Mean curvature

 ( Differential Geometry ) 

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 embedding surface in some ambient space such as Euclidean space.

The concept was used by Sophie Germain in her work on elasticity theory.Marie-Louise Dubreil-Jacotin on Sophie Germain Jean Baptiste Marie Meusnier used it in 1776, in his studies of minimal surface. It is important in the analysis of minimal surfaces, which have mean curvature zero, and in the analysis of physical interfaces between fluids (such as ) which, for example, have constant mean curvature in static flows, by the Young–Laplace equation.

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Definition Let $p$ be a point on the surface $S$ inside the three dimensional Euclidean space . Each plane through $p$ containing the normal line to $S$ cuts $S$ in a (plane) curve. Fixing a choice of unit normal gives a signed curvature to that curve. As the plane is rotated by an angle $\backslash theta$ (always containing the normal line) that curvature can vary. The maximal curvature $\backslash kappa\_1$ and minimal curvature $\backslash kappa\_2$ are known as the principal curvatures of $S$.

The mean curvature at $p\backslash in\; S$ is then the average of the signed curvature over all angles $\backslash theta$:

$H\; =\; \backslash frac\{1\}\{2\backslash pi\}\backslash int\_0^\{2\backslash pi\}\; \backslash kappa(\backslash theta)\; \backslash ;d\backslash theta$.

By applying Euler's theorem, this is equal to the average of the principal curvatures :

$H\; =\; \{1\; \backslash over\; 2\}\; (\backslash kappa\_1\; +\; \backslash kappa\_2).$

More generally , for a hypersurface $T$ the mean curvature is given as

$H=\backslash frac\{1\}\{n\}\backslash sum\_\{i=1\}^\{n\}\; \backslash kappa\_\{i\}.$

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 $H$ may be written in terms of the covariant derivative $\backslash nabla$ as

$H\backslash vec\{n\}\; =\; g^\{ij\}\backslash nabla\_i\backslash nabla\_j\; X,$

using the Gauss-Weingarten relations, where $X(x)$ is a smoothly embedded hypersurface, $\backslash vec\{n\}$ a unit normal vector, and $g\_\{ij\}$ 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 $S$, is said to obey a heat 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".

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Surfaces in 3D space For a surface defined in 3D space, the mean curvature is related to the divergence of a unit Surface normal of the surface:

$2\; H\; =\; -\backslash nabla\; \backslash cdot\; \backslash hat\; n$

where the sign of the curvature depends on the choice of normal (inward or outward): 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.

Mean curvature may also be calculated as

$2\; H\; =\; \backslash text\{Trace\}((\backslash mathrm\{II\})(\backslash mathrm\{I\}^\{-1\}))$

where I and II denote first and second fundamental forms, respectively.

If $S(x,y)$ is a parametrization of the surface and $u,\; v$ are two linearly independent vectors in parameter space then the mean curvature can be written in terms of the first and second quadratic form matrices as $$\backslash frac\{l\; G-2\; m\; F\; +\; n\; E\}\{2\; (\; E\; G\; -\; F^2)\}$$ where $E\; =\; \backslash mathrm\{I\}(u,u)$, $F\; =\; \backslash mathrm\{I\}(u,v)$, $G\; =\; \backslash mathrm\{I\}(v,v)$, $l\; =\; \backslash mathrm\{II\}(u,u)$, $m\; =\; \backslash mathrm\{II\}(u,v)$, $n\; =\; \backslash mathrm\{II\}(v,v)$.

For the special case of a surface defined as a function of two coordinates (a bivariate function), e.g. $z\; =\; S(x,\; y)$, and using the upward pointing normal, the (doubled) mean curvature expression is

$\backslash begin\{align\}$

2 H & = -\nabla \cdot \left(\frac{\nabla(z-S)}\right) \\ & = \nabla \cdot \left(\frac{\nabla S-\nabla z} {\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} - 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}

In particular at a point where $\backslash nabla\; S=0$, the mean curvature is half the trace of the Hessian matrix of $S$.

