1. CMB Online first
||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
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
2. CMB 2011 (vol 55 pp. 723)
||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
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
3. CMB 2004 (vol 47 pp. 492)