{"title":"Bundles with Non-multiplicativeÂ-Genus and Spaces of Metrics with Lower Curvature Bounds","authors":"Georg Frenck, Jens Reinhold","doi":"10.1093/IMRN/RNAA361","DOIUrl":"https://doi.org/10.1093/IMRN/RNAA361","url":null,"abstract":"We construct smooth bundles with base and fiber products of two spheres whose total spaces have non-vanishing $hat{A}$-genus. We then use these bundles to locate non-trivial rational homotopy groups of spaces of Riemannian metrics with lower curvature bounds for all Spin-manifolds of dimension six or at least ten which admit such a metric and are a connected sum of some manifold and $S^n times S^n$ or $S^n times S^{n+1}$, respectively. We also construct manifolds $M$ whose spaces of Riemannian metrics of positive scalar curvature have homotopy groups that contain elements of infinite order which lie in the image of the orbit map induced by the push-forward action of the diffeomorphism group of $M$.","PeriodicalId":8430,"journal":{"name":"arXiv: Differential Geometry","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86060733","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}
{"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}
{"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}
{"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}
{"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}
{"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}