Guofang Wang, C. Xia
{"title":"Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball","authors":"Julian Scheuer, Guofang Wang, C. Xia","doi":"10.4310/jdg/1645207496","DOIUrl":"https://doi.org/10.4310/jdg/1645207496","url":null,"abstract":"In this paper we first introduce quermassintegrals for free boundary hypersurfaces in the $(n+1)$-dimensional Euclidean unit ball. Then we solve some related isoperimetric type problems for convex free boundary hypersurfaces, which lead to new Alexandrov-Fenchel inequalities. In particular, for $n=2$ we obtain a Minkowski-type inequality and for $n=3$ we obtain an optimal Willmore-type inequality. To prove these estimates, we employ a specifically designed locally constrained inverse harmonic mean curvature flow with free boundary.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42972548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 23
Existence of solutions to the even dual Minkowski problem 偶对偶Minkowski问题解的存在性
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-11-01 DOI: 10.4310/JDG/1542423629
Yiming Zhao
{"title":"Existence of solutions to the even dual Minkowski problem","authors":"Yiming Zhao","doi":"10.4310/JDG/1542423629","DOIUrl":"https://doi.org/10.4310/JDG/1542423629","url":null,"abstract":"Recently, Huang, Lutwak, Yang & Zhang discovered the duals of Federer’s curvature measures within the dual Brunn-Minkowski theory and stated the “Minkowski problem” associated with these new measures. As they showed, this dual Minkowski problem has as special cases the Aleksandrov problem (when the index is 0) and the logarithmic Minkowski problem (when the index is the dimension of the ambient space) — two problems that were never imagined to be connected in any way. Huang, Lutwak, Yang & Zhang established sufficient conditions to guarantee existence of solution to the dual Minkowski problem in the even setting. In this work, existence of solution to the even dual Minkowski problem is established under new sufficiency conditions. It was recently shown by Böröczky, Henk & Pollehn that these new sufficiency conditions are also necessary.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4310/JDG/1542423629","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41599438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 90
Local entropy and generic multiplicity one singularities of mean curvature flow of surfaces 曲面平均曲率流的局部熵和一般多重性一奇点
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-10-18 DOI: 10.4310/jdg/1685121322
Ao Sun
{"title":"Local entropy and generic multiplicity one singularities of mean curvature flow of surfaces","authors":"Ao Sun","doi":"10.4310/jdg/1685121322","DOIUrl":"https://doi.org/10.4310/jdg/1685121322","url":null,"abstract":"In this paper we prove that the generic singularity of mean curvature flow of closed embedded surfaces in $mathbb R^3$ modelled by closed self-shrinkers with multiplicity has multiplicity one. Together with the previous result by Colding-Minicozzi in [CM12], we conclude that the only generic singularity of mean curvature flow of closed embedded surfaces in $mathbb R^3$ modelled by closed self-shrinkers is a multiplicity one sphere. We also construct particular perturbation of the flow to avoid those singularities with multiplicity higher than one. Our result partially addresses the well-known multiplicity one conjecture by Ilmanen.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44989525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 42
Unfolded Seiberg–Witten Floer spectra, II: Relative invariants and the gluing theorem 展开Seiberg-Witten Floer谱，II:相对不变量和胶合定理
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-09-24 DOI: 10.4310/jdg/1686931602
Tirasan Khandhawit, Jianfeng Lin, H. Sasahira
{"title":"Unfolded Seiberg–Witten Floer spectra, II: Relative invariants and the gluing theorem","authors":"Tirasan Khandhawit, Jianfeng Lin, H. Sasahira","doi":"10.4310/jdg/1686931602","DOIUrl":"https://doi.org/10.4310/jdg/1686931602","url":null,"abstract":"We use the construction of unfolded Seiberg-Witten Floer spectra of general 3-manifolds defined in our previous paper to extend the notion of relative Bauer-Furuta invariants to general 4-manifolds with boundary. One of the main purposes of this paper is to give a detailed proof of the gluing theorem for the relative invariants.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48752093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
Adiabatic limits of anti-self-dual connections on collapsed $K3$ surfaces 坍塌$K3$表面上反自对偶连接的绝热极限
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-09-23 DOI: 10.4310/JDG/1622743140
V. Datar, Adam Jacob, Yuguang Zhang
{"title":"Adiabatic limits of anti-self-dual connections on collapsed $K3$ surfaces","authors":"V. Datar, Adam Jacob, Yuguang Zhang","doi":"10.4310/JDG/1622743140","DOIUrl":"https://doi.org/10.4310/JDG/1622743140","url":null,"abstract":"We prove a convergence result for a family of Yang-Mills connections over an elliptic $K3$ surface $M$ as the fibers collapse. In particular, assume $M$ is projective, admits a section, and has singular fibers of Kodaira type $I_1$ and type $II$. Let $Xi_{t_k}$ be a sequence of $SU(n)$ connections on a principal $SU(n)$ bundle over $M$, that are anti-self-dual with respect to a sequence of Ricci flat metrics collapsing the fibers of $M$. Given certain non-degeneracy assumptions on the spectral covers induced by $barpartial_{Xi_{t_k}}$, we show that away from a finite number of fibers, the curvature $F_{Xi_{t_k}}$ is locally bounded in $C^0$, the connections converge along a subsequence (and modulo unitary gauge change) in $L^p_1$ to a limiting $L^p_1$ connection $Xi_0$, and the restriction of $Xi_0$ to any fiber is $C^{1,alpha}$ gauge equivalent to a flat connection with holomorphic structure determined by the sequence of spectral covers. Additionally, we relate the connections $Xi_{t_k}$ to a converging family of special Lagrangian multi-sections in the mirror HyperK\"ahler structure, addressing a conjecture of Fukaya in this setting.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42240777","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Improvements for eigenfunction averages: An application of geodesic beams 本征函数平均值的改进：测地光束的应用
