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Mean Curvature


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

 H=1/2(kappa_1+kappa_2)
(1)

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,

 H=1/2(1/(R_1)+1/(R_2))=(R_1+R_2)/(2R_1R_2).
(2)

In terms of the Gaussian curvature K,

 H=1/2(R_1+R_2)K.
(3)

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

 H(p)=1/2Tr(S(p)),
(4)

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

 H=((1+h_v^2)h_(uu)-2h_uh_vh_(uv)+(1+h_u^2)h_(vv))/(2(1+h_u^2+h_v^2)^(3/2))
(5)

(Gray 1997, p. 399).

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

 H=(eG-2fF+gE)/(2(EG-F^2)),
(6)

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

 H=(det(x_(uu)x_ux_v)|x_v|^2-2det(x_(uv)x_ux_v)(x_u·x_v))/(2[|x_u|^2|x_v|^2-(x_u·x_v)^2]^(3/2)) 
 +(det(x_(vv)x_ux_v)|x_u|^2)/(2[|x_u|^2|x_v|^2-(x_u·x_v)^2]^(3/2))
(7)

Gray (1997, p. 380).

The Gaussian and mean curvature satisfy

 H^2>=K,
(8)

with equality only at umbilic points, since

 H^2-K=1/4(kappa_1-kappa_2)^2.
(9)

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

 S(v_(p))xw_(p)+v_(p)xS(w_(p))=2H(p)v_(p)xw_(p).
(10)

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

 H=-(Z·(D_VZxW+VxD_WZ))/(2|Z|^3)
(11)

(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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References

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." http://www.gang.umass.edu/gallery/cmc/.Wente, 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. https://mathworld.wolfram.com/MeanCurvature.html

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