Mean curvature

# Mean curvature

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the mean curvature of a surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

is an extrinsic measure of curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

that comes from differential geometry and that locally describes the curvature of an embedded
Embedding
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....

surface in some ambient space such as Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

.

The concept was introduced by Sophie Germain
Sophie Germain
Marie-Sophie Germain was a French mathematician, physicist, and philosopher. Despite initial opposition from her parents and difficulties presented by a gender-biased society, she gained education from books in her father's library and from correspondence with famous mathematicians such as...

in her work on elasticity theory.

## Definition

Let be a point on the surface . Consider all curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

s on passing through . Every such has an associated curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

given at . Of those curvatures , at least one is characterized as maximal
Maxima and minima
In mathematics, the maximum and minimum of a function, known collectively as extrema , are the largest and smallest value that the function takes at a point either within a given neighborhood or on the function domain in its entirety .More generally, the...

and one as minimal
Minimal
Minimal can mean:* In mathematics, see Minimal element.* In music, see Minimalist music.* In electronic music, see Minimal techno.* In art see Minimalism.* In computing, see Computing minimalism.* Names:...

, and these two curvatures are known as the principal curvature
Principal curvature
In differential geometry, the two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point. They measure how the surface bends by different amounts in different directions at that point.-Discussion:...

s
of .

The mean curvature at is then the average of the principal curvatures , hence the name:

More generally , for a hypersurface
Hypersurface
In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...

the mean curvature is given as

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 may be written in terms of the covariant derivative
Covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...

as
using the Gauss-Weingarten relations, where is a family of smoothly embedded hypersurfaces, a unit normal vector, and the metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

.

A surface is a minimal surface
Minimal surface
In mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....

if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

the mean curvature is zero. Furthermore, a surface which evolves under the mean curvature of the surface , is said to obey a heat-type equation
Heat equation
The heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...

called the mean curvature flow
Mean curvature flow
In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold . Intuitively, a family of surfaces evolves under mean curvature flow if the velocity of which a point on the surface moves is given by the mean...

equation.

The sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

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
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...

of the surface:

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 "away" from the normal. The formula above holds for surfaces in 3D space defined in any manner, as long as the divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...

of the unit normal may be calculated.

For the special case of a surface defined as a function of two coordinates, eg , and using downward pointing normal the (doubled) mean curvature expression is

If the surface is additionally known to be axisymmetric with ,

where comes from the derivative of .

## Mean curvature in fluid mechanics

An alternate definition is occasionally used in fluid mechanics
Fluid mechanics
Fluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...

to avoid factors of two:.

This results in the pressure according to the Young-Laplace equation inside an equilibrium spherical droplet being surface tension
Surface tension
Surface tension is a property of the surface of a liquid that allows it to resist an external force. It is revealed, for example, in floating of some objects on the surface of water, even though they are denser than water, and in the ability of some insects to run on the water surface...

times ; the two curvatures are equal to the reciprocal of the droplet's radius.

## Minimal surfaces

A minimal surface is a surface which has zero mean curvature at all points. Classic examples include the catenoid
Catenoid
A catenoid is a three-dimensional surface made by rotating a catenary curve about its directrix. Not counting the plane, it is the first minimal surface to be discovered. It was found and proved to be minimal by Leonhard Euler in 1744. Early work on the subject was published also by Jean Baptiste...

, helicoid
Helicoid
The helicoid, after the plane and the catenoid, is the third minimal surface to be known. It was first discovered by Jean Baptiste Meusnier in 1776. Its name derives from its similarity to the helix: for every point on the helicoid there is a helix contained in the helicoid which passes through...

and Enneper surface
Enneper surface
In mathematics, in the fields of differential geometry and algebraic geometry, the Enneper surface is a surface that can be described parametrically by:It was introduced by Alfred Enneper in connection with minimal surface theory....

. Recent discoveries include Costa's minimal surface
Costa's minimal surface
In mathematics, Costa's minimal surface is an embedded minimal surface and was discovered in 1982 by the Brazilian mathematician Celso Costa. It is also a surface of finite topology, which means that it can be formed by puncturing a compact surface...

and the Gyroid
Gyroid
"Gyroid" redirects here. For the creature, see Animal Crossing .A gyroid is a certain infinitely connected triply periodic minimal surface discovered by Alan Schoen in 1970.The gyroid has space group Iad...

.

An extension of the idea of a minimal surface are surfaces of constant mean curvature.

• Gaussian curvature
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...

• Mean curvature flow
Mean curvature flow
In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold . Intuitively, a family of surfaces evolves under mean curvature flow if the velocity of which a point on the surface moves is given by the mean...

• Inverse mean curvature flow
Inverse mean curvature flow
In the field of differential geometry in mathematics, inverse mean curvature flow is an example of a geometric flow of hypersurfaces a Riemannian manifold...

• First variation of area formula
First variation of area formula
In Riemannian geometry, the first variation of area formula relates the mean curvature of a hypersurface to the rate of change of its area as it evolves in the outward normal direction....