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Search: MSC category 53C23 ( Global geometric and topological methods (a la Gromov); differential geometric analysis on metric spaces )

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1. CMB Online first

Bačák, Miroslav; Kovalev, Leonid V.
Lipschitz retractions in Hadamard spaces via gradient flow semigroups
Let $X(n),$ for $n\in\mathbb{N},$ be the set of all subsets of a metric space $(X,d)$ of cardinality at most $n.$ The set $X(n)$ equipped with the Hausdorff metric is called a finite subset space. In this paper we are concerned with the existence of Lipschitz retractions $r\colon X(n)\to X(n-1)$ for $n\ge2.$ It is known that such retractions do not exist if $X$ is the one-dimensional sphere. On the other hand L. Kovalev has recently established their existence in case $X$ is a Hilbert space and he also posed a question as to whether or not such Lipschitz retractions exist for $X$ being a Hadamard space. In the present paper we answer this question in the positive.

Keywords:finite subset space, gradient flow, Hadamard space, Lie-Trotter-Kato formula, Lipschitz retraction
Categories:53C23, 47H20, 54E40, 58D07

2. CMB 2015 (vol 58 pp. 787)

Kitabeppu, Yu; Lakzian, Sajjad
Non-branching RCD$(0,N)$ Geodesic Spaces with Small Linear Diameter Growth have Finitely Generated Fundamental Groups
In this paper, we generalize the finite generation result of Sormani to non-branching $RCD(0,N)$ geodesic spaces (and in particular, Alexandrov spaces) with full support measures. This is a special case of the Milnor's Conjecture for complete non-compact $RCD(0,N)$ spaces. One of the key tools we use is the Abresch-Gromoll type excess estimates for non-smooth spaces obtained by Gigli-Mosconi.

Keywords:Milnor conjecture, non negative Ricci curvature, curvature dimension condition, finitely generated, fundamental group, infinitesimally Hilbertian
Categories:53C23, 30L99

3. CMB 2011 (vol 55 pp. 723)

Gigli, Nicola; Ohta, Shin-Ichi
First Variation Formula in Wasserstein Spaces over Compact Alexandrov Spaces
We extend results proved by the second author (Amer. J. Math., 2009) for nonnegatively curved Alexandrov spaces to general compact Alexandrov spaces $X$ with curvature bounded below. The gradient flow of a geodesically convex functional on the quadratic Wasserstein space $(\mathcal P(X),W_2)$ satisfies the evolution variational inequality. Moreover, the gradient flow enjoys uniqueness and contractivity. These results are obtained by proving a first variation formula for the Wasserstein distance.

Keywords:Alexandrov spaces, Wasserstein spaces, first variation formula, gradient flow
Categories:53C23, 28A35, 49Q20, 58A35

4. CMB 2004 (vol 47 pp. 492)

Boumuki, Nobutaka
Isotropic Immersions with Low Codimension of Complex Space Forms into Real Space Forms
The main purpose of this paper is to determine isotropic immersions of complex space forms into real space forms with low codimension. This is an improvement of a result of S. Maeda.

Categories:53B25, 53C235

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