The Plateau Problem: the Architecture of Douglas' Proof

Derives Proposition A43.3. Pointwise on \(B\), the arithmetic–geometric mean inequality \(\left(\sqrt{E}-\sqrt{G}\right)^{2}\ge0\) gives \(E+G\ge2\sqrt{EG}\), with equality if and only if \(E=G\); and \(\sqrt{EG}\ge\sqrt{EG-F^{2}}\), with equality if and only if \(F=0\), since \(F^{2}\ge0\) and both radicands are nonnegative — indeed \(EG-F^{2}=\abs{\vect{X}_{u}}^{2}\abs{\vect{X}_{v}}^{2} -\left(\vect{X}_{u}\cdot\vect{X}_{v}\right)^{2} =\abs{\vect{X}_{u}\times\vect{X}_{v}}^{2}\ge0\) by Lagrange's identity. Chaining the two,

and integrating over \(B\) gives \(D\ge A\). Equality of the integrals of two continuous functions ordered pointwise forces equality of the functions almost everywhere, and by the two equality cases just identified that means \(E=G\) and \(F=0\) almost everywhere, i.e.\ conformality.

Derives Lemma A43.4. Write \(\vect{Y}=\vect{X}\circ\tau\). By the chain rule (Proposition 11.31), for each of the three components,

the derivatives of \(\vect{X}\) being evaluated at \(\tau\left(u,v\right)\). The Cauchy–Riemann equations for \(\tau\) read \(\sigma_{u}=\varrho_{v}\) and \(\sigma_{v}=-\varrho_{u}\), so with \(J=\sigma_{u}^{2}+\varrho_{u}^{2}\) — which is \(\abs{\tau'}^{2}\) and also the Jacobian determinant \(\sigma_{u}\varrho_{v}-\sigma_{v}\varrho_{u}\) — one finds

because \(\sigma_{u}^{2}+\sigma_{v}^{2} =\varrho_{u}^{2}+\varrho_{v}^{2}=J\) and \(\sigma_{u}\varrho_{u}+\sigma_{v}\varrho_{v} =\sigma_{u}\varrho_{u}-\varrho_{u}\sigma_{u}=0\) by Cauchy–Riemann. Hence the integrand of \(D\left[\vect{Y}\right]\) is the integrand of \(D\left[\vect{X}\right]\) composed with \(\tau\) and multiplied by the Jacobian \(J\), and the change-of-variables formula for a bijective \(C^{1}\) substitution gives \(D\left[\vect{Y}\right]=D\left[\vect{X}\right]\).

Derives Theorem A43.7. Throughout, \(\set{\left(2\pi\right)^{-1/2}\ee^{\ii n\theta}}_{n\in\Z}\) is the standard orthonormal basis of \(L^{2}\left(S^{1}\right)\) (Section 13.7.4), and Parseval's identity Equation (16.19) is applied componentwise to the three components of \(\vect{g}\), so that \(\abs{\vect{c}_{n}}^{2}\) denotes \(\sum_{k=1}^{3}\abs{c_{n}^{k}}^{2}\).

Step 1: the Dirichlet integral of the extension. In polar coordinates \(\dd u\,\dd v=\rho\,\dd\rho\,\dd\theta\) and

Differentiating Equation (A43.5) term by term — legitimate on every disc \(\rho\le\rho_{0}<1\), where the series and each of its derived series are dominated termwise by \(\abs{n}^{2}\abs{\vect{c}_{n}}\rho_{0}^{\abs{n}-2}\), whose sum converges because \(\abs{\vect{c}_{n}}\) is bounded and \(\rho_{0}<1\), so that the Weierstrass \(M\)-test Lemma 13.6 gives uniform convergence — and integrating in \(\theta\) by orthogonality,

The two are equal — a first sign that the extension is conformally balanced — so

the term-by-term integration in \(\rho\) being legitimate because every term is nonnegative.

Step 2: the Douglas integral. Substitute \(\theta=\varphi+t\) and integrate first in \(\varphi\) at fixed \(t\). The Fourier coefficients of \(\varphi\longmapsto\vect{g}\left(\varphi+t\right)-\vect{g}(\varphi)\) are \(\vect{c}_{n}\left(\ee^{\ii nt}-1\right)\), so Parseval Equation (16.19) gives

whence

the interchange of sum and integral being legitimate because every term is nonnegative.

