00934nas a2200133 4500008004100000245005700041210005100098260005100149520050200200100002100702700001700723700002400740856003600764 2012 en d00aOn 2-step, corank 2 nilpotent sub-Riemannian metrics0 a2step corank 2 nilpotent subRiemannian metrics bSociety for Industrial and Applied Mathematics3 aIn this paper we study the nilpotent 2-step, corank 2 sub-Riemannian metrics\\r\\nthat are nilpotent approximations of general sub-Riemannian metrics. We exhibit optimal syntheses for these problems. It turns out that in general the cut time is not equal to the first conjugate time but has a simple explicit expression. As a byproduct of this study we get some smoothness properties of the spherical Hausdorff measure in the case of a generic 6 dimensional, 2-step corank 2 sub-Riemannian metric.1 aBarilari, Davide1 aBoscain, Ugo1 aGauthier, Jean-Paul uhttp://hdl.handle.net/1963/606501520nas a2200133 4500008004300000245008600043210006900129520106500198100002501263700001701288700002401305700002101329856003601350 2009 en_Ud 00aThe intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups0 aintrinsic hypoelliptic Laplacian and its heat kernel on unimodul3 aWe present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector, using the Popp\\\'s volume form introduced by Montgomery. This definition generalizes the one of the Laplace-Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that it coincides with the usual sum of squares.\\nWe then extend a method (first used by Hulanicki on the Heisenberg group) to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular Lie group of type I. The main tool is the noncommutative Fourier transform. We then study some relevant cases: SU(2), SO(3), SL(2) (with the metrics inherited by the Killing form), and the group SE(2) of rototranslations of the plane.\\nOur study is motivated by some recent results about the cut and conjugate loci on these sub-Riemannian manifolds. The perspective is to understand how singularities of the sub-Riemannian distance reflect on the kernel of the corresponding hypoelliptic heat equation.1 aAgrachev, Andrei, A.1 aBoscain, Ugo1 aGauthier, Jean-Paul1 aRossi, Francesco uhttp://hdl.handle.net/1963/266900435nas a2200121 4500008004100000245008600041210006900127260001800196100001700214700002200231700002400253856003600277 2002 en d00aOn the K+P problem for a three-level quantum system: optimality implies resonance0 aKP problem for a threelevel quantum system optimality implies re bSISSA Library1 aBoscain, Ugo1 aChambrion, Thomas1 aGauthier, Jean-Paul uhttp://hdl.handle.net/1963/160100364nas a2200109 4500008004100000245005900041210005100100260001800151100002500169700002400194856003600218 2001 en d00aOn the subanalyticity of Carnot-Caratheodory distances0 asubanalyticity of CarnotCaratheodory distances bSISSA Library1 aAgrachev, Andrei, A.1 aGauthier, Jean-Paul uhttp://hdl.handle.net/1963/1483