Minimal Surfaces, the Plateau Problem & Stability

From geodesics to surfaces: vary the area

The previous guide ran the variational machine on a one-dimensional object — a curve — and found that making energy critical produces a geodesic. Now raise the dimension by one. Replace the curve by an immersed surface Sigma sitting inside a Riemannian manifold M (think of a soap film floating in R^3), replace fixed endpoints by a fixed boundary curve Gamma = partial-Sigma, and replace length by area. The whole subject is governed by the same two questions you already trust: what does the first derivative of area single out, and what does the second derivative say about stability? The grammar is identical; only the geometry of the answer is richer.

Set up the variation exactly as before. Take a smooth normal variation of Sigma — a family Sigma_s of surfaces with Sigma_0 = Sigma, moved at infinitesimal velocity V = f N, where N is the unit normal and f is a function on Sigma that vanishes on the boundary Gamma (so the wire frame stays pinned). Only the normal component matters: pushing a surface tangentially merely reparametrizes it without changing its area, the surface analogue of the reparametrization symmetry that pestered length last time. So the honest degrees of freedom are encoded in the single scalar f, and the area functional A(Sigma_s) = integral over Sigma_s of dA becomes an ordinary function of s to differentiate.

First variation of area: H = 0

Now differentiate. The area element dA depends on the first fundamental form (the induced metric) of Sigma_s, so moving the surface in the normal direction changes the metric, and that change is governed by how the surface bends — precisely the second fundamental form you met two rungs back. Carrying out the s-derivative of dA at s = 0 produces a clean, memorable result: the rate of change of area equals minus the pairing of the variation against the mean curvature vector. In coordinates, H is the trace of the shape operator (equivalently the trace of the second fundamental form with respect to the induced metric), so it sums the principal curvatures.

First variation of area, normal variation  V = f N  ( f = 0 on the boundary Gamma ):

  d/ds |_(s=0)  A(Sigma_s)  =  - integral_Sigma  f * H  dA

  where  H = trace of the shape operator = kappa_1 + kappa_2 + ... + kappa_(n-1)   (sum of principal curvatures)

Vanishes for ALL admissible f   <=>   H = 0  everywhere on Sigma     ( MINIMAL )

Graph case  Sigma = graph of u(x,y) in R^3,  H = 0  becomes the minimal surface equation:

  (1 + u_y^2) u_xx  -  2 u_x u_y u_xy  +  (1 + u_x^2) u_yy  =  0

  i.e.   div ( grad u / sqrt(1 + |grad u|^2) )  =  0

First variation of area: its integrand pairs the normal speed f against the mean curvature H. Demanding it vanish for every f gives H = 0, which for a graph z = u(x,y) is the (nonlinear, but only mildly) minimal surface equation.

So a surface is a critical point of area for every boundary-fixing variation if and only if its mean curvature vanishes identically — this is the definition of a minimal surface, and the higher-dimensional version is a minimal submanifold. Beware the single most common misconception in the subject: 'minimal' means critical, not minimizing. The name is historical and misleading; H = 0 only says the first derivative of area is zero, exactly as 'geodesic' meant critical-length, not shortest. A small piece of soap film really does locally minimize, but a large minimal surface can be a saddle of the area functional, and deciding which is the entire content of the stability discussion below. Written for a graph z = u(x,y) in R^3, H = 0 unfolds into the second-order minimal surface equation shown above — nonlinear, but in divergence form, which is what makes the modern existence theory possible.

Two faces of minimality: soap films and harmonic coordinates

Before chasing existence, build intuition with the canonical examples, because in this subject a worked surface teaches more than a general theorem. The simplest nonplanar minimal surface in R^3 is the catenoid: spin a catenary y = cosh(x) about the x-axis. Its two principal curvatures are equal in magnitude and opposite in sign at every point, so they cancel and H = 0, even though it is visibly curved. The helicoid — the staircase ruled surface traced by a horizontal line screwing up the z-axis — is also minimal, and remarkably it is locally isometric to the catenoid: you can bend one into the other through a one-parameter family of minimal surfaces without stretching. These are the surfaces a soap film actually forms; the catenoid is the film spanning two coaxial rings.

There is a second, deeper face of minimality that quietly connects this guide to the next one. Choose isothermal (conformal) parameters on the surface — coordinates in which the induced metric is a positive function times the flat metric, which always exist locally on a surface by conformal parameterization. In such coordinates the condition H = 0 becomes exactly the statement that the immersion map Sigma -> R^3 is harmonic: each coordinate function is annihilated by the Laplacian. So a conformally parametrized minimal surface is precisely a conformal harmonic map into R^3. This is no coincidence — it is the bridge to the next guide, where the Dirichlet energy and harmonic maps take center stage; minimal surfaces are their first and most tangible incarnation.

Mind the sign and trace conventions — sources disagree sharply here. Some authors define H as the SUM of the principal curvatures (the trace of the shape operator), others as their AVERAGE (the trace divided by n-1); some take H to be a scalar relative to a chosen normal, others a normal-vector-valued quantity. The minimal condition H = 0 is unaffected by any of these, which is a mercy. But the moment you compute a nonzero H, the mean curvature of the unit sphere, or a stability inequality, the constant and sign you inherit depend entirely on the book. We use H = trace of the shape operator (the sum); state your convention before you compute, and translate carefully when reading a second source.

The Plateau problem: does a film always exist?

