Mean Curvature

Let kappa_1 and kappa_2 be the principal curvatures, then their mean


is called the mean curvature. Let R_1 and R_2 be the radii corresponding to the principal curvatures, then the multiplicative inverse of the mean curvature H is given by the multiplicative inverse of the harmonic mean,


In terms of the Gaussian curvature K,


The mean curvature of a regular surface in R^3 at a point p is formally defined as


where S is the shape operator and Tr(S) denotes the matrix trace. For a Monge patch with z=h(u,v),


(Gray 1997, p. 399).

If x:U->R^3 is a regular patch, then the mean curvature is given by


where E, F, and G are coefficients of the first fundamental form and e, f, and g are coefficients of the second fundamental form (Gray 1997, p. 377). It can also be written


Gray (1997, p. 380).

The Gaussian and mean curvature satisfy


with equality only at umbilic points, since


If p is a point on a regular surface M subset R^3 and v_(p) and w_(p) are tangent vectors to M at p, then the mean curvature of M at p is related to the shape operator S by


Let Z be a nonvanishing vector field on M which is everywhere perpendicular to M, and let V and W be vector fields tangent to M such that VxW=Z, then


(Gray 1997, p. 410).

Wente (1985, 1986, 1987) found a nonspherical finite surface with constant mean curvature, consisting of a self-intersecting three-lobed toroidal surface. A family of such surfaces exists.

See also

Gaussian Curvature, Lagrange's Equation, Minimal Surface, Principal Curvatures, Shape Operator Explore this topic in the MathWorld classroom

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Gray, A. "The Gaussian and Mean Curvatures." §16.5 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 373-380, 1997.Isenberg, C. The Science of Soap Films and Soap Bubbles. New York: Dover, p. 108, 1992.Peterson, I. The Mathematical Tourist: Snapshots of Modern Mathematics. New York: W. H. Freeman, pp. 69-70, 1988.Schmidt, N. "GANG | Constant Mean Curvature Surfaces.", H. C. "A Counterexample in 3-Space to a Conjecture of H. Hopf." In Workshop Bonn 1984, Proceedings of the 25th Mathematical Workshop Held at the Max-Planck Institut für Mathematik, Bonn, June 15-22, 1984 (Ed. F. Hirzebruch, J. Schwermer, and S. Suter). New York: Springer-Verlag, pp. 421-429, 1985.Wente, H. C. "Counterexample to a Conjecture of H. Hopf." Pac. J. Math. 121, 193-243, 1986.Wente, H. C. "Immersed Tori of Constant Mean Curvature in R^3." In Variational Methods for Free Surface Interfaces, Proceedings of a Conference Held in Menlo Park, CA, Sept. 7-12, 1985 (Ed. P. Concus and R. Finn). New York: Springer-Verlag, pp. 13-24, 1987.

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Mean Curvature

Cite this as:

Weisstein, Eric W. "Mean Curvature." From MathWorld--A Wolfram Web Resource.

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