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, published 1831. 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} inside the three dimensional Euclidean space R3. 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 κ 1 {\displaystyle \kappa _{1}} and minimal curvature κ 2 {\displaystyle \kappa _{2}} are known as the principal curvatures of S {\displaystyle S} . The mean curvature at p ∈ S {\displaystyle p\in S} is then the average of the signed curvature over all angles θ {\displaystyle \theta } :
H = 1 2 π ∫ 0 2 π κ ( θ ) d θ {\displaystyle H={\frac {1}{2\pi }}\int _{0}^{2\pi }\kappa (\theta )\;d\theta } . By applying Euler's theorem, this is equal to the average of the principal curvatures (Spivak 1999, Volume 3, Chapter 2):
H = 1 2 ( κ 1 + κ 2 ) . {\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 = 1 n ∑ i = 1 n κ i . {\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
H n → = g i j ∇ i ∇ j X , {\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 g i j {\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".
Surfaces in 3D space For a surface defined in 3D space, the mean curvature is related to the divergence of a unit normal of the surface:
2 H = − ∇ ⋅ n ^ {\displaystyle 2H=-\nabla \cdot {\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
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