The mean curvature is the mean of the principal curvatures 
and 
,
 |
(1)
|
If 
and 
are 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.
." In