Let
and
be the principal curvatures , then their mean

(1)

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

(2)

In terms of the Gaussian curvature ,

(3)

The mean curvature of a regular surface in at a point is formally defined as

(4)

where
is the shape operator and denotes the matrix trace .
For a Monge patch with ,

(5)

(Gray 1997, p. 399).

If
is a regular patch , then the mean curvature is given
by

(6)

where ,
, and are coefficients of the first fundamental
form and ,
, and are coefficients of the second fundamental
form (Gray 1997, p. 377). It can also be written

(7)

Gray (1997, p. 380).

The Gaussian and mean curvature satisfy

(8)

with equality only at umbilic points , since

(9)

If
is a point on a regular surface and and are tangent vectors to at , then the mean curvature of at is related to the shape operator by

(10)

Let
be a nonvanishing vector field on which is everywhere perpendicular
to ,
and let
and
be vector fields tangent to such that , then

(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 ." 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. Referenced on Wolfram|Alpha 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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