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
![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))](/images/equations/MeanCurvature/NumberedEquation7.svg) |
(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.
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