Talk:Mean curvature

Discussion[edit]

This article needs some serious rewording, but I don't know enough about the subject to write it. 65.25.226.98 22:07, 8 October 2005 (UTC)[reply]

The formula for axissymmetric 3D-curvature is not correct, or is it? $1/r$ should not appear in the formula, maybe it is $1/S(r)$ ? 131.188.102.71 (talk) 16:11, 18 December 2007 (UTC)[reply]

I agree with the above. Formula dimensions do not look correct. —Preceding unsigned comment added by 72.200.111.2 (talk) 03:17, 25 March 2009 (UTC)[reply]

Can someone discuss minimal surfaces which are constrained by volume?

For instance a sphere with two equivalent circular plates at each end would deflate but maintain a 'circular' curvature until deflating into a cilindar and then into a rotated ctenary (I think). Or if the plates are asku then the surface would be a sphere only at one volume.

Since these can be entierly convex what defines the surface?

Email me User:Timothy Sheridan

The Description of the Principal Curvature in Fact is wrong. If one takes the Max. or Min. over all the curvatures of curves passing through q, this terms are not even defined. They occur as Inf. and Sup. and are always zero resp. infinity. One has to look at curves that "occur" by intersecting the surface with a 2-plane in q, and this 2-plane has to be perpendicular to the tangent space at q. I am not able to correct this on my own, perhaps someone else would like to. —Preceding unsigned comment added by 80.143.251.206 (talk) 20:12, 29 May 2008 (UTC)[reply]
- You're right - I'll amend it. (I guess it's not very important as this has taken 4 years!) Jamontaldi (talk) 15:16, 29 October 2012 (UTC)[reply]

Additional def. of H[edit]

I think, there's something wrong with the equation $H{\vec {n}}=g^{ij}\nabla _{i}\nabla _{j}X$ . On the RHS of this euqation there's a vector, ${\vec {n}}=n^{i}$ , whereas on the LHS it is summed over all i,j (X is not defined as vector but as parametrisation of the submanifold/ hypersurface; the choice of the letter X may be missleading; remember that in general relativity vector fields are often denoted by X or Y!). I think the problem might be solved by either cancelling the normal vector on the RHS or by replacing the inner product $g(X,X)$ by an cross product $X\times X$ 86.32.173.12 (talk) 19:18, 28 April 2011 (UTC)[reply]

The equation is OK, it means $Hn^{k}=g^{ij}\nabla _{i}\nabla _{j}X^{k}$ $Hn^{k}=g^{ij}\nabla _{i}\nabla _{j}X^{k}$ (for each $k$ $k$ ). In fact I think the best approach is to define the mean curvature vector to be $H^{k}=g^{ij}\nabla _{i}\nabla _{j}X^{k}$ $H^{k}=g^{ij}\nabla _{i}\nabla _{j}X^{k}$ . That is ${\vec {H}}=H{\vec {n}}$ ${\vec {H}}=H{\vec {n}}$ , but it has the advantage of being independent of any choice of orientation (choice of ${\vec {n}}$ ${\vec {n}}$ ). Equivalently ${\vec {H}}$ ${\vec {H}}$ is the projection into the normal space of the vector ${\frac {1}{2}}\left(X_{11}^{k}+X_{22}^{k}\right)$ ${\frac {1}{2}}\left(X_{11}^{k}+X_{22}^{k}\right)$ , which can be shown to be independent of the embedding $X$ $X$ . Jamontaldi (talk) 16:19, 29 October 2012 (UTC)[reply]
- There is still a problem, since the LHS is obviously normal, while the RHS is a sum of covariant derivatives which are as far as I understand them by definition tangential. 79.224.79.161 (talk) 17:11, 5 February 2013 (UTC)[reply]

Sophie Germain vs Young and Laplace[edit]

I'm a little confused by the history here. According to Sophie Germain, her work on elasticity did not begin until 1809, and our mean curvature article states that it was this work that introduced mean curvature. However, according to Young–Laplace equation, Young and Laplace developed this equation in 1805 and 1806, and it uses mean curvature in an essential way. Can anyone else help untangle this? —David Eppstein (talk) 18:36, 7 September 2012 (UTC)[reply]

Surfaces in 3D space[edit]

Surely the sign of H is wrong. It should be $H{\vec {n}}=-\nabla \cdot {\vec {n}}$ . If no-one disagrees, I'll change it. It then follows that $H$ is positive if the surface curves towards the normal - surely more reasonable! James Montaldi 20:51, 31 October 2012 (UTC)

A few serious problems with this article[edit]

1. The first sentence reads:

"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."

This neglects to mention whether the embedded surface needs to be a hypersurface (as is implied but not clearly stated elsewhere in the article). Otherwise, does "surface" here mean submanifold, or does it specifically mean a 2-dimensional submanifold (of n-space)? Not clear.

Also, "some ambient space such as Euclidean space" suggests that the space need not be a manifold. But for all remotely common situations I've ever heard of, differential geometry takes place inside manifolds. So this word ought to be used instead of "space".

Better idea: The *first* sentence could just refer to (by default, ordinary) surfaces in 3-space. The next paragraph could mention the more general setting of hypersurfaces (or submanifolds?) of Euclidean space, or more generally, of Riemannian manifolds.

2. The Definition section says nothing about whether it is *initially* discussing an (ordinary) surface in 3-space -- as it appears to be doing. If so, it needs to make that clear. Then it mentions a hypersurface, but never says where the hypersurface is embedded.

3. The only subsection of the Definition section is titled "Surfaces in 3D space", which further makes it unclear whether that was what was just being discussed. Wasn't it? Clarity is needed here.

4. The Mean curvature in fluid mechanics section mentions the utterly trivial fact that in fluid mechanics, "mean curvature" is often used to mean the sum of the principal curvatures at a point, without the denominator (of 2, or n, as the case may be). (A whole section for this???????)

This oddly omits the fact that *everyone* who uses the concept of mean curvature frequently may do the same thing -- above all, differential geometers.

5. There is no reason that the hypersurface (or submanifold?) must be *embedded* in its ambient manifold. It needs only to be locally embedded (i.e., immersed). This ought to be mentioned.

6. The initial sentences mention that a minimal surface has mean curvature zero. It should not leave the impression that there might be other surfaces that also have mean curvature zero (even if that is made explicit later in the article). It should be stated *there* that surfaces with mean curvature zero are precisely the minimal surfaces.Daqu (talk) 22:32, 13 December 2012 (UTC)[reply]

"Constant mean curvature" in fluid mechanics[edit]

The lead states that, "It is important ... in the analysis of physical interfaces between fluids (such as soap films) which by the Young–Laplace equation have constant mean curvature." The Young-Laplace equation applies to hydrostatic fluids, and for those fluids the statement is true; in general however, the Young-Laplace pressure difference can be included as a correction term for the inclusion of surface tension in the Navier-Stokes equations, which include non-hydrostatic flow. It is not hard to see that an unsteady flow can support a non-uniform mean curvature in a fluid interface. An intuitive example is a vibrating soap bubble. I propose the correction: "...which, for example, have constant mean curvature in static flows, by the Young-Laplace equation." Osmanthus22 (talk) 14:06, 22 May 2018 (UTC)[reply]