Differential Geometry
A Note on hypersurfaces of a Euclidean space
Comptes Rendus. Mathématique, Volume 351 (2013) no. 15-16, pp. 631-634.

In this short Note, we consider a compact and connected orientable hypersurface M of the Euclidean space Rn+1 with non-negative support function and Minkowskiʼs integrand σ, and show that the mean curvature function α is the solution of the Poisson equation Δφ=σ if and only if M is isometric to n-sphere Sn(c) of constant curvature c. A similar result is proved for a hypersurface with scalar curvature satisfying the Poisson equation Δφ=σ.

Dans cette courte Note, nous considérons une hypersurface compacte, connexe orientable M de lʼespace euclidien Rn+1, de fonction support positive ou nulle et dʼintégrande de Minkowski σ. Nous montrons que la fonction courbure moyenne α est la solution de lʼéquation de Poisson Δφ=σ si et seulement si M est isométrique à une sphère Sn(c) de dimension n et courbure constante égale à c. Un résultat similaire est démontré pour une hypersurface de courbure scalaire satisfaisant lʼéquation de Poisson Δφ=σ.

Published online:
DOI: 10.1016/j.crma.2013.09.003
Deshmukh, Sharief 1

1 Department of Mathematics, College of Science, King Saud University, P. O. Box-2455, Riyadh 11451, Saudi Arabia
     author = {Deshmukh, Sharief},
     title = {A {Note} on hypersurfaces of a {Euclidean} space},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {631--634},
     publisher = {Elsevier},
     volume = {351},
     number = {15-16},
     year = {2013},
     doi = {10.1016/j.crma.2013.09.003},
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Deshmukh, Sharief. A Note on hypersurfaces of a Euclidean space. Comptes Rendus. Mathématique, Volume 351 (2013) no. 15-16, pp. 631-634. doi : 10.1016/j.crma.2013.09.003. http://www.numdam.org/articles/10.1016/j.crma.2013.09.003/

[1] Donaldson, S. Geometric analysis lecture notes http://www2.imperial.ac.uk/~skdona/ (available online at)

[2] Li, P. Lecture Notes on Geometric Analysis, Global Analysis Research Center, Seoul National University, Korea, 1993

[3] Li, P. Curvature and function theory on Riemannian manifolds, Surveys in Differential Geometry: Papers Dedicated to Atiyah, Bott, Hirzebruch, and Singer, vol. VII, International Press, 2000, pp. 375-432

Cited by Sources:

This work is sponsored by the Distinguished Scientist Fellowship Program (DSFP), King Saud University, Riyadh, Saudi Arabia.