Step 3: the kernel integral. Using \(\sin^{2}\left(t/2\right)=\left(1-\cos t\right)/2\) and, for \(n\ge1\), the Dirichlet kernel \(\Delta_{n}(t)=\sum_{j=0}^{n-1}\ee^{\ii jt} =\left(\ee^{\ii nt}-1\right)/\left(\ee^{\ii t}-1\right)\),

where \(\abs{\ee^{\ii\alpha}-1}^{2}=2\left(1-\cos\alpha\right)\) has been used twice. By orthogonality of the exponentials, \(\int_{0}^{2\pi}\abs{\Delta_{n}}^{2}\dd t=2\pi n\), so the integral in Equation (A43.8) equals \(4\pi\abs{n}\) — the case \(n=0\) giving \(0\), and negative \(n\) the same value as \(\abs{n}\) because \(\cos\) is even. Substituting,

which with Equation (A43.7) is Equation (A43.6). Both computations produce the same nonnegative series, so one side is infinite exactly when the other is.

Derives Lemma A43.9. The integrand of Equation (A43.4) is nonnegative, and for almost every pair \(\left(\theta,\varphi\right)\) it converges to the integrand built from \(\vect{g}\), the denominator being independent of \(k\) and the numerator continuous in the values of the map. Fatou's lemma — quoted, as throughout this appendix (Remark A41.12) — applied on the square \(\left[0,2\pi\right]^{2}\) gives the claim.

Derives Lemma A43.10. The circle is preserved. For \(\abs{z}=1\), \(\overline{z}=z^{-1}\), so

whence \(\abs{M_{\alpha,\vartheta}(z)}=1\). The denominator never vanishes on \(\overline{B}\), since \(\abs{\overline{\alpha}z} \le\abs{\alpha}<1\), so \(M_{\alpha,\vartheta}\) is holomorphic on a neighbourhood of \(\overline{B}\) and continuous on \(\overline{B}\).

The disc is preserved, and the map is bijective. A direct computation gives, for \(\abs{z}<1\),

the middle equality by expanding both squared moduli, so \(M_{\alpha,\vartheta}\) maps \(B\) into \(B\). Its inverse is \(M_{\beta,\eta}\) with \(\beta=-\ee^{-\ii\vartheta}\alpha\) suitably normalised — explicitly, solving \(w=\ee^{\ii\vartheta}\left(z-\alpha\right)/ \left(1-\overline{\alpha}z\right)\) for \(z\) gives \(z=\left(\ee^{-\ii\vartheta}w+\alpha\right)/ \left(1+\overline{\alpha}\ee^{-\ii\vartheta}w\right)\), which is again of the form Equation (A43.10); so each \(M_{\alpha,\vartheta}\) is a bijection of \(B\) and of \(S^{1}\), and the family is closed under inversion. It is closed under composition as well, since a composition of two quotients of linear polynomials is again one and it maps \(B\) bijectively onto \(B\) — that such a map must again be of the form Equation (A43.10) is the classification quoted in the fourth assertion.

Non-compactness. Take \(\vartheta=0\) and \(\alpha=1-1/k\) along the real axis. For any fixed \(z\in S^{1}\) with \(z\ne1\),

whereas \(M_{\alpha,0}(1)=1\) for every \(k\). The sequence therefore has no subsequence converging uniformly on \(S^{1}\) to a homeomorphism: its pointwise limit collapses the whole circle except one point onto the single value \(-1\). This is precisely the degeneration a minimising sequence for \(\mathcal{D}\) may undergo at no cost, since by Lemma A43.4 and Equation (A43.6) the value of \(\mathcal{D}\) is unchanged by such a reparametrisation of the boundary.

