Abstract
Horizontal points of smooth submanifolds in stratified groups play the role of singular points with respect to the Carnot-Carathéodory distance. When we consider hypersurfaces, they coincide with the well known characteristic points. In two step groups, we obtain pointwise estimates for the Riemannian surface measure at all horizontal points of C 1,1 smooth submanifolds. As an application, we establish an integral formula to compute the spherical Hausdorff measure of any C 1,1 submanifold. Our technique also shows that C 2 smooth submanifolds everywhere admit an intrinsic blow-up and that the limit set is an intrinsically homogeneous algebraic variety.
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References
Ambrosio, L.: Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces. Adv. Math. 159, 51–67 (2001)
Ambrosio, L., Tilli, P.: Selected Topics on Analysis in Metric Spaces. Oxford University Press, London (2003)
Ambrosio, L., Kleiner, B., Le Donne, E.: Rectifiability of sets of finite perimeter in Carnot groups: existence of a tangent hyperplane. J. Geom. Anal. 19(3), 509–540 (2009)
Arena, G., Serapioni, R.: Intrinsic regular submanifolds in Heisenberg groups are differentiable graphs. Calc. Var. Partial Differ. Equ. 35(4), 517–536 (2009)
Balogh, Z.M., Tyson, J.T., Warhurst, B.: Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups. Adv. Math. 220(2), 560–619 (2009)
Barone Adesi, V., Serra Cassano, F., Vittone, D.: The Bernstein problem for intrinsic graphs in Heisenberg groups and calibrations. Calc. Var. Partial Differ. Equ. 30(1), 17–49 (2007)
Bigolin, F., Serra Cassano, F.: Intrinsic regular graphs in Heisenberg groups vs. weak solutions of non linear first-order PDEs. Adv. Calc. Var. (to appear)
Capogna, L., Garofalo, N.: Boundary behavior of nonnegative solutions of subelliptic equations in NTA domains for Carnot-Carathéodory metrics. J. Fourier Anal. Appl. 4, 403–432 (1998)
Capogna, L., Garofalo, N.: Ahlfors type estimates for perimeter measures in Carnot-Carathéodory spaces. J. Geom. Anal. 16(3), 455–497 (2006)
Capogna, L., Pauls, S.D., Tyson, J.: Convexity and horizontal second fundamental forms for hypersurfaces in Carnot groups. Trans. Am. Math. Soc. (to appear)
Cheeger, J., Kleiner, B.: Differentiating maps into L 1 and the geometry of BV functions. Ann. Math. (to appear)
Corwin, L., Greenleaf, F.P.: Representation of Nilpotent Lie Groups and Their Applications. Part 1: Basic Theory and Examples. Cambridge University Press, Cambridge (1990)
Danielli, D., Garofalo, N., Nhieu, D.M.: Non-doubling Ahlfors measures, Perimeter measures, and the characterization of the trace spaces of Sobolev functions in Carnot-Carathéodory spaces. Mem. Am. Math. Soc. 182, 857 (2006)
Danielli, D., Garofalo, N., Nhieu, D.M.: Sub-Riemannian calculus on hypersurfaces in Carnot groups. Adv. Math. 215(1), 292–378 (2007)
Danielli, D., Garofalo, N., Nhieu, D.M.: A notable family of entire intrinsic minimal graphs in the Heisenberg group which are not perimeter minimizing. Am. J. Math. 130(2), 317–339 (2008)
Danielli, D., Garofalo, N., Nhieu, D.M., Pauls, S.D.: Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group ℍ1. J. Differ. Geom. 81(2), 251–295 (2009)
Federer, H.: Geometric Measure Theory. Springer, Berlin (1969)
Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous Groups. Princeton University Press, Princeton (1982)
Franchi, B., Serapioni, R., Serra Cassano, F.: On the structure of finite perimeter sets in step 2 Carnot groups. J. Geom. Anal. 13(3), 421–466 (2003)
Franchi, B., Serapioni, R., Serra Cassano, F.: Regular submanifolds, graphs and area formula in Heisenberg groups. Adv. Math. 211(1), 152–203 (2007)
Garofalo, N., Nhieu, D.M.: Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces. Commun. Pure Appl. Math. 49, 1081–1144 (1996)
Gromov, M.: Carnot-Carathéodory spaces seen from within. In: Bellaïche, A., Risler, J. (eds.) Subriemannian Geometry. Progress in Mathematics, vol. 144. Birkhäuser, Basel (1996)
Hajlasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 145 (2000)
Jerison, D.: The Poincaré inequality for vector fields satisfying Hörmander’s condition. Duke Math. J. 53(2), 503–523 (1986)
Le Donne, E., Magnani, V.: Measure of submanifolds in the Engel group. Rev. Mat. Iberoam. 26(1), 333–346 (2010)
Magnani, V.: A blow-up theorem for regular hypersurfaces on nilpotent groups. Manuscr. Math. 110(1), 55–76 (2003)
Magnani, V.: Characteristic points, rectifiability and perimeter measure on stratified groups. J. Eur. Math. Soc. 8(4), 585–609 (2006)
Magnani, V.: Non-horizontal submanifolds and coarea formula. J. Anal. Math. 106, 95–127 (2008)
Magnani, V., Vittone, D.: An intrinsic measure for submanifolds in stratified groups. J. Reine Angew. Math. 619, 203–232 (2008)
Monti, R., Morbidelli, D.: Regular domains in homogeneous groups. Trans. Am. Math. Soc. 357(8), 2975–3011 (2005)
Monti, R., Rickly, M.: Convex isoperimetric sets in the Heisenberg group. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8(2), 391–415 (2009)
Ritorè, M., Rosales, C.: Area-stationary surfaces in the Heisenberg group ℍ1. Adv. Math. 219(2), 633–671 (2008)
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Communicated by Mike Wolf.
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Magnani, V. Blow-Up Estimates at Horizontal Points and Applications. J Geom Anal 20, 705–722 (2010). https://doi.org/10.1007/s12220-010-9124-5
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DOI: https://doi.org/10.1007/s12220-010-9124-5