Mean curvature

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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.

The concept was introduced by Sophie Germain in her work on elasticity theory.[1][2] 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 by the Young–Laplace equation have constant mean curvature.

Definition[edit]

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 (always containing the normal line) that curvature can vary, and the maximal curvature \kappa_1 and minimal curvature \kappa_2 are known as the principal curvatures of S.

The mean curvature at p\in S is then the average of the principal curvatures (Spivak 1999, Volume 3, Chapter 2), hence the name:

H = {1 \over 2} (\kappa_1 + \kappa_2).

More generally (Spivak 1999, Volume 4, Chapter 7), for a hypersurface T the mean curvature is given as

H=\frac{1}{n}\sum_{i=1}^{n} \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 \nabla as

H\vec{n} = g^{ij}\nabla_i\nabla_j X,

using the Gauss-Weingarten relations, where  X(x) is a smoothly embedded hypersurface, \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-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".[3]

Surfaces in 3D space[edit]

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

2 H = -\nabla \cdot \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

 H = \text{Trace}( (II)(I^{-1}))

where I and II denote first and second quadratic form matrices, respectively.

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

\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} - 
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 \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 = \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}},

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

Mean curvature in fluid mechanics[edit]

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

H_f = (\kappa_1 + \kappa_2) \,.

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

\kappa_1 = \kappa_2 = r^{-1} \,.

Minimal surfaces[edit]

A rendering of Costa's minimal surface.
Main article: 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. The surfaces of unit constant mean curvature in hyperbolic space are called Bryant surfaces.[4]

See also[edit]

Notes[edit]

  1. ^ Dubreil-Jacotin on Sophie Germain
  2. ^ Lodder, J. (2003). "Curvature in the Calculus Curriculum". The American Mathematical Monthly 110 (7): 593–605. doi:10.2307/3647744. JSTOR 3647744.  edit
  3. ^ http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pjm/1102702809
  4. ^ Rosenberg, Harold (2002), "Bryant surfaces", The global theory of minimal surfaces in flat spaces (Martina Franca, 1999), Lecture Notes in Math. 1775, Berlin: Springer, pp. 67–111, doi:10.1007/978-3-540-45609-4_3, MR 1901614 .

References[edit]

  • 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) .