ترغب بنشر مسار تعليمي؟ اضغط هنا

The Fundamental Groups of Open Manifolds with Nonnegative Ricci Curvature

122   0   0.0 ( 0 )
 نشر من قبل Jiayin Pan
 تاريخ النشر 2020
  مجال البحث
والبحث باللغة English
 تأليف Jiayin Pan




اسأل ChatGPT حول البحث

We survey the results on fundamental groups of open manifolds with nonnegative Ricci curvature. We also present some open questions on this topic.



قيم البحث

اقرأ أيضاً

236 - Michael Munn 2009
Let $M^n$ be a complete, open Riemannian manifold with $Ric geq 0$. In 1994, Grigori Perelman showed that there exists a constant $delta_{n}>0$, depending only on the dimension of the manifold, such that if the volume growth satisfies $alpha_M := lim _{r to infty} frac{Vol(B_p(r))}{omega_n r^n} geq 1-delta_{n}$, then $M^n$ is contractible. Here we employ the techniques of Perelman to find specific lower bounds for the volume growth, $alpha(k,n)$, depending only on $k$ and $n$, which guarantee the individual $k$-homotopy group of $M^n$ is trivial.
148 - Xian-Tao Huang 2021
Suppose $(M,g)$ is a Riemannian manifold having dimension $n$, nonnegative Ricci curvature, maximal volume growth and unique tangent cone at infinity. In this case, the tangent cone at infinity $C(X)$ is an Euclidean cone over the cross-section $X$. Denote by $alpha=lim_{rrightarrowinfty}frac{mathrm{Vol}(B_{r}(p))}{r^{n}}$ the asymptotic volume ratio. Let $h_{k}=h_{k}(M)$ be the dimension of the space of harmonic functions with polynomial growth of growth order at most $k$. In this paper, we prove a upper bound of $h_{k}$ in terms of the counting function of eigenvalues of $X$. As a corollary, we obtain $lim_{krightarrowinfty}k^{1-n}h_{k}=frac{2alpha}{(n-1)!omega_{n}}$. These results are sharp, as they recover the corresponding well-known properties of $h_{k}(mathbb{R}^{n})$. In particular, these results hold on manifolds with nonnegative sectional curvature and maximal volume growth.
165 - Jon Wolfson 2019
Let M be a Riemannian n-manifold with n greater than or equal to 3. For k between 1 and n, we say M has k-positive Ricci curvature if at every point of M the sum of any k eigenvalues of the Ricci curvature is strictly positive. In particular, one pos itive Ricci curvature is equivalent to positive Ricci curvature and n-positive Ricci curvature is equivalent to positive scalar curvature. Let G be the fundamental group of the closed manifold M. We say that G is virtually free if G contains a free subgroup of finite index, or equivalently, if some finite cover of M has a fundamental group that is a free group. In this paper we will prove: Let M be a closed Riemannian n-manifold, with n greater than or equal to 3, such that (n-1)-eigenvalues of the Ricci curvature are strictly positive. Then the fundamental group of M is virtually free. As an immediate consequence we have: Let M be a closed Riemannian n-manifold, with n greater than or equal to 3, with 2-positive Ricci curvature. Then the fundamental group of M is virtually free.
116 - Huihong Jiang , Yi-Hu Yang 2019
In a previous paper, we constructed complete manifolds of positive Ricci curvature with quadratically asymptotically nonnegative curvature and infinite topological type but dimension $ge 6$. The purpose of the present paper is to use a different way to exhibit a family of complete $I$-dimensinal ($Ige5$) Riemannian manifolds of positive Ricci curvature, quadratically asymptotically nonnegative sectional curvature, and certain infinite Betti number $b_j$ ($2le jle I-2$).
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا