Pub Date : 2020-03-25DOI: 10.4310/sdg.2018.v23.n1.a2
R. Berman
This is an invitation to the probabilistic approach for constructing Kahler-Einstein metrics on complex projective algebraic manifolds X. The metrics in question emerge in the large N-limit from a canonical way of sampling N points on X, i.e. from random point processes on X, defined in terms of algebro-geometric data. The proof of the convergence towards Kahler-Einstein metrics with negative Ricci curvature is explained. In the case of positive Ricci curvature a variational approach is introduced to prove the conjectural convergence, which can be viewed as a probabilistic constructive analog of the Yau-Tian-Donaldson conjecture. The variational approach reveals, in particular, that the convergence holds under the hypothesis that there is no phase transition, which - from the algebro-geometric point of view - amounts to an analytic property of a certain Archimedean zeta function.
{"title":"An invitation to K\"ahler-Einstein metrics and random point processes","authors":"R. Berman","doi":"10.4310/sdg.2018.v23.n1.a2","DOIUrl":"https://doi.org/10.4310/sdg.2018.v23.n1.a2","url":null,"abstract":"This is an invitation to the probabilistic approach for constructing Kahler-Einstein metrics on complex projective algebraic manifolds X. The metrics in question emerge in the large N-limit from a canonical way of sampling N points on X, i.e. from random point processes on X, defined in terms of algebro-geometric data. The proof of the convergence towards Kahler-Einstein metrics with negative Ricci curvature is explained. In the case of positive Ricci curvature a variational approach is introduced to prove the conjectural convergence, which can be viewed as a probabilistic constructive analog of the Yau-Tian-Donaldson conjecture. The variational approach reveals, in particular, that the convergence holds under the hypothesis that there is no phase transition, which - from the algebro-geometric point of view - amounts to an analytic property of a certain Archimedean zeta function.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"114 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85067876","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate for the first time the curve shortening flow in the metric-affine plane and prove that under simple geometric condition it shrinks a closed convex curve to a "round point" in finite time. This generalizes the classical result by M. Gage and R.S. Hamilton about convex curves in Euclidean plane.
{"title":"The Curve Shortening Flow in the Metric-Affine Plane","authors":"V. Rovenski","doi":"10.3390/math8050701","DOIUrl":"https://doi.org/10.3390/math8050701","url":null,"abstract":"We investigate for the first time the curve shortening flow in the metric-affine plane and prove that under simple geometric condition it shrinks a closed convex curve to a \"round point\" in finite time. This generalizes the classical result by M. Gage and R.S. Hamilton about convex curves in Euclidean plane.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89452765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-03-22DOI: 10.2140/PJM.2020.309.421
Ali Maalaoui
Given a three dimensional pseudo-Einstein CR manifold $(M,T^{1,0}M,theta)$, we establish an expression for the difference of determinants of the Paneitz type operators $A_{theta}$, related to the problem of prescribing the $Q'$-curvature, under the conformal change $thetamapsto e^{w}theta$ with $win P$ the space of pluriharmonic functions. This generalizes the expression of the functional determinant in four dimensional Riemannian manifolds established in cite{BO2}. We also provide an existence result of maximizers for the scaling invariant functional determinant as in cite{CY}.
{"title":"Functional determinant on pseudo-Einstein 3-manifolds","authors":"Ali Maalaoui","doi":"10.2140/PJM.2020.309.421","DOIUrl":"https://doi.org/10.2140/PJM.2020.309.421","url":null,"abstract":"Given a three dimensional pseudo-Einstein CR manifold $(M,T^{1,0}M,theta)$, we establish an expression for the difference of determinants of the Paneitz type operators $A_{theta}$, related to the problem of prescribing the $Q'$-curvature, under the conformal change $thetamapsto e^{w}theta$ with $win P$ the space of pluriharmonic functions. This generalizes the expression of the functional determinant in four dimensional Riemannian manifolds established in cite{BO2}. We also provide an existence result of maximizers for the scaling invariant functional determinant as in cite{CY}.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"81 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72985652","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider an expanding flow of closed, smooth, uniformly convex hypersurface in Euclidean mathbb{R}^{n+1} with speed u^alpha f^beta (alpha, betainmathbb{R}^1), where u is support function of the hypersurface, f is a smooth, symmetric, homogenous of degree one, positive function of the principal curvature radii of the hypersurface. If alpha leq 0
{"title":"A class of curvature flows expanded by support function and curvature function","authors":"S. Ding, Guanghan Li","doi":"10.1090/proc/15189","DOIUrl":"https://doi.org/10.1090/proc/15189","url":null,"abstract":"In this paper, we consider an expanding flow of closed, smooth, uniformly convex hypersurface in Euclidean mathbb{R}^{n+1} with speed u^alpha f^beta (alpha, betainmathbb{R}^1), where u is support function of the hypersurface, f is a smooth, symmetric, homogenous of degree one, positive function of the principal curvature radii of the hypersurface. If alpha leq 0<betaleq 1-alpha, we prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalization, to a round sphere centered at the origin.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84505282","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-03-14DOI: 10.7546/CRABS.2020.11.03
G. Nakova
We introduce a class of half lightlike submanifolds of almost contact B-metric manifolds and prove that such submanifolds are semi-Riemannian with respect to the associated B-metric. Object of investigations are also minimal of the considered submanifolds and a non-trivial example for them is given.
{"title":"Radical Screen Transversal Half Lightlike Submanifolds of Almost Contact B-metric Manifolds","authors":"G. Nakova","doi":"10.7546/CRABS.2020.11.03","DOIUrl":"https://doi.org/10.7546/CRABS.2020.11.03","url":null,"abstract":"We introduce a class of half lightlike submanifolds of almost contact B-metric manifolds and prove that such submanifolds are semi-Riemannian with respect to the associated B-metric. Object of investigations are also minimal of the considered submanifolds and a non-trivial example for them is given.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"80 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79318444","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-03-12DOI: 10.1007/978-3-030-03009-4_87-1
Martin Bauer, N. Charon, E. Klassen, Alice Le Brigant
{"title":"Intrinsic Riemannian Metrics on Spaces of Curves: Theory and Computation","authors":"Martin Bauer, N. Charon, E. Klassen, Alice Le Brigant","doi":"10.1007/978-3-030-03009-4_87-1","DOIUrl":"https://doi.org/10.1007/978-3-030-03009-4_87-1","url":null,"abstract":"","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"375 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76897507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the radial parts of the Brownian motions on K"ahler and quaternion K"ahler manifolds. Thanks to sharp Laplacian comparison theorems, we deduce as a consequence a sharp Cheeger-Yau type lower bound for the heat kernels of such manifolds and also sharp Cheng's type estimates for the Dirichlet eigenvalues of metric balls.
{"title":"Brownian Motions and Heat Kernel Lower Bounds on Kähler and Quaternion Kähler Manifolds","authors":"Fabrice Baudoin, Guang Yang","doi":"10.1093/imrn/rnaa199","DOIUrl":"https://doi.org/10.1093/imrn/rnaa199","url":null,"abstract":"We study the radial parts of the Brownian motions on K\"ahler and quaternion K\"ahler manifolds. Thanks to sharp Laplacian comparison theorems, we deduce as a consequence a sharp Cheeger-Yau type lower bound for the heat kernels of such manifolds and also sharp Cheng's type estimates for the Dirichlet eigenvalues of metric balls.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76255377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A consequence of the Cheeger-Gromoll splitting theorem states that for any open manifold $(M,x)$ of nonnegative Ricci curvature, if all the minimal geodesic loops at $x$ that represent elements of $pi_1(M,x)$ are contained in a bounded ball, then $pi_1(M,x)$ is virtually abelian. We generalize the above result: if these minimal representing geodesic loops of $pi_1(M,x)$ escape from any bounded metric balls at a sublinear rate with respect to their lengths, then $pi_1(M,x)$ is virtually abelian.
{"title":"On the escape rate of geodesic loops in an open manifold with nonnegative Ricci curvature","authors":"Jiayin Pan","doi":"10.2140/GT.2021.25.1059","DOIUrl":"https://doi.org/10.2140/GT.2021.25.1059","url":null,"abstract":"A consequence of the Cheeger-Gromoll splitting theorem states that for any open manifold $(M,x)$ of nonnegative Ricci curvature, if all the minimal geodesic loops at $x$ that represent elements of $pi_1(M,x)$ are contained in a bounded ball, then $pi_1(M,x)$ is virtually abelian. We generalize the above result: if these minimal representing geodesic loops of $pi_1(M,x)$ escape from any bounded metric balls at a sublinear rate with respect to their lengths, then $pi_1(M,x)$ is virtually abelian.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84077306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","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：