In mathematics, the mean curvature of a surface 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.
^Marie-Louise Dubreil-Jacotin on Sophie Germain Archived 2008-02-23 at the Wayback Machine
^Lodder, J. (2003). "Curvature in the Calculus Curriculum". The American Mathematical Monthly. 110 (7): 593–605. doi:10.2307/3647744. JSTOR 3647744.
In mathematics, the meancurvature H {\displaystyle H} of a surface S {\displaystyle S} is an extrinsic measure of curvature that comes from differential...
manifold. This leads to the concepts of maximal curvature, minimal curvature, and meancurvature. In Tractatus de configurationibus qualitatum et motuum, the...
In the field of differential geometry in mathematics, meancurvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold (for...
within 100 m (330 ft) of mean sea level (neglecting geoid height). Additionally, the radius can be estimated from the curvature of the Earth at a point...
that locally minimizes its area. This is equivalent to having zero meancurvature (see definitions below). The term "minimal surface" is used because...
the principal curvatures. Their average is called the meancurvature of the surface, and their product is called the Gaussian curvature. There are many...
mathematical fields of differential geometry and geometric analysis, inverse meancurvature flow (IMCF) is a geometric flow of submanifolds of a Riemannian or pseudo-Riemannian...
equations. He is known for foundational contributions to the theory of the meancurvature flow, including Huisken's monotonicity formula, which is named after...
curvatures is the Gaussian curvature, K, and the average (k1 + k2)/2 is the meancurvature, H. If at least one of the principal curvatures is zero at every point...
that this transformation formula is for the meancurvature vector, and the formula for the meancurvature H {\displaystyle H} in the hypersurface case...
positive meancurvature. Other such immersed surfaces as minimal surfaces have constant meancurvature. The sphere has constant positive Gaussian curvature. Gaussian...
Gaussian curvature or Gauss curvature Κ of a smooth surface in three-dimensional space at a point is the product of the principal curvatures, κ1 and κ2...
Huisken's analysis of meancurvature flow. Robert Bartnik and Simon considered the problem of prescribing the boundary and meancurvature of a spacelike hypersurface...
to the meancurvature, as seen in the Young–Laplace equation. For an open soap film, the pressure difference is zero, hence the meancurvature is zero...
}}\int _{\Sigma }H^{2}da\right),} where H {\displaystyle H} is the meancurvature of Σ {\displaystyle \Sigma } . In the Schwarzschild metric, the Hawking...
Meancurvature flow, as in soap films; critical points are minimal surfaces Curve-shortening flow, the one-dimensional case of the meancurvature flow...
invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization. The simplest...
it is embedded, and in 1997 Große-Brauckmann provided CMC (constant meancurvature) variants of the gyroid and made further numerical investigations about...
Craig Evans as supervisor. Ilmanen and Gerhard Huisken used inverse meancurvature flow to prove the Riemannian Penrose conjecture, which is the fifteenth...
is a prime p such that 2p + 1 is also prime. The Germain curvature (also called meancurvature) is ( k 1 + k 2 ) / 2 {\displaystyle (k_{1}+k_{2})/2} ,...
{c^{2}}{\left(a^{2}+\left(c^{2}-a^{2}\right)\cos ^{2}\beta \right)^{2}}},} and its meancurvature is H ( β , λ ) = c ( 2 a 2 + ( c 2 − a 2 ) cos 2 β ) 2 a ( a 2 + (...
liquid-vapor boundary is due to Laplace pressure, which is proportional to the meancurvature. Solving the above equation for both convex and concave surfaces yields:...
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