Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTopological regularity of spaces with an upper curvature bound
124
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We prove that a locally compact space with an upper curvature bound is a topological manifold if and only if all of its spaces of directions are homotopy equivalent and not contractible. We discuss applications to homology manifolds, limits of Riemannian manifolds and deduce a sphere theorem.
rate research
Read More
We study geometric and topological properties of locally compact, geodesically complete spaces with an upper curvature bound. We control the size of singular subsets, discuss homotopical and measure-theoretic stratifications and regularity of the metric structure on a large part.
Let $M^n$ be a closed immersed hypersurface lying in a contractible ball $B(p,R)$ of the ambient $(n+1)$-manifold $N^{n+1}$. We prove that, by pinching Heintze-Reillys inequality via sectional curvature upper bound of $B(p,R)$, 1st eigenvalue and mean curvature of $M$, not only $M$ is Hausdorff close to a geodesic sphere $S(p_0,R_0)$ in $N$, but also the ``enclosed ball $B(p_0,R_0)$ is close to be of constant curvature, provided with a uniform control on the volume and mean curvature of $M$. We raise a conjecture for $M$ to be a diffeomorphic sphere, and give some positive partial answer.
The aim of this paper is to give not only an explicit upper bound of the total Q-curvature but also an induced isoperimetric deficit formula for the complete conformal metrics on $mathbb R^n$, $nge 3$ with scalar curvature being nonnegative near infinity and Q-curvature being absolutely convergent.
We show that any space with a positive upper curvature bound has in a small neighborhood of any point a closely related metric with a negative upper curvature bound.
We obtain new topological information about the local structure of collapsing under a lower sectional curvature bound. As an application we prove a new sphere theorem and obtain a partial result towards the conjecture that not every Alexandrov space can be obtained as a limit of a sequence of Riemannian manifolds with sectional curvature bounded from below.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