If the surface is additionally known to be axisymmetric with $z\; =\; S(r)$,

$2\; H\; =\; \backslash frac\{\backslash frac\{\backslash partial^2\; S\}\{\backslash partial\; r^2\}\}\{\backslash left(1\; +\; \backslash left(\backslash frac\{\backslash partial\; S\}\{\backslash partial\; r\}\backslash right)^2\backslash right)^\{3/2\}\}\; +\; \{\backslash frac\{\backslash partial\; S\}\{\backslash partial\; r\}\}\backslash frac\{1\}\{r\; \backslash left(1\; +\; \backslash left(\backslash frac\{\backslash partial\; S\}\{\backslash partial\; r\}\backslash right)^2\backslash right)^\{1/2\}\},$

where $\{\backslash frac\{\backslash partial\; S\}\{\backslash partial\; r\}\}\; \backslash frac\{1\}\{r\}$ comes from the derivative of $z\; =\; S(r)\; =\; S\backslash left(\backslash sqrt\{x^2\; +\; y^2\}\; \backslash right)$.

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Implicit form of mean curvature The mean curvature of a surface specified by an equation $F(x,y,z)=0$ can be calculated by using the gradient $\backslash nabla\; F=\backslash left(\; \backslash frac\{\backslash partial\; F\}\{\backslash partial\; x\},\; \backslash frac\{\backslash partial\; F\}\{\backslash partial\; y\},\; \backslash frac\{\backslash partial\; F\}\{\backslash partial\; z\}\; \backslash right)$ and the Hessian matrix

$\backslash textstyle\; \backslash mbox\{Hess\}(F)=$

\begin{pmatrix} \frac{\partial^2 F}{\partial x^2} & \frac{\partial^2 F}{\partial x\partial y} & \frac{\partial^2 F}{\partial x\partial z} \\ \frac{\partial^2 F}{\partial y\partial x} & \frac{\partial^2 F}{\partial y^2} & \frac{\partial^2 F}{\partial y\partial z} \\ \frac{\partial^2 F}{\partial z\partial x} & \frac{\partial^2 F}{\partial z\partial y} & \frac{\partial^2 F}{\partial z^2} \end{pmatrix} . The mean curvature is given by:

$H\; =\; \backslash frac\{\; \backslash nabla\; F\backslash \; \backslash mbox\{Hess\}(F)\; \backslash \; \backslash nabla\; F^\{\backslash mathsf\; \{T\}\}\; -\; |\backslash nabla\; F|^2\backslash ,\; \backslash text\{Trace\}(\backslash mbox\{Hess\}(F))\; \}\; \{\; 2|\backslash nabla\; F|^3\; \}$

Another form is as the divergence of the unit normal. A unit normal is given by $\backslash frac\{\backslash nabla\; F\}$

and the mean curvature is

$H\; =\; -\{\backslash frac\{1\}\{2\}\}\backslash nabla\backslash cdot\; \backslash left(\backslash frac\{\backslash nabla\; F\}$\right).

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In fluid mechanics An alternate definition is occasionally used in fluid mechanics to avoid factors of two:

$H\_f\; =\; (\backslash kappa\_1\; +\; \backslash kappa\_2)\; \backslash ,$.

This results in the pressure according to the Young–Laplace equation inside an equilibrium spherical droplet being surface tension times $H\_f$; the two curvatures are equal to the reciprocal of the droplet's radius

$\backslash kappa\_1\; =\; \backslash kappa\_2\; =\; r^\{-1\}\; \backslash ,$.

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Minimal surfaces 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.

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CMC surfaces An extension of the idea of a minimal surface are surfaces of constant mean curvature. The surfaces of unit constant mean curvature in hyperbolic space are called ..

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See also

• Gaussian curvature
• Mean curvature flow
• Inverse mean curvature flow
• First variation of area formula
• Stretched grid method

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Notes

• .
• P.Grinfeld (2025). 9781461478669, Springer. ISBN 9781461478669

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