448 research outputs found
Convex hull property and exclosure theorems for H-minimal hypersurfaces in carnot groups
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 âexclosure theoremsâ for H-minimal hypersurfaces of class C2 satisfying a
Hörmander-type condition
Geometric inequalities in Carnot groups
Let \GG be a sub-Riemannian -step Carnot group of homogeneous dimension
. In this paper, we shall prove several geometric inequalities concerning
smooth hypersurfaces (i.e. codimension one submanifolds) immersed in \GG,
endowed with the \HH-perimeter measure.Comment: 26 page
Dark Energy, Extra Dimensions, and the Swampland
Perhaps the greatest challenge for fundamental theories based on
compactification from extra dimensions is accommodating a period of accelerated
cosmological expansion. Previous studies have identified constraints imposed by
the existence of dark energy for two overlapping classes of compactified
theories: (1) those in which the higher dimensional picture satisfies certain
metric properties selected to reproduce known low energy phenomenology; or (2)
those derived from string theory assuming they satisfy the Swampland
conjectures. For either class, the analyses showed that dark energy is only
possible if it takes the form of quintessence. In this paper, we explore the
consequences for theories that belong to both classes and show that the joint
constraints are highly restrictive, leaving few options.Comment: 12 pages, 3 figure
Hypersurfaces and variational formulas in sub-Riemannian Carnot groups
AbstractIn this paper we study smooth immersed non-characteristic submanifolds (with or without boundary) of k-step sub-Riemannian Carnot groups, from a differential-geometric point of view. The methods of exterior differential forms and moving frames are extensively used. Particular emphasis is given to the case of hypersurfaces. We state divergence-type theorems and integration by parts formulas with respect to the intrinsic measure ÏHnâ1 on hypersurfaces. General formulas for the first and the second variation of the measure ÏHnâ1 are proved
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