Pub Date : 2020-11-23DOI: 10.1142/S0219887821500766
Yong Wang
In this paper, we introduce semi-symmetric metric Koszul forms and semi-symmetric non-metric Koszul forms on singular semi-Riemannian manifolds. Semi-symmetric metric Koszul forms and semi-symmetric non-metric Koszul forms and their curvature of semi-regular warped products are expressed in terms of those of the factor manifolds. We also introduce Koszul forms associated to the almost product structure on singular almost product semi-Riemannian manifolds. Koszul forms associated to the almost product structure and their curvature of semi-regular almost product warped products are expressed in terms of those of the factor manifolds. Furthermore, we generalize the results in cite{St2} to singular multiply warped products.
{"title":"Affine connections on singular warped products","authors":"Yong Wang","doi":"10.1142/S0219887821500766","DOIUrl":"https://doi.org/10.1142/S0219887821500766","url":null,"abstract":"In this paper, we introduce semi-symmetric metric Koszul forms and semi-symmetric non-metric Koszul forms on singular semi-Riemannian manifolds. Semi-symmetric metric Koszul forms and semi-symmetric non-metric Koszul forms and their curvature of semi-regular warped products are expressed in terms of those of the factor manifolds. We also introduce Koszul forms associated to the almost product structure on singular almost product semi-Riemannian manifolds. Koszul forms associated to the almost product structure and their curvature of semi-regular almost product warped products are expressed in terms of those of the factor manifolds. Furthermore, we generalize the results in cite{St2} to singular multiply warped products.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89586655","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 : 2020-11-17DOI: 10.1142/S0129167X2150049X
M. Domínguez-Vázquez, Olga Perez-Barral
We complete the classification of ruled real hypersurfaces with shape operator of constant norm in nonflat complex space forms by showing the existence of a unique inhomogeneous example in the complex hyperbolic space.
{"title":"Existence and uniqueness of inhomogeneous ruled hypersurfaces with shape operator of constant norm in the complex hyperbolic space","authors":"M. Domínguez-Vázquez, Olga Perez-Barral","doi":"10.1142/S0129167X2150049X","DOIUrl":"https://doi.org/10.1142/S0129167X2150049X","url":null,"abstract":"We complete the classification of ruled real hypersurfaces with shape operator of constant norm in nonflat complex space forms by showing the existence of a unique inhomogeneous example in the complex hyperbolic space.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"2014 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87748892","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 obtain a comparison of Steklov eigenvalues and Laplacian eigenvalues on graphs and discuss its rigidity and some of its applications.
{"title":"Comparison of Steklov eigenvalues and Laplacian eigenvalues on graphs","authors":"Yongjie Shi, Chengjie Yu","doi":"10.1090/proc/15866","DOIUrl":"https://doi.org/10.1090/proc/15866","url":null,"abstract":"In this paper, we obtain a comparison of Steklov eigenvalues and Laplacian eigenvalues on graphs and discuss its rigidity and some of its applications.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86497449","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 establish lower bound for the first nonzero eigenvalue of the Laplacian on a closed K"ahler manifold in terms of dimension, diameter, and lower bounds of holomorphic sectional curvature and orthogonal Ricci curvature. On compact K"ahler manifolds with boundary, we prove lower bounds for the first nonzero Neumann or Dirichlet eigenvalue in terms of geometric data. Our results are K"ahler analogues of well-known results for Riemannian manifolds.
