Abstract
We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measure. We explicitly compute this formula in some simple examples and we present a lower semicontinuity result for the spherical Hausdorff measure with respect to the weak convergence of currents. Another application is the proof of an intrinsic coarea formula for vector-valued mappings on the Heisenberg group.
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References
L. Ambrosio: “Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces”, Adv. Math., Vol. 159, (2001), pp. 51–67.
Z.M. Balogh: “Size of characteristic sets and functions with prescribed gradients”, J. Reine Angew. Math., Vol. 564, (2003), pp. 63–83.
A. Bellaïche and J.J. Risler (Eds.): Sub-Riemannian geometry, Progress in Mathematics, Vol. 144, Birkhäuser Verlag, Basel, 1996.
Y.D. Burago and V.A. Zalgaller: Geometric inequalities, Grundlehren Math. Springer, Berlin.
H. Federer: Geometric Measure Theory, Springer, 1969.
G.B. Folland and E.M. Stein: Hardy Spaces on Homogeneous groups, Princeton University Press, 1982.
B. Franchi, R. Serapioni and F. Serra Cassano: “Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields”, Houston Jour. Math., Vol. 22, (1996), pp. 859–889.
B. Franchi, R. Serapioni and F. Serra Cassano: “Rectifiability and Perimeter in the Heisenberg group”, Math. Ann., Vol. 321(3), (2001).
B. Franchi, R. Serapioni and F. Serra Cassano: “Regular hypersurfaces, intrinsic perimeter and implicit function theorem in Carnot groups”, Comm. Anal. Geom., Vol. 11(5), (2003), pp. 909–944.
B. Franchi, R. Serapioni and F. Serra Cassano: Regular submanifolds, graphs and area formula in Heisenberg groups, preprint, (2004).
M. Gromov: “Carnot-Carathéodory spaces seen from within”, In: A. Bellaiche and J. Risler (Eds.): Subriemannian Geometry, Progress in Mathematics, Vol. 144, Birkhauser Verlag, Basel, 1996.
N. Garofalo and D.M. Nhieu: “Isoperimetric and Sobolev Inequalities for Carnot-Carathéodory Spaces and the Existence of Minimal Surfaces”, Comm. Pure Appl. Math., Vol. 49, (1996), pp. 1081–1144.
P. Hajlasz and P. Koskela: “Sobolev met Poincaré”, Mem. Amer. Math. Soc., Vol. 145, (2000).
B. Kirchheim and F. Serra Cassano: “Rectifiability and parametrization of intrinsic regular surfaces in the Heisenberg group”, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Vol. 3(4), (2004), pp. 871–896.
A. Korányi: “Geometric properties of Heisenberg-type groups”, Adv. Math., Vol. 56(1), (1985), pp. 28–38.
I. Kupka: “Géométrie sous-riemannienne”, Astérisque, Vol. 241(817,5), (1997), pp. 351–380.
V. Magnani: “Differentiability and Area formula on stratified Lie groups”, Houston Jour. Math., Vol. 27(2), (2001), pp. 297–323.
V. Magnani: “On a general coarea inequality and applications”, Ann. Acad. Sci. Fenn. Math., Vol. 27, (2002), pp. 121–140.
V. Magnani: “A Blow-up Theorem for regular hypersurfaces on nilpotent groups”, Manuscripta Math., Vol. 110(1), (2003), pp. 55–76.
V. Magnani: “The coarea formula for real-valued Lipschitz maps on stratified groups”, Math. Nachr., Vol. 278(14), (2005), pp. 1–17.
V. Magnani: “Note on coarea formulae in the Heisenberg group”, Publ. Mat., Vol. 48(2), (2004), pp. 409–422.
V. Magnani: “Characteristic points, rectifiability and perimeter measure on stratified groups”, J. Eur. Math. Soc., to appear.
R. Montgomery: A Tour of Subriemannian Geometries, Their Geodesics and Applications, Mathematical Surveys and Monographs, Vol. 91, American Mathematical Society, Providence, 2002.
P.Pansu, Geometrie du Group d'Heisenberg, Thesis (PhD), 3rd ed., Université Paris VII, 1982.
P. Pansu, “Une inégalité isoperimetrique sur le groupe de Heisenberg”, C.R. Acad. Sc. Paris, Série I, Vol. 295, (1982), pp. 127–130.
P. Pansu: “Métriques de Carnot-Carathéodory quasiisométries des espaces symétriques de rang un”, Ann. Math., Vol. 129, (1989), pp. 1–60.
E.M. Stein: Harmonic Analysis, Princeton University Press, 1993.
N.Th. Varopoulos, L. Saloff-Coste and T. Coulhon: Analysis and Geometry on Groups, Cambridge University Press, Cambridge, 1992.
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Magnani, V. Blow-up of regular submanifolds in Heisenberg groups and applications. centr.eur.j.math. 4, 82–109 (2006). https://doi.org/10.1007/s11533-005-0006-1
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DOI: https://doi.org/10.1007/s11533-005-0006-1