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Register a new userThe Soul Conjecture in Alexandrov Geometry in dimension 4
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In this paper, we prove the Soul Conjecture in Alexandrov geometry in dimension $4$, i.e. if $X$ is a complete non-compact $4$-dimensional Alexandrov space of non-negative curvature and positive curvature around one point, then a soul of $X$ is a point.
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Given two $n_i$-dimensional Alexandrov spaces $X_i$ of curvature $ge 1$, the join of $X_1$ and $X_2$ is an $(n_1+n_2+1)$-dimensional Alexandrov space $X$ of curvature $ge 1$, which contains $X_i$ as convex subsets such that their points are $frac pi2$ apart. If a group acts isometrically on a join that preserves $X_i$, then the orbit space is called quotient of join. We show that an $n$-dimensional Alexandrov space $X$ with curvature $ge 1$ is isometric to a finite quotient of join, if $X$ contains two compact convex subsets $X_i$ without boundary such that $X_1$ and $X_2$ are at least $frac pi2$ apart and $dim(X_1)+dim(X_2)=n-1$.
Positively curved Alexandrov spaces of dimension 4 with an isometric circle action are classified up to equivariant homeomorphism, subject to a certain additional condition on the infinitesimal geometry near fixed points which we conjecture is always satisfied. As a corollary, positively curved Riemannian orbifolds of dimension 4 with an isometric circle action are also classified.
We show that every finite-dimensional Alexandrov space X with curvature bounded from below embeds canonically into a product of an Alexandrov space with the same curvature bound and a Euclidean space such that each affine function on X comes from an affine function on the Euclidean space.
We show that, in the sense of Baire category, most Alexandrov surfaces with curvature bounded below by $kappa$ have no conical points. We use this result to prove that at most points of such surfaces, the lower and the upper Gaussian curvatures are equal to $kappa$ and $infty$ respectively.
Let $(X,d)$ be an $n$-dimensional Alexandrov space whose Hausdorff measure $mathcal{H}^n$ satisfies a condition giving the metric measure space $(X,d,mathcal{H}^n)$ a notion of having nonnegative Ricci curvature. We examine the influence of large volume growth on these spaces and generalize some classical arguments from Riemannian geometry showing that when the volume growth is sufficiently large, then $(X,d,mathcal{H}^n)$ has finite topological type.
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