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In mathematics, the mean curvature H{displaystyle H} of a surface S{displaystyle 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.^[1][2]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{displaystyle p} be a point on the surface S{displaystyle S}. Each plane through p{displaystyle p} containing the normal line to S{displaystyle S} cuts S{displaystyle 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 θ{displaystyle theta } (always containing the normal line) that curvature can vary. The maximal curvature κ1{displaystyle kappa _{1}} and minimal curvature κ2{displaystyle kappa _{2}} are known as the principal curvatures of S{displaystyle S}.

The mean curvature at p∈S{displaystyle pin S} is then the average of the signed curvature over all angles θ{displaystyle theta }:

H=12π∫02πκ(θ)dθ{displaystyle H={frac {1}{2pi }}int _{0}^{2pi }kappa (theta );dtheta }.

By applying Euler's theorem, this is equal to the average of the principal curvatures (Spivak 1999, Volume 3, Chapter 2):

H=12(κ1+κ2).{displaystyle H={1 over 2}(kappa _{1}+kappa _{2}).}

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

H=1n∑i=1nκi.{displaystyle 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{displaystyle H} may be written in terms of the covariant derivative ∇{displaystyle nabla } as

Hn→=gij∇i∇jX,{displaystyle H{vec {n}}=g^{ij}nabla _{i}nabla _{j}X,}

using the Gauss-Weingarten relations, where X(x){displaystyle X(x)} is a smoothly embedded hypersurface, n→{displaystyle {vec {n}}} a unit normal vector, and gij{displaystyle 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{displaystyle 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

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

2H=−∇⋅n^{displaystyle 2H=-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

2H=Trace((II)(I−1)){displaystyle 2H={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){displaystyle z=S(x,y)}, and using the upward pointing normal the (doubled) mean curvature expression is

2H=−∇⋅(∇(z−S)|∇(z−S)|)=∇⋅(∇S−∇z1+|∇S|2)=(1+(∂S∂x)2)∂2S∂y2−2∂S∂x∂S∂y∂2S∂x∂y+(1+(∂S∂y)2)∂2S∂x2(1+(∂S∂x)2+(∂S∂y)2)3/2.{displaystyle {begin{aligned}2H&=-nabla cdot left({frac {nabla (z-S)}{|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 xpartial 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{aligned}}}

In particular at a point where ∇S=0{displaystyle nabla S=0}, the mean curvature is half the trace of the Hessian matrix of S{displaystyle S}.

If the surface is additionally known to be axisymmetric with z=S(r){displaystyle z=S(r)},

2H=∂2S∂r2(1+(∂S∂r)2)3/2+∂S∂r1r(1+(∂S∂r)2)1/2,{displaystyle 2H={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}{rleft(1+left({frac {partial S}{partial r}}right)^{2}right)^{1/2}}},}

where ∂S∂r1r{displaystyle {frac {partial S}{partial r}}{frac {1}{r}}} comes from the derivative of z=S(r)=S(x2+y2){displaystyle z=S(r)=Sleft(scriptstyle {sqrt {x^{2}+y^{2}}}right)}.

Implicit form of mean curvature

The mean curvature of a surface specified by an implicit equation F(x,y,z)=0{displaystyle F(x,y,z)=0} can be calculated by using the gradient ∇F=(∂F∂x,∂F∂y,∂F∂z){displaystyle nabla F=left({frac {partial F}{partial x}},{frac {partial F}{partial y}},{frac {partial F}{partial z}}right)} and the Hessian matrix

Hess(F)=(∂2F∂x2∂2F∂x∂y∂2F∂x∂z∂2F∂y∂x∂2F∂y2∂2F∂y∂z∂2F∂z∂x∂2F∂z∂y∂2F∂z2).{displaystyle textstyle {mbox{Hess}}(F)={begin{pmatrix}{frac {partial ^{2}F}{partial x^{2}}}&{frac {partial ^{2}F}{partial xpartial y}}&{frac {partial ^{2}F}{partial xpartial z}}\{frac {partial ^{2}F}{partial ypartial x}}&{frac {partial ^{2}F}{partial y^{2}}}&{frac {partial ^{2}F}{partial ypartial z}}\{frac {partial ^{2}F}{partial zpartial x}}&{frac {partial ^{2}F}{partial zpartial y}}&{frac {partial ^{2}F}{partial z^{2}}}end{pmatrix}}.}

The mean curvature is given by:^[4][5]

H=∇F Hess(F) ∇FT−|∇F|2Trace(Hess(F))2|∇F|3{displaystyle H={frac {nabla F {mbox{Hess}}(F) nabla F^{mathsf {T}}-|nabla F|^{2},{text{Trace}}({mbox{Hess}}(F))}{2|nabla F|^{3}}}}

Another form is as the divergence of the unit normal. A unit normal is given by ∇F|∇F|{displaystyle {frac {nabla F}{|nabla F|}}} and the mean curvature is

H=−12∇⋅(∇F|∇F|).{displaystyle H=-{frac {1}{2}}nabla cdot left({frac {nabla F}{|nabla F|}}right).}

Mean curvature in fluid mechanics

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

Hf=(κ1+κ2){displaystyle 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 Hf{displaystyle H_{f}}; the two curvatures are equal to the reciprocal of the droplet's radius

κ1=κ2=r−1{displaystyle kappa _{1}=kappa _{2}=r^{-1},}.

Minimal surfaces

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.^[6]

See also

• Gaussian curvature


• Mean curvature flow


• Inverse mean curvature flow


• First variation of area formula


• Stretched grid method

Notes

References

• 
  Spivak, Michael (1999), A comprehensive introduction to differential geometry (Volumes 3-4) (3rd ed.), Publish or Perish Press, ISBN 978-0-914098-72-0, (Volume 3), (Volume 4).


• 
  P.Grinfeld (2014). Introduction to Tensor Analysis and the Calculus of Moving Surfaces. Springer. ISBN 978-1-4614-7866-9.