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.
The concept was used by Sophie Germain in her work on elasticity theory. Jean Baptiste Marie Meusnier used it in 1776, in his studies of minimal surfaces. 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 soap films) which, for example, have constant mean curvature in static flows, by the Young-Laplace equation.

Definition

Let $p$ be a point on the surface $S$. 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 $\backslash theta$: :$H\; =\; \backslash frac\backslash int\_0^\; \backslash kappa(\backslash theta)\; \backslash ;d\backslash theta$. By applying Euler's theorem, this is equal to the average of the principal curvatures : :$H\; =\; (\backslash kappa\_1\; +\; \backslash kappa\_2).$ More generally , for a hypersurface $T$ the mean curvature is given as :$H=\backslash frac\backslash sum\_^\; \backslash kappa\_.$ 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 :$H\backslash vec\; =\; g^\backslash nabla\_i\backslash nabla\_j\; X,$ using the ''Gauss-Weingarten relations,'' where $X(x)$ is a smoothly embedded hypersurface, $\backslash vec$ a unit normal vector, and $g\_$ 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-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".

Surfaces in 3D space

For a surface defined in 3D space, the mean curvature is related to a unit normal of the surface: :$2\; H\; =\; -\backslash nabla\; \backslash cdot\; \backslash hat\; n$ 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. Mean Curvature may also be calculated :$2\; H\; =\; \backslash text((\backslash mathrm)(\backslash mathrm^))$ where I and II denote first and second quadratic form matrices, 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 fundamental forms as $$\backslash frac$$ where $E=\backslash mathrm(u,u),\; F=\backslash mathrm(u,v),\; G=\backslash mathrm(v,v),\; l=\backslash mathrm(u,u),\; m=\backslash mathrm(u,v),\; n=\backslash mathrm(v,v)$. For the special case of a surface defined as a function of two coordinates, e.g. $z\; =\; S(x,\; y)$, and using the upward pointing normal the (doubled) mean curvature expression is :$\backslash begin2\; H\; \&\; =\; -\backslash nabla\; \backslash cdot\; \backslash left(\backslash frac\backslash right)\; \backslash \backslash \; \&\; =\; \backslash nabla\; \backslash cdot\; \backslash left(\backslash frac\; \backslash right)\; \backslash \backslash \; \&\; =\; \backslash frac.\; \backslash end$ 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 $z\; =\; S(r)$, :$2\; H\; =\; \backslash frac\; +\; \backslash frac,$ where $\backslash frac$ comes from the derivative of $z\; =\; S(r)=S\backslash left(\backslash scriptstyle\; \backslash sqrt\; \backslash right)$.

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 $\backslash nabla\; F=\backslash left(\; \backslash frac,\; \backslash frac,\; \backslash frac\; \backslash right)$ and the Hessian matrix :$\backslash textstyle\; \backslash mbox(F)=\; \backslash begin\; \backslash frac\; \&\; \backslash frac\; \&\; \backslash frac\; \backslash \backslash \; \backslash frac\; \&\; \backslash frac\; \&\; \backslash frac\; \backslash \backslash \; \backslash frac\; \&\; \backslash frac\; \&\; \backslash frac\; \backslash end\; .$ The mean curvature is given by: :$H\; =\; \backslash frac$ Another form is as the divergence of the unit normal. A unit normal is given by $\backslash frac$ and the mean curvature is :$H\; =\; -\backslash nabla\backslash cdot\; \backslash left(\backslash frac\backslash right).$

Mean curvature 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 :$\backslash kappa\_1\; =\; \backslash kappa\_2\; =\; r^\; \backslash ,$.

Minimal surfaces

175px|A rendering of Costa's 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.

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 Bryant surfaces..

See also

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

Notes

References

*. * {{curvature Category:Differential geometry Category:Differential geometry of surfaces Category:Surfaces Category:Curvature (mathematics)

Definition

Let $p$ be a point on the surface $S$. 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 $\backslash theta$: :$H\; =\; \backslash frac\backslash int\_0^\; \backslash kappa(\backslash theta)\; \backslash ;d\backslash theta$. By applying Euler's theorem, this is equal to the average of the principal curvatures : :$H\; =\; (\backslash kappa\_1\; +\; \backslash kappa\_2).$ More generally , for a hypersurface $T$ the mean curvature is given as :$H=\backslash frac\backslash sum\_^\; \backslash kappa\_.$ 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 :$H\backslash vec\; =\; g^\backslash nabla\_i\backslash nabla\_j\; X,$ using the ''Gauss-Weingarten relations,'' where $X(x)$ is a smoothly embedded hypersurface, $\backslash vec$ a unit normal vector, and $g\_$ 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-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".

Surfaces in 3D space

For a surface defined in 3D space, the mean curvature is related to a unit normal of the surface: :$2\; H\; =\; -\backslash nabla\; \backslash cdot\; \backslash hat\; n$ 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. Mean Curvature may also be calculated :$2\; H\; =\; \backslash text((\backslash mathrm)(\backslash mathrm^))$ where I and II denote first and second quadratic form matrices, 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 fundamental forms as $$\backslash frac$$ where $E=\backslash mathrm(u,u),\; F=\backslash mathrm(u,v),\; G=\backslash mathrm(v,v),\; l=\backslash mathrm(u,u),\; m=\backslash mathrm(u,v),\; n=\backslash mathrm(v,v)$. For the special case of a surface defined as a function of two coordinates, e.g. $z\; =\; S(x,\; y)$, and using the upward pointing normal the (doubled) mean curvature expression is :$\backslash begin2\; H\; \&\; =\; -\backslash nabla\; \backslash cdot\; \backslash left(\backslash frac\backslash right)\; \backslash \backslash \; \&\; =\; \backslash nabla\; \backslash cdot\; \backslash left(\backslash frac\; \backslash right)\; \backslash \backslash \; \&\; =\; \backslash frac.\; \backslash end$ 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 $z\; =\; S(r)$, :$2\; H\; =\; \backslash frac\; +\; \backslash frac,$ where $\backslash frac$ comes from the derivative of $z\; =\; S(r)=S\backslash left(\backslash scriptstyle\; \backslash sqrt\; \backslash right)$.

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 $\backslash nabla\; F=\backslash left(\; \backslash frac,\; \backslash frac,\; \backslash frac\; \backslash right)$ and the Hessian matrix :$\backslash textstyle\; \backslash mbox(F)=\; \backslash begin\; \backslash frac\; \&\; \backslash frac\; \&\; \backslash frac\; \backslash \backslash \; \backslash frac\; \&\; \backslash frac\; \&\; \backslash frac\; \backslash \backslash \; \backslash frac\; \&\; \backslash frac\; \&\; \backslash frac\; \backslash end\; .$ The mean curvature is given by: :$H\; =\; \backslash frac$ Another form is as the divergence of the unit normal. A unit normal is given by $\backslash frac$ and the mean curvature is :$H\; =\; -\backslash nabla\backslash cdot\; \backslash left(\backslash frac\backslash right).$

Mean curvature 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 :$\backslash kappa\_1\; =\; \backslash kappa\_2\; =\; r^\; \backslash ,$.

Minimal surfaces

175px|A rendering of Costa's 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.

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 Bryant surfaces..

See also

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

Notes

References

*. * {{curvature Category:Differential geometry Category:Differential geometry of surfaces Category:Surfaces Category:Curvature (mathematics)