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In mathematics, the mean curvature $H$ of a surface $S$ is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space.

Notes

References

Spivak, Michael (1999), A comprehensive introduction to differential geometry (Volumes 3-4) (3rd ed.), Publish or Perish Press, ISBN 0-914098-72-1, (Volume 3), (Volume 4).

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 In [[mathematics]], the '''mean curvature''' <math>H</math> of a [[surface]] <math> S</math> is an ''extrinsic'' measure of [[curvature]] that comes from [[differential geometry]] and that locally describes the curvature of an [[embedding|embedded]] surface in some ambient space such as [[Euclidean space]].
-The concept was introduced by [[Sophie Germain]] in her work on [[elasticity theory]].<ref>[http://www-groups.dcs.st-and.ac.uk/~history/Extras/Dubreil-Jacotin_Germain.html Dubreil-Jacotin on Sophie Germain<!-- Bot generated title -->]</ref><ref>{{Cite jstor | 3647744}}</ref> It is important in the analysis of [[minimal surface]]s, which have mean curvature zero, and in the analysis of physical interfaces between fluids (such as [[soap film]]s) which by the [[Young–Laplace equation]] have constant mean curvature.
-==Definition==
-Let <math>p</math> be a point on the surface <math>S</math>. Each plane through <math>p</math> containing the normal line to <math>S</math> cuts <math>S</math> in a (plane) curve. Fixing a choice of unit normal gives a signed curvature to that curve. As the plane is rotated (always containing the normal line) that curvature can vary, and the [[maxima and minima|maximal]] curvature <math>\kappa_1</math> and [[maxima and minima|minimal]] curvature <math>\kappa_2</math> are known as the ''[[principal curvature]]s'' of <math>S</math>.
-The '''mean curvature''' at <math>p\in S</math> is then the average of the principal curvatures {{harv|Spivak|1999|loc=Volume 3, Chapter 2}}, hence the name:
-:<math>H = {1 \over 2} (\kappa_1 + \kappa_2).</math>
-More generally {{harv|Spivak|1999|loc=Volume 4, Chapter 7}}, for a [[hypersurface]] <math>T</math> the mean curvature is given as
-:<math>H=\frac{1}{n}\sum_{i=1}^{n} \kappa_{i}.</math>
-More abstractly, the mean curvature is the trace of the [[second fundamental form]] divided by ''n'' (or equivalently, the [[shape operator]]).
-Additionally, the mean curvature <math>H</math>  may be written in terms of the [[covariant derivative]] <math>\nabla</math> as
-:<math>H\vec{n} = g^{ij}\nabla_i\nabla_j X,</math>
-using the ''Gauss-Weingarten relations,'' where <math> X(x) </math> is a smoothly embedded hypersurface, <math>\vec{n}</math> a unit normal vector, and <math>g_{ij}</math> the [[metric tensor]].
-A surface is a [[minimal surface]] [[if and only if]] the mean curvature is zero.  Furthermore, a surface which evolves under the mean curvature of the surface <math> S</math>, is said to obey a [[heat equation|heat-type equation]] called the [[mean curvature flow]] equation.
-The [[sphere]] is the only embedded surface of constant positive mean curvature without boundary or singularities.  However, the result is not true when the condition "embedded surface" is weakened to "immersed surface".<ref>http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pjm/1102702809</ref>
-===Surfaces in 3D space===
-For a surface defined in 3D space, the mean curvature is related to a unit [[Surface normal|normal]] of the surface:
-:<math>2 H = -\nabla \cdot \hat n</math>
-where the normal chosen affects the sign of the curvature. The sign of the curvature depends on the choice of normal: the curvature is positive if the surface curves "towards" the normal. The formula above holds for surfaces in 3D space defined in any manner, as long as the [[divergence]] of the unit normal may be calculated.
-For the special case of a surface defined as a function of two coordinates, e.g. <math>z = S(x, y)</math>, and using the upward pointing normal the (doubled) mean curvature expression is
-:<math>\begin{align}2 H & = -\nabla \cdot \left(\frac{\nabla(z-S)}{|\nabla(z - S)|}\right) \\
-& = \nabla \cdot \left(\frac{\nabla S}
-{\sqrt{1 + |\nabla S|^2}}\right) \\
-& =
-\frac{
-\left(1 + \left(\frac{\partial S}{\partial x}\right)^2\right) \frac{\partial^2 S}{\partial y^2} -
-\frac{\partial S}{\partial x} \frac{\partial S}{\partial y} \frac{\partial^2 S}{\partial x \partial y} +
-\left(1 + \left(\frac{\partial S}{\partial y}\right)^2\right) \frac{\partial^2 S}{\partial x^2}
-}{\left(1 + \left(\frac{\partial S}{\partial x}\right)^2 + \left(\frac{\partial S}{\partial y}\right)^2\right)^{3/2}}.
-\end{align}
-</math>
-In particular at a point where <math>\nabla S=0</math>, the mean curvature is half the trace of the Hessian matrix of <math>S</math>.
-If the surface is additionally known to be [[axisymmetric]] with <math>z = S(r)</math>,
-:<math>2 H = \frac{\frac{\partial^2 S}{\partial r^2}}{\left(1 + \left(\frac{\partial S}{\partial r}\right)^2\right)^{3/2}} + {\frac{\partial S}{\partial r}}\frac{1}{r \left(1 + \left(\frac{\partial S}{\partial r}\right)^2\right)^{1/2}},</math>
-where <math>{\frac{\partial S}{\partial r}}\frac{1}{r}</math> comes from the derivative of <math>z = S(r)=S\left(\scriptstyle \sqrt{x^2+y^2} \right)</math>.
-==Mean curvature in fluid mechanics==
-An alternate definition is occasionally used in [[fluid mechanics]] to avoid factors of two:
-:<math>H_f = (\kappa_1 + \kappa_2) \,</math>.
-This results in the pressure according to the [[Young-Laplace equation]] inside an equilibrium spherical droplet being [[surface tension]] times <math>H_f</math>; the two curvatures are equal to the reciprocal of the droplet's radius
-:<math>\kappa_1 = \kappa_2 = r^{-1} \,</math>.
-==Minimal surfaces==
-[[Image:Costa minimal surface.jpg|right|thumb|175px|A rendering of Costa's minimal surface.]]
-{{main|Minimal surface}}
-A '''minimal surface''' is a surface which has zero mean curvature at all points. Classic examples include the [[catenoid]], [[helicoid]] and [[Enneper surface]]. Recent discoveries include [[Costa's minimal surface]] and the [[Gyroid]].
-An extension of the idea of a minimal surface are surfaces of [[constant mean curvature]].
-==See also==
-* [[Gaussian curvature]]
-* [[Mean curvature flow]]
-* [[Inverse mean curvature flow]]
-* [[First variation of area formula]]
-* [[Stretched grid method]]
 ==Notes==
 {{reflist}}

v t e Various notions of curvature defined in differential geometry
Differential geometry of curves	Curvature Torsion of a curve Frenet–Serret formulas Radius of curvature (applications) Affine curvature Total curvature Total absolute curvature
Differential geometry of surfaces	Principal curvatures Gaussian curvature Mean curvature Darboux frame Gauss–Codazzi equations First fundamental form Second fundamental form Third fundamental form
Riemannian geometry	Curvature of Riemannian manifolds Riemann curvature tensor Ricci curvature Scalar curvature Sectional curvature
Curvature of connections	Curvature form Torsion tensor Cocurvature Holonomy