Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRemark on an inequality for closed hypersurfaces in complete manifolds with nonnegative Ricci curvature
89
0
0.0
(
0
)
Authors
Xiaodong Wang
Ask ChatGPT about the research
No Arabic abstract
We give a simple proof of a recent result due to Agostiniani, Fogagnolo and Mazzieri.
rate research
Read More
In this paper we provide some local and global splitting results on complete Riemannian manifolds with nonnegative Ricci curvature. We achieve the splitting through the analysis of some pointwise inequalities of Modica type which hold true for every bounded solution to a semilinear Poisson equation. More precisely, we prove that the existence of a nonconstant bounded solution $u$ for which one of the previous inequalities becomes an equality at some point leads to the splitting results as well as to a classification of such a solution $u$.
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.
We survey the results on fundamental groups of open manifolds with nonnegative Ricci curvature. We also present some open questions on this topic.
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.
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 positive 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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