Derives Lemma A43.12. The class \(\mathcal{C}^{\ast}\) is contained in the larger class, so the left infimum is at least the right one. Conversely let \(\vect{g}\) be any monotone parametrisation. The three points \(\vect{g}^{-1}\left(\vect{Q}_{i}\right)\) may be chosen as three points \(\zeta_{1},\zeta_{2},\zeta_{3}\) of \(S^{1}\) in positive cyclic order, because \(\vect{g}\) is monotone of degree one; by the simple transitivity in Lemma A43.10 there is a conformal automorphism \(M\) of \(B\) carrying \(\left(1,\ii,-1\right)\) to \(\left(\zeta_{1},\zeta_{2},\zeta_{3}\right)\). Then \(\vect{g}\circ M\) is again a monotone parametrisation, now satisfying Equation (A43.11), and its harmonic extension is \(\vect{h}\circ M\) — the composition of a harmonic map with a holomorphic one is harmonic, since \(\nabla^{2}\left(\vect{h}\circ M\right) =\abs{M'}^{2}\left(\nabla^{2}\vect{h}\right)\circ M\) by the computation of Lemma A43.4 applied to second derivatives, and its boundary values are \(\vect{g}\circ M\), so it is the harmonic extension of \(\vect{g}\circ M\) by the uniqueness of the solution of the Dirichlet problem (Corollary 14.76). Hence, by Equation (A43.6) and Lemma A43.4,

so every value attained on the larger class is attained on \(\mathcal{C}^{\ast}\).

Derives Lemma A43.13. By definition a Jordan curve is the image of a homeomorphism \(\gamma\) of \(S^{1}\), so \(\gamma^{-1}\) exists and is continuous, and \(\Gamma\) is compact, being a continuous image of a compact set.

Both maps are uniformly continuous. For \(\gamma\) this is Theorem 11.25 applied componentwise to the angle parametrisation on \(\left[0,2\pi\right]\). For \(\gamma^{-1}\), suppose not: there are \(\varepsilon_{0}>0\) and points \(\vect{P}_{k},\vect{P}'_{k}\in\Gamma\) with \(\abs{\vect{P}_{k}-\vect{P}'_{k}}\longrightarrow0\) and \(\abs{\gamma^{-1}\left(\vect{P}_{k}\right) -\gamma^{-1}\left(\vect{P}'_{k}\right)}\ge\varepsilon_{0}\). The circle is compact, hence sequentially compact (Theorem 10.31), so along a subsequence \(\gamma^{-1}\left(\vect{P}_{k}\right)\longrightarrow\zeta\) and \(\gamma^{-1}\left(\vect{P}'_{k}\right)\longrightarrow\zeta'\) with \(\abs{\zeta-\zeta'}\ge\varepsilon_{0}\), so \(\zeta\ne\zeta'\). Applying the continuous \(\gamma\), \(\vect{P}_{k}\longrightarrow\gamma\left(\zeta\right)\) and \(\vect{P}'_{k}\longrightarrow\gamma\left(\zeta'\right)\), and \(\abs{\vect{P}_{k}-\vect{P}'_{k}}\longrightarrow0\) then forces \(\gamma\left(\zeta\right)=\gamma\left(\zeta'\right)\), contradicting injectivity.

Given \(\varepsilon>0\), uniform continuity of \(\gamma\) supplies \(\eta>0\) such that arcs of \(S^{1}\) of length below \(\eta\) have images of diameter below \(\varepsilon\); uniform continuity of \(\gamma^{-1}\) then supplies \(\delta>0\) such that \(\abs{\vect{P}-\vect{P}'}<\delta\) forces \(\gamma^{-1}\left(\vect{P}\right)\) and \(\gamma^{-1}\left(\vect{P}'\right)\) to be at distance below \(\eta/\left(2\pi\right)\) along \(S^{1}\), so that the shorter of the two arcs they determine has length below \(\eta\). Its image is one of the two arcs of \(\Gamma\), of diameter below \(\varepsilon\).

Derives Lemma A43.14. Use polar coordinates \(\left(\varsigma,\omega\right)\) centred at \(z_{0}\), in which \(\dd u\,\dd v=\varsigma\,\dd\varsigma\,\dd\omega\) and \(\abs{\nabla\vect{h}}^{2} =\abs{\pp_{\varsigma}\vect{h}}^{2} +\varsigma^{-2}\abs{\pp_{\omega}\vect{h}}^{2}\). Writing \(I_{\varsigma}\) for the set of angles \(\omega\) with \(z_{0}+\varsigma\ee^{\ii\omega}\in B\) and discarding the radial term,

Since \(\int_{\delta}^{\sqrt{\delta}}\dd\varsigma/\varsigma =\tfrac12\log\left(1/\delta\right)\), the mean value of \(\zeta\) against the measure \(\dd\varsigma/\varsigma\) on that range is at most \(4M/\log\left(1/\delta\right)\), so there exists \(\varsigma\in\left(\delta,\sqrt{\delta}\right)\) with \(\zeta\left(\varsigma\right)\le4M/\log\left(1/\delta\right)\) — otherwise the integral would exceed \(2M\). For that radius the length of the image curve is bounded by the Cauchy–Schwarz inequality Equation (A41.4),

which is Equation (A43.12), and the diameter of a curve is at most its length.