{"title":"Lower bounds for the first eigenvalue of the Laplacian on Kähler manifolds","authors":"Xiaolong Li, Kui Wang","doi":"10.1090/tran/8434","DOIUrl":"https://doi.org/10.1090/tran/8434","url":null,"abstract":"We establish lower bound for the first nonzero eigenvalue of the Laplacian on a closed K\"ahler manifold in terms of dimension, diameter, and lower bounds of holomorphic sectional curvature and orthogonal Ricci curvature. On compact K\"ahler manifolds with boundary, we prove lower bounds for the first nonzero Neumann or Dirichlet eigenvalue in terms of geometric data. Our results are K\"ahler analogues of well-known results for Riemannian manifolds.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78915854","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 short note we prove that a Kahler manifold with lower Ricci curvature bound and almost maximal volume is Gromov-Hausdorff close to the projective space with the Fubini-Study metric. This is done by combining the recent results of Kewei Zhang and Yuchen Liu on holomorphic rigidity of such Kahler manifolds with the structure theorem of Tian-Wang for almost Einstein manifolds. This can be regarded as the complex analog of the result on Colding on the shape of Riemannian manifolds with almost maximal volume
{"title":"Metric rigidity of Kähler manifolds with lower Ricci bounds and almost maximal volume","authors":"V. Datar, H. Seshadri, Jian Song","doi":"10.1090/PROC/15473","DOIUrl":"https://doi.org/10.1090/PROC/15473","url":null,"abstract":"In this short note we prove that a Kahler manifold with lower Ricci curvature bound and almost maximal volume is Gromov-Hausdorff close to the projective space with the Fubini-Study metric. This is done by combining the recent results of Kewei Zhang and Yuchen Liu on holomorphic rigidity of such Kahler manifolds with the structure theorem of Tian-Wang for almost Einstein manifolds. This can be regarded as the complex analog of the result on Colding on the shape of Riemannian manifolds with almost maximal volume","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82032243","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 introduce Condition W $,$(1.2) for a smooth differential form $omega$ on a complete noncompact Riemannian manifold $M$. We prove that $omega$ is a harmonic form on $M$ if and only if $omega$ is both closed and co-closed on $M, ,$ where $omega$ has $2$-balanced growth either for $q=2$, or for $1 < q(ne 2) < 3, $ with $omega$ satisfying Condition W $,$(1.2). In particular, every $L^2$ harmonic form, or every $L^q$ harmonic form, $1
{"title":"Growth estimates for generalized harmonic\u0000 forms on noncompact manifolds with geometric\u0000 applications","authors":"S. Wei","doi":"10.1090/conm/756/15215","DOIUrl":"https://doi.org/10.1090/conm/756/15215","url":null,"abstract":"We introduce Condition W $,$(1.2) for a smooth differential form $omega$ on a complete noncompact Riemannian manifold $M$. We prove that $omega$ is a harmonic form on $M$ if and only if $omega$ is both closed and co-closed on $M, ,$ where $omega$ has $2$-balanced growth either for $q=2$, or for $1 < q(ne 2) < 3, $ with $omega$ satisfying Condition W $,$(1.2). In particular, every $L^2$ harmonic form, or every $L^q$ harmonic form, $1<q(ne 2)<3, $ satisfying Condition W $,$(1.2) is both closed and co-closed (cf. Theorem 1.1). This generalizes the work of A. Andreotti and E. Vesentini [AV] for every $L^2$ harmonic form $omega, .$ In extending $omega$ in $L^2$ to $L^q$, for $q ne 2$, Condition W $,$(1.2) has to be imposed due to counter-examples of D. Alexandru-Rugina$big($ [AR] p. 81, Remarque 3$big).$ We then study nonlinear partial differential inequalities for differential forms $ langleomega, Delta omegarangle ge 0, $ in which solutions $omega$ can be viewed as generalized harmonic forms. We prove that under the same growth assumption on $omega, $ (as in Theorem 1.1, or 1.2, or 1.3), the following six statements: (i) $langleomega, Delta omegarangle ge 0, ,$ (ii) $Delta omega = 0, ,$ $($iii$)$$quad d, omega = d^{star}omega = 0, ,$ (iv) $langle star, omega, Delta star, omegarangle ge 0, ,$ (v) $Delta star, omega = 0, ,$ and (vi) $d, star, omega = d^{star} star, omega = 0, $ are equivalent (cf. Theorem 4.1). We also study As geometric applications, we employ the theory in [DW] and [W3], solve constant Dirichlet problems for generalized harmonic $1$-forms and $F$-harmomic maps (cf. Theorems 10.3 and 10.2), derive monotonicity formulas for $2$-balanced solutions, and vanishing theorems for $2$-moderate solutions of $langleomega, Delta omegarangle ge 0, $ on $M$ (cf. Theorem 8.2 and Theorem 9.3).","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84402831","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}
For an $H>0$ rotationally symmetric embedded torus $N_{0} subset mathbb{R}^{3}$, evolved by Inverse Mean Curvature Flow, we show that the total curvature $|A|$ remains bounded up to the singular time $T_{max}$. We then show convergence of the $N_{t}$ to a $C^{1}$ rotationally symmetric embedded torus $N_{T_{max}}$ as $t rightarrow T_{max}$ without rescaling. Later, we observe a scale-invariant $L^{2}$ energy estimate on any embedded solution of the flow in $mathbb{R}^{3}$ that may be useful in ruling out curvature blowup near singularities in general.
{"title":"The Limit of The Inverse Mean Curvature Flow on a Torus","authors":"Brian Harvie","doi":"10.1090/proc/15812","DOIUrl":"https://doi.org/10.1090/proc/15812","url":null,"abstract":"For an $H>0$ rotationally symmetric embedded torus $N_{0} subset mathbb{R}^{3}$, evolved by Inverse Mean Curvature Flow, we show that the total curvature $|A|$ remains bounded up to the singular time $T_{max}$. We then show convergence of the $N_{t}$ to a $C^{1}$ rotationally symmetric embedded torus $N_{T_{max}}$ as $t rightarrow T_{max}$ without rescaling. Later, we observe a scale-invariant $L^{2}$ energy estimate on any embedded solution of the flow in $mathbb{R}^{3}$ that may be useful in ruling out curvature blowup near singularities in general.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"165 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78018552","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 give a sufficient condition for a positive constant scalar curvature metric on a manifold with boundary to be a relative Yamabe metric, which is a natural relative version of the classical Yamabe metric. We also give examples of non-Einstein relative Yamabe metrics with positive scalar curvature.
{"title":"Non-Einstein relative Yamabe metrics","authors":"Shota Hamanaka","doi":"10.2996/kmj44202","DOIUrl":"https://doi.org/10.2996/kmj44202","url":null,"abstract":"In this paper, we give a sufficient condition for a positive constant scalar curvature metric on a manifold with boundary to be a relative Yamabe metric, which is a natural relative version of the classical Yamabe metric. We also give examples of non-Einstein relative Yamabe metrics with positive scalar curvature.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"187 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74426428","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 construct a family of flat isotropic non-homogeneous tori in $mathbb{H}^n$ and $mathbb{C} mathrm{P}^{2n+1}$ and find necessary and sufficient conditions for their Hamiltonian minimality.
{"title":"О гамильтоновой минимальностиизотропных неоднородных\u0000торов в $mathbb H^n$ и $mathbb Cmathrm P^{2n+1}$","authors":"М А Овчаренко, Mikhail Aleksandrovich Ovcharenko","doi":"10.4213/MZM12475","DOIUrl":"https://doi.org/10.4213/MZM12475","url":null,"abstract":"We construct a family of flat isotropic non-homogeneous tori in $mathbb{H}^n$ and $mathbb{C} mathrm{P}^{2n+1}$ and find necessary and sufficient conditions for their Hamiltonian minimality.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"19 1","pages":"119-129"},"PeriodicalIF":0.0,"publicationDate":"2020-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73232328","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 construct a one-parameter family of complete metrics of Horowitz-Myers type with the negative constant scalar curvature. We also verify a positive energy conjecture of Horowitz-Myers for these metrics.
{"title":"Metrics of Horowitz–Myers type with the negative constant scalar curvature","authors":"Zhuobin Liang, Xiao Zhang","doi":"10.1063/5.0032241","DOIUrl":"https://doi.org/10.1063/5.0032241","url":null,"abstract":"We construct a one-parameter family of complete metrics of Horowitz-Myers type with the negative constant scalar curvature. We also verify a positive energy conjecture of Horowitz-Myers for these metrics.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77585497","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}
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