Comments on: Second variation formula for minimal surfaces

By: The Willmore conjecture after Fernando Coda Marques and Andre Neves « Disquisitiones Mathematicae

Mon, 23 Apr 2012 09:50:23 +0000

[...] of . For a discussion of the index and second variation formulas for surfaces, see e.g. this post of Danny [...]

By: RoulettGK

RoulettGK — Sat, 12 Jun 2010 22:34:40 +0000

Great idea, thanks for this post!

By: Danny Calegari

Danny Calegari — Thu, 24 Sep 2009 00:04:10 +0000

Hi Olivier – if is a codimension one minimal surface in some Euclidean space, the Ricci term goes away, and the stability operator reduces to . Here is just the sum of the squares of the principal curvatures, i.e. and is the metric Laplacian on , acting on functions on , thought of as measuring the size of a (normal) variation. So for example if is very close to being totally flat, the first term is almost zero, and the only relevant term is . In other words, the “eigenvariations” of a totally flat (two-dimensional) membrane with fixed boundary are the eigenfunctions of the Laplacian in the domain. For an almost flat surface, thought of as a graph over some domain, the metric Laplacian on the surface will be well-approximated by the ordinary Laplacian in the domain the surface is a graph over; I’m guessing this is the case for a water meniscus?

Best,

Danny

By: Anonymous

Anonymous — Wed, 23 Sep 2009 18:29:36 +0000

Could you explicit your second variation formula in the case where S is Euclidian (say RxR)?

I’m just an experimental physicist trying to calculate the parametric surface that describes the shape of a water meniscus, so keeping the technical mathematical terminology to a minimum would be apreciated.

Thanks,
Olivier

By: Danny Calegari

Danny Calegari — Fri, 28 Aug 2009 17:04:13 +0000

Dear Terry – thanks for the LaTeX fix (I used \begin{matrix} instead of \begin{pmatrix}) and for the comments on the relationship between first/second variation formula and Ricci flow/GR.

I guess this is a manifestation of the idea that Ricci flow is like an “intrinsic” version of mean curvature flow, and the mean curvature is the gradient of the area functional (by the first variation formula). The fixed points for Ricci flow are Einstein manifolds, whereas the fixed points for mean curvature flow are minimal surfaces; thus in GR (where one studies Einstein manifolds – albeit with a Lorentz signature!) the second order terms are the leading ones, and one can imagine a role for the second variation formula.

Best,

Danny

By: Terence Tao

Terence Tao — Fri, 28 Aug 2009 07:26:25 +0000

WordPress LaTeX doesn’t support environments such as align*. I suppose something like a \begin{pmatrix} \end{pmatrix} might work instead.

The first variation formula is essential in the Colding-Minicozzi argument showing finite time extinction for Ricci flow, see e.g.

http://terrytao.wordpress.com/2008/04/11/285g-lecture-4-finite-time-extinction-of-the-second-homotopy-group/

The second variation formula isn’t used though, I guess because Ricci flow is first order in time rather than second order. One could speculate though that it could play a role in general relativity, which could crudely be viewed as a variant of Ricci flow that is second order in time.

(The second variation formula for geodesics, on the other hand, is used all over the place in Riemannian geometry, including multiple places in the proof of the Poincare conjecture. This is, of course, the 1-dimensional version of the minimal surfaces second variation formula.)

Incidentally, see you soon in Oz!

By: Danny Calegari

Danny Calegari — Thu, 27 Aug 2009 14:29:45 +0000

Dear anonymous – not quite sure what you mean: applying the exterior derivative twice to anything gives zero (which is certainly coordinate independent :).

In general, to differentiate a 1-form, you need a connection on the cotangent bundle. If is a 1-form, then is a section of the second tensor power of , but one cannot generally say any more than that. If a connection is torsion free (i.e. if the induced connection on satisfies ) then the antisymmetric part of is equal to ; hence is always symmetric (but depends on a choice of torsion-free connection). The point is that at a critical point of , the expression does not depend on a choice of connection.

Incidentally, I don’t know what happened to your latex code; are there any readers familiar with latex in wordpress who can say what to fix?

By: Anonymous

Anonymous — Thu, 27 Aug 2009 09:50:06 +0000

Is it also possible to define the Hessian by applying the exterior derivative twice, ? In two-dimensional coordinates the calculation would look like

And then observing that this is coordinate independent at critical points.

By: Anonymous

Anonymous — Thu, 27 Aug 2009 09:48:46 +0000

Is it also possible to define the Hessian by applying the exterior derivative twice, latex \displaystyle\begin{align*}d^2\!f &= d\left(\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy\right)\\ &= \frac{\partial^2\!f}{\partial x^2}dx^2+\frac{\partial^2\!f}{\partial x\partial y}dxdy+\frac{\partial^2\!f}{\partial y\partial x}dydx+\frac{\partial^2\!f}{\partial y^2}dy^2\\ &= \frac{\partial^2\!f}{\partial x^2}dx^2+2\frac{\partial^2\!f}{\partial x\partial y}dxdy+\frac{\partial^2\!f}{\partial y^2}dy^2\end{align*}$

And then observing that this is coordinate independent at critical points.