AN INTRODUCTION TO THE HEISENBERG GROUP AND SUB-RIEMANNIAN GEOMETRY

By V. CAVID

CONTENTS THE ISOPERIMETRIC PROBLEM IN EUCLIDEAN SPACE 1 1 NOTES THE ?RST HEISENBERG GROUP H THE HORIZONTAL DISTRIBUTION IN H HIGHER-DIMENSIONAL HEISENBERG GROUPS HN CARNOT GROUPS CARNOT–CARATH´EODORY DISTANCE CONSTRAINED DYNAMICS SUB-RIEMANNIAN STRUCTURE GEODESICS AND BUBBLE SETS RIEMANNIAN APPROXIMANTS TO THE HEISENBERG GROUP THE GL METRICS LEVI-CIVITA CONNECTION AND CURVATURE GROMOV–HAUSDOR?CONVERGENCE CARNOT–CARATH´EODORY GEODESICS RIEMANNIAN APPROXIMANTS TO HN AND CARNOT GROUPS NOTES JET SPACES APPLIED MODELS NONHOLONOMIC PATH PLANNING GEOMETRY OF THE VISUAL CORTEX CR STRUCTURES BOUNDARY OF COMPLEX HYPERBOLIC SPACE GROMOV HYPERBOLIC SPACES GROMOV BOUNDARY AND VISUAL METRIC C AND ITS BOUNDARY AT IN?NITY THE BERGMAN METRIC C AND THE HEISENBERG GROUP FURTHER RESULTS: GEODESICS IN THE ROTO-TRANSLATION SPACE INVARIANCE OF THE SUB-RIEMANNIAN METRIC WITH RESPECT TO RIEMANNIAN EXTENSIONS THE SECOND FUNDAMENTAL FORM IN (R3, GL) HORIZONTAL GEOMETRY OF HYPERSURFACES IN H HORIZONTAL GEOMETRY IN HN LEGENDRIAN FOLIATIONS ANALYSIS AT THE CHARACTERISTIC SET AND ?NE REGULARITY OF SURFACES THE LEGENDRIAN FOLIATION NEAR NON-ISOLATED POINTS OF THE CHARACTERISTIC LOCUS THE LEGENDRIAN FOLIATION NEAR ISOLATED POINTS OF THE CHARACTERISTIC LOCUS FURTHER RESULTS: INTRINSICALLY REGULAR SURFACES AND THE RUMIN COMPLEX SOBOLEV SPACES, PERIMETER MEASURE AND TOTAL VARIATION RIEMANNIAN PERIMETER APPROXIMATION A SUB-RIEMANNIAN GREEN’S FORMULA AND THE FUNDAMENTAL SOLUTION OF THE HEISENBERG LAPLACIAN EMBEDDING THEOREMS FOR THE SOBOLEV AND BV SPACES THE GEOMETRIC CASE (SOBOLEV–GAGLIARDO–NIRENBERG INEQUALITY) THE SUBCRITICAL CASE THE SUPERCRITICAL CASE ON JOHN DOMAINS FURTHER RESULTS: SOBOLEV EMBEDDING THEOREMS AREA AND CO-AREA FORMULAS PANSU–RADEMACHER THEOREM EQUIVALENCE OF PERIMETER AND MINKOWSKI CONTENT FIRST VARIATION OF THE PERIMETER PARAMETRIC SURFACES AND NONCHARACTERISTIC VARIATIONS GENERAL VARIATIONS QUASICONFORMAL MAPPINGS ON H EQUIVALENCE OF THE ISOPERIMETRIC AND GEOMETRIC SOBOLEV INEQUALITIES ISOPERIMETRIC INEQUALITIES IN HADAMARD MANIFOLDS PANSU’S PROOF OF THE ISOPERIMETRIC INEQUALITY IN H PANSU’S CONJECTURE EXISTENCE OF MINIMIZERS ISOPERIMETRIC PRO?LE HAS CONSTANT MEAN CURVATURE PARAMETRIZATION OF C2 CMC T-GRAPHS IN H MINIMIZERS WITH SYMMETRIES THE C2 ISOPERIMETRIC PRO?LE IN H THE CONVEX ISOPERIMETRIC PRO?LE OF H OTHER APPROACHES RIEMANNIAN APPROXIMATION APPROACH FAILURE OF THE BRUNN–MINKOWSKI APPROACH TO ISOPERIMETRY IN H HORIZONTAL MEAN CURVATURE ?OW FURTHER RESULTS THE ISOPERIMETRIC PROBLEM IN THE GRUSHIN PLANE THE CLASSI?CATION OF SYMMETRIC CMC SURFACES IN HN L2-SOBOLEV EMBEDDING THEOREM MOSER–TRUDINGER INEQUALITY HARDY INEQUALITY