Steiner Formula and Gaussian Curvature in the Heisenberg Group

Eugenio Vecchi


The classical Steiner formula expresses the volume of the ∈-neighborhood Ωof a bounded and regular domain  Ω⊂Rn as a polynomial of degree n in ∈. In particular, the coefficients of this polynomial are the integrals of functions of the curvatures of the boundary ∂Ω. The aim of this note is to present the Heisenberg counterpart of this result. The original motivation for studying this kind of extension is to try to identify a suitable candidate for the notion of horizontal Gaussian curvature. The results presented in this note are contained in the paper [4] written in collaboration with Zoltàn Balogh, Fausto Ferrari, Bruno Franchi and Kevin Wildrick


Heisenberg group; Steiner's formula

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N. Arcozzi, F. Ferrari. Metric normal and distance function in the Heisenberg group. Math. Z., 256 (2007), 661-684.

N. Arcozzi, F. Ferrari. The Hessian of the distance from a surface in the Heisenberg group. Ann. Acad. Sci. Fenn. Math., 33 (2008), 35-63.

N. Arcozzi, F. Ferrari, F. Montefalcone. Regularity of the distance function to smooth hypersurfaces in some two-step Carnot groups. Ann. Acad. Sci. Fenn. Math., 42 (2017), 1-18.

Z. M. Balogh, F. Ferrari, B. Franchi, E. Vecchi, K. Wildrick. Steiner's formula in the Heisenberg group. Nonlinear Anal., 126 (2015), 201-217.

Z. M. Balogh, J. T. Tyson, E. Vecchi. Intrinsic curvature of curves and surfaces and a Gauss-Bonnet theorem in the Heisenberg group. Math. Z. (2016). doi:10.1007/s00209-016-1815-6.

A. Bonfiglioli, E. Lanconelli, F. Uguzzoni. Stratified Lie groups and potential theory for their sub-Laplacians, Springer Monographs in Mathematics, Springer, Berlin (2007).

L. Capogna, D. Danielli, S. D. Pauls, J. T. Tyson. An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem, Birkhäuser Verlag, vol. 259, Basel (2007).

L. Capogna, S. D. Pauls, J. T. Tyson. Convexity and horizontal second fundamental forms for hypersurfaces in Carnot groups, Trans. Amer. Math. Soc., 362 (2010), 4045-4062.

D. Danielli, N. Garofalo, D. M. Nhieu. Notions of convexity in Carnot groups, Comm. Anal. Geom., 11 (2003), 263-341.

D. Danielli, N. Garofalo, D. M. Nhieu. Sub-Riemannian calculus on hypersurfaces in Carnot groups, Adv. Math., 215 (2007), 292-378.

M.M. Diniz, J.M.M. Veloso. Gauss-Bonnet Theorem in Sub-Riemannian Heisenberg Space. J. Dyn. Control Syst., 22 (2016), 807-820.

H. Federer. Curvature measures. Trans. Amer. Math. Soc. 93 (1959), 418-491.

F. Ferrari. A Steiner formula in the Heisenberg group for Carnot-Charath_eodory balls. Subelliptic PDE's and applications to geometry and finance, Semin. Interdiscip. Mat. (S.I.M.), Potenza, 6 (2007), 133-143.

R. Foote. Regularity of the distance function. Proc. Amer. Math. Soc. 92 (1984), no 1., 153-155.

B. Franchi, R. Serapioni, F. Serra Cassano. Rectifiability and perimeter in the Heisenberg group. Math. Ann., 321 (2001), 479-531.

D. Gilbarg, N. S. Trudinger. Elliptic partial differential equations of second order, Grundlehrer der Math. Wiss. vol. 224, Springer-Verlag, New York (1977).

E. Haller Martin. Horizontal Gauss curvature ow of graphs in Carnot groups, Indiana Univ. Math. J., 60 (2011), 1267-1302.

S. Krantz, H. R. Parks. Distance to Ck hypersurfaces. J. Di_erential Equations, 40 (1981), no 1., 116-120.

R. Monti, F. Serra Cassano. Surface measures in Carnot-Carath_eodory spaces. Calc. Var. Partial Differential Equations, 13 (2001), 339-376.

S.D. Pauls. Minimal surfaces in the Heisenberg group. Geom. Dedicata, 104 (2004), 201-231.

R. C. Reilly. On the Hessian of a function and the curvatures of its graph. Michigan Math. J., 20 (1973), 373-383.

R. C. Reilly. Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J. Differential Geometry, 8 (1973), 465-477.

H. Weyl. On the Volume of Tubes. Amer. J. Math., 61 (1939), 461-472.

DOI: 10.6092/issn.2240-2829/6693


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