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-09-17 DOI: 10.4310/jdg/1689262062
Y. Canzani, J. Galkowski
{"title":"Improvements for eigenfunction averages: An application of geodesic beams","authors":"Y. Canzani, J. Galkowski","doi":"10.4310/jdg/1689262062","DOIUrl":"https://doi.org/10.4310/jdg/1689262062","url":null,"abstract":"Let $(M,g)$ be a smooth, compact Riemannian manifold and ${phi_lambda }$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-Delta_gphi_lambda =lambda^2 phi_lambda$. Given a smooth submanifold $H subset M$ of codimension $kgeq 1$, we find conditions on the pair $(M,H)$, even when $H={x}$, for which $$ Big|int_Hphi_lambda dsigma_HBig|=OBig(frac{lambda^{frac{k-1}{2}}}{sqrt{log lambda}}Big)qquad text{or}qquad |phi_lambda(x)|=OBig(frac{lambda ^{frac{n-1}{2}}}{sqrt{log lambda}}Big), $$ as $lambdato infty$. These conditions require no global assumption on the manifold $M$ and instead relate to the structure of the set of recurrent directions in the unit normal bundle to $H$. Our results extend all previously known conditions guaranteeing improvements on averages, including those on sup-norms. For example, we show that if $(M,g)$ is a surface with Anosov geodesic flow, then there are logarithmically improved averages for any $Hsubset M$. We also find weaker conditions than having no conjugate points which guarantee $sqrt{log lambda}$ improvements for the $L^infty$ norm of eigenfunctions. Our results are obtained using geodesic beam techniques, which yield a mechanism for obtaining general quantitative improvements for averages and sup-norms.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42279484","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
Closed geodesics on connected sums and $3$-manifolds 连通和和$3$-流形上的闭测地线
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-09-12 DOI: 10.4310/jdg/1649953350
H. Rademacher, I. Taimanov
{"title":"Closed geodesics on connected sums and $3$-manifolds","authors":"H. Rademacher, I. Taimanov","doi":"10.4310/jdg/1649953350","DOIUrl":"https://doi.org/10.4310/jdg/1649953350","url":null,"abstract":"We study the asymptotics of the number N(t) of geometrically distinct closed geodesics of a Riemannian or Finsler metric on a connected sum of two compact manifolds of dimension at least three with non-trivial fundamental groups and apply this result to the prime decomposition of a three-manifold. In particular we show that the function N(t) grows at least like the prime numbers on a compact 3-manifold with infinite fundamental group. It follows that a generic Riemannian metric on a compact 3-manifold has infinitely many geometrically distinct closed geodesics. We also consider the case of a connected sum of a compact manifold with positive first Betti number and a simply-connected manifold which is not homeomorphic to a sphere.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47842181","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
An optimal $L^2$ extension theorem on weakly pseudoconvex Kähler manifolds 弱伪凸Kähler流形上的最优L^2扩展定理
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-09-01 DOI: 10.4310/JDG/1536285628
Xiangyu Zhou, Langfeng Zhu
{"title":"An optimal $L^2$ extension theorem on weakly pseudoconvex Kähler manifolds","authors":"Xiangyu Zhou, Langfeng Zhu","doi":"10.4310/JDG/1536285628","DOIUrl":"https://doi.org/10.4310/JDG/1536285628","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"1 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4310/JDG/1536285628","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70470792","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 43
The $L_p$-Aleksandrov problem for $L_p$-integral curvature $L_p$积分曲率的$L_p$-Alekandrov问题
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-09-01 DOI: 10.4310/JDG/1536285625
Yong Huang, E. Lutwak, Deane Yang, Gaoyong Zhang
{"title":"The $L_p$-Aleksandrov problem for $L_p$-integral curvature","authors":"Yong Huang, E. Lutwak, Deane Yang, Gaoyong Zhang","doi":"10.4310/JDG/1536285625","DOIUrl":"https://doi.org/10.4310/JDG/1536285625","url":null,"abstract":"It is shown that within the Lp-Brunn–Minkowski theory that Aleksandrov’s integral curvature has a natural Lp extension, for all real p. This raises the question of finding necessary and sufficient conditions on a given measure in order for it to be the Lp-integral curvature of a convex body. This problem is solved for positive p and is answered for negative p provided the given measure is even.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4310/JDG/1536285625","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49578384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 47
A note on Selberg’s lemma and negatively curved Hadamard manifolds 关于Selberg引理和负弯曲Hadamard流形的注解
IF 2.5 1区 数学
Journal of Differential Geometry Pub Date : 2018-08-05 DOI: 10.4310/jdg/1649953550
M. Kapovich
{"title":"A note on Selberg’s lemma and negatively curved Hadamard manifolds","authors":"M. Kapovich","doi":"10.4310/jdg/1649953550","DOIUrl":"https://doi.org/10.4310/jdg/1649953550","url":null,"abstract":"Author(s): Kapovich, Michael | Abstract: Answering a question by Margulis we prove that the conclusion of Selberg's Lemma fails for discrete isometry groups of negatively curved Hadamard manifolds.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43789812","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
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