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A survey on Inverse mean curvature flow in ROSSes

100%
Pipoli G.
Complex Manifolds
|
2017
|
tom 4
|
nr 1
245-262
EN
In this survey we discuss the evolution by inverse mean curvature flow of star-shaped mean convex hypersurfaces in non-compact rank one symmetric spaces. We show similarities and differences between the case considered, with particular attention to how the geometry of the ambient manifolds influences the behaviour of the evolution. Moreover we try, when possible, to give an unified approach to the results present in literature.
2
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A Formula for Popp’s Volume in Sub-Riemannian Geometry

100%
Barilari D., Rizzi L.
Analysis and Geometry in Metric Spaces
|
2013
|
tom 1
42-57
EN
For an equiregular sub-Riemannian manifold M, Popp’s volume is a smooth volume which is canonically associated with the sub-Riemannian structure, and it is a natural generalization of the Riemannian one. In this paper we prove a general formula for Popp’s volume, written in terms of a frame adapted to the sub-Riemannian distribution. As a first application of this result, we prove an explicit formula for the canonical sub- Laplacian, namely the one associated with Popp’s volume. Finally, we discuss sub-Riemannian isometries, and we prove that they preserve Popp’s volume. We also show that, under some hypotheses on the action of the isometry group of M, Popp’s volume is essentially the unique volume with such a property.
3
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Convex Hull Property and Exclosure Theorems for H-Minimal Hypersurfaces in Carnot Groups

88%
Montefalcone F.
Analysis and Geometry in Metric Spaces
|
2016
|
tom 4
|
nr 1
EN
In this paper, we generalize to sub-Riemannian Carnot groups some classical results in the theory of minimal submanifolds. Our main results are for step 2 Carnot groups. In this case, we will prove the convex hull property and some “exclosure theorems” for H-minimal hypersurfaces of class C2 satisfying a Hörmander-type condition.
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