Derives Theorem A43.15. Equicontinuity. Let \(\varepsilon>0\) and take \(\delta_{0}\) from Lemma A43.13 for this \(\varepsilon\); shrink \(\varepsilon\) if necessary so that \(2\varepsilon\) is smaller than the mutual distances of \(\vect{Q}_{1},\vect{Q}_{2},\vect{Q}_{3}\) and of the arcs between them. Choose \(\delta\in(0,1)\) with \(\left(4\pi M/\log\left(1/\delta\right)\right)^{1/2}<\delta_{0}\) and \(\sqrt{\delta}\) smaller than one third of the least of those arc lengths. Let \(\vect{h}_{k}\) be the harmonic extension of \(\vect{g}_{k}\), which by Equation (A43.6) has \(D\left[\vect{h}_{k}\right]=\mathcal{D}\left[\vect{g}_{k}\right] \le M\), and which is continuous on \(\overline{B}\) with boundary values \(\vect{g}_{k}\) by the quoted solvability of the Dirichlet problem (Theorem A43.16). For \(z_{0}\in S^{1}\), Lemma A43.14 produces a radius \(\varsigma_{k}\in\left(\delta,\sqrt{\delta}\right)\) for which the arc \(C_{\varsigma_{k}}\) has image of diameter below \(\delta_{0}\); its two endpoints lie on \(S^{1}\) and their images \(\vect{P},\vect{P}'\) are values of \(\vect{g}_{k}\) at distance below \(\delta_{0}\). By Lemma A43.13 one of the two arcs of \(\Gamma\) they cut off has diameter below \(\varepsilon\); it is the one whose preimage under the monotone map \(\vect{g}_{k}\) is the short boundary arc cut off by \(C_{\varsigma_{k}}\), because the other boundary arc has length above \(2\pi-2\sqrt{\delta}\) and its image therefore contains at least two of \(\vect{Q}_{1},\vect{Q}_{2},\vect{Q}_{3}\) — which by the choice of \(\varepsilon\) cannot lie in a set of diameter \(\varepsilon\). Hence \(\vect{g}_{k}\) oscillates by less than \(\varepsilon\) on the boundary arc cut off at \(z_{0}\), which contains all points of \(S^{1}\) within \(\delta\) of \(z_{0}\); and \(\delta\) was chosen independently of \(k\) and of \(z_{0}\). That is equicontinuity.

Extraction of a limit. Write \(\vect{g}_{k}=\gamma\circ\psi_{k}\) as in Definition A43.11. Equicontinuity of \(\vect{g}_{k}\) transfers to \(\psi_{k}\) through the uniform continuity of \(\gamma^{-1}\) (Lemma A43.13 and its proof), and the \(\psi_{k}\) are uniformly bounded once normalised by \(\psi_{k}(0)\in\left[0,2\pi\right)\). The Arzelà–Ascoli theorem Lemma A41.7 gives a uniformly convergent subsequence \(\psi_{k}\longrightarrow\psi\); the limit is continuous, nondecreasing and satisfies \(\psi\left(2\pi\right)=\psi(0)+2\pi\), these being closed conditions under uniform convergence. Hence \(\vect{g}=\gamma\circ\psi\) is again a monotone parametrisation, it satisfies Equation (A43.11) because each \(\vect{g}_{k}\) does and convergence is pointwise, and \(\vect{g}_{k}\longrightarrow\vect{g}\) uniformly by the uniform continuity of \(\gamma\).

Derives Proposition A43.18. Work at a point where \(E=G=\Lambda^{2}>0\) and \(F=0\), with unit normal \(\hat n\) (Equation (17.80)). Differentiating the conformality relations,

using \(F\equiv0\) and \(G\equiv E\). Adding, \(\left(\vect{X}_{uu}+\vect{X}_{vv}\right)\cdot\vect{X}_{u}=0\), and the same computation with \(u\) and \(v\) exchanged gives \(\left(\vect{X}_{uu}+\vect{X}_{vv}\right)\cdot\vect{X}_{v}=0\). So the Laplacian \(\nabla^{2}\vect{X}=\vect{X}_{uu}+\vect{X}_{vv}\) is orthogonal to both tangent vectors, hence parallel to \(\hat n\), and by Equation (17.80) its normal component is