Soap films pose a sharp existence question, named for the blind physicist who catalogued them experimentally. The Plateau problem asks: given a closed curve Gamma in R^3 (a bent wire loop), does there exist a surface of least area spanning it — equivalently a minimal surface with boundary Gamma? The physical answer is obviously yes; the mathematical proof is hard and instructive, because the naive strategy fails in a revealing way. Take a minimizing sequence of surfaces whose areas approach the infimum, and try to extract a limit. Area is not coercive in any classical function space: surfaces in the sequence can grow long thin tentacles, oscillate wildly, or pinch — the limit may not be a smooth surface at all, and the area can drop in the limit. The honest difficulty is that the space of surfaces is too floppy to be compact.

There are two great resolutions, and a learner should know that they answer slightly different questions. The classical Douglas-Rado solution (1930s) restricts to surfaces of disk type and exploits the harmonic-map face from the previous section: minimize the Dirichlet energy of conformal maps from a fixed disk, which is well-behaved where area is not, then recover a minimal surface from the energy-minimizer. This produces a disk spanning Gamma but says nothing about other topological types and tolerates self-intersections. The modern resolution enlarges the very notion of 'surface': in geometric measure theory one minimizes over currents or varifolds, generalized surfaces that DO form a compact space, obtains a minimizer cheaply by compactness, and then does the genuinely hard work — regularity theory — to prove the abstract minimizer is a smooth surface away from a small singular set.

Be honest about what is proved and what is surveyed. That a least-area minimizer EXISTS as a varifold follows fairly directly from the compactness of these generalized surfaces — that part is a clean theorem. That the minimizer is actually a smooth manifold is the deep part, and its conclusion is dimension-dependent: an area-minimizing hypersurface is smooth away from a singular set of codimension at least 7, so in R^3, R^4, ..., up through R^7 it is everywhere smooth, but in R^8 genuine singularities appear (the Simons cone is the first). This dimensional threshold is a real theorem of Almgren, De Giorgi, Federer, and others — a course in itself, not a guide. We state and motivate it, never proving it here; treat any one-line 'minimizers are always smooth' claim as false in high dimensions.

Second variation and stability

Existence of a critical surface settles nothing about whether it is a minimum, exactly as a critical geodesic could be a saddle. So differentiate area a second time, AT a minimal surface (where the first-order term drops out), and read the second variation. The structure precisely mirrors the geodesic index form from the previous guide, now with two curvature contributions instead of one. For a normal variation f N with f vanishing on the boundary, the second variation of area takes the form below: a positive 'bending cost' integral of |grad f|^2, fighting two destabilizing terms — the squared norm of the second fundamental form |A|^2, which is how much Sigma itself curves, and the ambient Ricci curvature of M in the normal direction.

Write down the form. For a normal variation f N at a minimal hypersurface Sigma in M (with f = 0 on the boundary), the second variation is Q(f,f) = integral_Sigma [ |grad_Sigma f|^2 - ( |A|^2 + Ric_M(N,N) ) f^2 ] dA — read it as one positive term fighting two negative ones.
The positive term |grad_Sigma f|^2 is the bending cost: making the variation field wiggle always costs area, exactly as the term |nabla_(gamma') V|^2 did in the geodesic index form. It is always nonnegative and works FOR stability.
The two subtracted terms work AGAINST stability: |A|^2 is the squared norm of the second fundamental form (how much Sigma curves into M), and Ric_M(N,N) is the ambient Ricci in the normal direction. Positive ambient curvature, like a sphere round Sigma, pushes toward instability.
Integrate by parts to read off the operator. Q(f,f) is the quadratic form of L f = - Delta_Sigma f - ( |A|^2 + Ric_M(N,N) ) f, the Jacobi (stability) operator — a Schrodinger-type operator. Stability is then exactly lambda_1(L) >= 0: the lowest eigenvalue is nonnegative.

A minimal surface is called stable when this quadratic form is nonnegative for every admissible f — that is, when no infinitesimal wiggle lowers area to second order. Notice the analytic payoff: after one integration by parts the form belongs to a Schrodinger-type operator L = -Delta - (|A|^2 + Ric), the Jacobi operator, and stability is exactly the statement that its lowest eigenvalue is nonnegative. Just as in the geodesic story, the spectrum of a linear operator decides the geometry, and its kernel — solutions of L f = 0, the surface's Jacobi fields — records the infinitesimal deformations through nearby minimal surfaces. Soap films are stable by construction (nature finds local minima); the catenoid spanning two rings is stable only while the rings are close enough, and snaps when pulled too far apart, the instant lambda_1(L) crosses zero.

This stability inequality is one of the most productive tools in modern geometry, and its power comes from reading it as a constraint, not a definition. Two examples set honest expectations. Bernstein's theorem says a minimal graph defined over all of R^2 (an entire solution of the minimal surface equation) must be a plane — and the slick modern proof feeds stability into a clever choice of test function f. The naive hope that 'this holds in every dimension' is FALSE: the analogue is true for entire minimal graphs in R^n only up to n = 8, and fails beyond, the same dimension-7 threshold that governed regularity. Second, Schoen-Yau used the stability inequality to prove the positive mass theorem in general relativity, by showing a stable minimal surface cannot live in a region with the wrong sign of scalar curvature. As always, we state these to show what the machine buys; each is a real proof, a paper or a course, not a paragraph.