Since \(F=0\) and \(E=G=\Lambda^{2}\), the inverse metric is \(g^{ij}=\Lambda^{-2}\delta^{ij}\) (Definition 17.23), so \(2H=g^{ij}b_{ij}=\Lambda^{-2}\left(b_{11}+b_{22}\right)\) and therefore

If \(\vect{X}\) is harmonic the left side vanishes and, \(\Lambda^{2}\) being positive, \(H=0\); conversely \(H=0\) makes the right side vanish. By Proposition 20.8 the vanishing of the mean curvature is the minimal-surface equation Equation (20.7) wherever the surface is a graph, so the image is a minimal surface in the sense of Section 20.1.3.

Proof of Theorem A43.2, granting the two quoted theorems. Derives Theorem A43.2. Let \(m=\inf\set{\mathcal{D}\left[\vect{g}\right]\mid \vect{g}\in\mathcal{C}^{\ast}\left(\Gamma\right)}\). It is finite: \(\Gamma\) is rectifiable, and a parametrisation proportional to arc length has \(\abs{\vect{g}(\theta)-\vect{g}(\varphi)}\le \ell\,\abs{\theta-\varphi}/\left(2\pi\right)\) with \(\ell\) the length of \(\Gamma\), so the integrand of Equation (A43.4) is bounded by \(\ell^{2}\abs{\theta-\varphi}^{2}/ \left(4\pi^{2}\sin^{2}\left(\left(\theta-\varphi\right)/2\right)\right)\), which is bounded on the square, and \(\mathcal{D}<\infty\) there; this is the only place where rectifiability is used, and it is used exactly to make the infimum finite.

Take a minimising sequence \(\left(\vect{g}_{k}\right)\subset\mathcal{C}^{\ast}\), which by discarding finitely many terms satisfies \(\mathcal{D}\left[\vect{g}_{k}\right]\le m+1\). By Theorem A43.15 a subsequence converges uniformly to some \(\vect{g}_{\ast}\in\mathcal{C}^{\ast}\left(\Gamma\right)\), and by lower semicontinuity Lemma A43.9, \(\mathcal{D}\left[\vect{g}_{\ast}\right]\le \liminf_{k}\mathcal{D}\left[\vect{g}_{k}\right]=m\); the reverse inequality holds because \(\vect{g}_{\ast}\in\mathcal{C}^{\ast}\). So the infimum is attained.

Let \(\vect{X}\) be the harmonic extension of \(\vect{g}_{\ast}\), which exists and is continuous on \(\overline{B}\) by Theorem A43.16. By Theorem A43.17 it is conformal, so by Proposition A43.3 \(A\left[\vect{X}\right]=D\left[\vect{X}\right] =\mathcal{D}\left[\vect{g}_{\ast}\right]=m\), using Equation (A43.6) for the last equality.

It remains to see that no disc-type surface spanning \(\Gamma\) has smaller area. Let \(\vect{Y}\) be one, with boundary parametrisation \(\vect{g}\), which by Lemma A43.12 may be taken in \(\mathcal{C}^{\ast}\left(\Gamma\right)\). The area is reparametrisation invariant, since \(\sqrt{EG-F^{2}}\) transforms with the Jacobian under any \(C^{1}\) change of variables, and by Proposition A43.3 \(A\left[\vect{Y}\right]=A\left[\vect{Y}\circ\tau\right] \le D\left[\vect{Y}\circ\tau\right]\) for every such \(\tau\); the quoted approximation step — item (4) of Remark A43.20 — upgrades this to the equality

the infimum being over the reparametrisations of the disc. Each \(\vect{Y}\circ\tau\) has the same boundary values as \(\vect{Y}\) up to a reparametrisation of \(S^{1}\), and by the Dirichlet principle of Theorem A43.16 its Dirichlet integral is at least that of the harmonic extension of those boundary values, which by Equation (A43.6) and Lemma A43.12 is \(\mathcal{D}\left[\vect{g}\right]\ge m\). Hence \(A\left[\vect{Y}\right]\ge m=A\left[\vect{X}\right]\): the surface \(\vect{X}\) minimises area among disc-type surfaces spanning \(\Gamma\). Finally Proposition A43.18 makes the image a minimal surface away from branch points.