数理解析研究所講究録
第 1628 巻 2009 年 109-115

109Isothermal coordinates on singular minimal surfaces

Sumio Yamada
Mathematical InstituteTohoku University
yamada@math.tohoku.ac.jp

1

Introduction

J.Taylor’s list $([T])$ of configurations for so-called $(M, 0, \delta)$ -minimal sets ([Alm]) in
include a
singularity type where three minimal surfaces are meetiiig along areal analytic ([KNS]) singularcurve with
degree angle. In this article we introduce alocal conformal parametrizationof such aconfiguration by a2-dimensional simplicial complex
consisting of three half discs
whose diameters are identified to form al-dimensional face. The parametrization functions as an
isothermal coordinate system of the neighborhood of the singular surface.
We introduce two different methods in order to construct such parametrizations, both utilizing
the real analyticity of the singular surface, and the Euclidean ambieiit geometry. What is required
to establish aconformal parametrization is aBeltrami equation locally defined on ahalf plane.
Then we point out that the conformal parameterization $hom$ the 2-dimensional simplicial
coinplex into the singular surface with the three balanced surfaces has amean value property.
There has been much work on the subject of harmonic analysis on Euclidean buildings where
harmonic functions are defined on the buildings. The conformal harmonic parameterization of
the singular minimal surfaces can be regarded as agraph of aharmonic function defined over thesimplcial complex .
The content of this article is apart of an ongoing project(cf. [MY],[MY2]) by C.Mese and theauthor.
$R^{3}$$120^{o}$$Y_{0}$$Y_{0}$

2

Identifying Beltrami equation

Let$\triangle^{+}=\{(x, y)\in R^{2}:x^{2}+y^{2}<1, y>0\}$and consider three copies of$\triangle^{+}$and label them$\triangle_{1}^{+},$$\triangle_{2}^{+},$$\triangle_{3}^{+}$

to distinguish one from another. Let$A_{i}=\{(x, y)\in R^{2}:-1<x<1,y=0\}\subset\overline{\triangle_{i}^{+}}.$

.

110

and
be the union of the 3by the identity map Id :and
Identify the points ofwith this identification on ’s and denote the ’s by . For a map from domain
half-discs.
and writeby, we will denote the restriction of to$A_{i}$$A_{i}arrow A_{j}$$A_{j}$$Y_{0}$$\triangle_{i}^{+}$$\alpha$$Y_{0}$$A$$A_{i}$$A_{i}$$\triangle_{i}$$\alpha$$\alpha=(\alpha_{1}, \alpha_{2}, \alpha_{n})$$\alpha_{i}$be a real-analytic curve and $\Sigma_{i}\subset R^{3}(i=1,2,3)$ be a real-analytic surface
Lemma 1 Let
is a real-analytic surface with boundary . We assume further that the three surfa ces
so that, there exists a$balanced_{J}$at 120’ degree angle. Then for every
meeting along
namelyare$Furthem\iota
ore,$
.
and
so that:real-analytic map
is a constant speed parametrization of in a neighborhood of , with the constant speed shared by$u_{1}|u|_{A}$ and.
$\Gamma\subset R^{3}$$\Gamma$$\Sigma_{i}\cup\Gamma$$p_{0}\in\Gamma$$\Gamma$$\triangle^{+}\cup Aarrow\Sigma_{i}\cup\Gamma$$u_{i}$$u(\triangle^{+})\subset\Sigma_{i}$$\Gamma$$p_{0}\in u_{i}(A)\subset\Gamma$$u_{i}|_{A}$$p_{0}$$u_{3}|_{A}$PROOF. [Construction 1] We $wiU$ present aversion of the proof where the regularity is optimal.
Then the statement for the real analytic data follows ffom the stronger statement. Let befor now. Let : $(-t_{0}, t_{0})arrow\Gamma$ be an arclength parametrization of so that $\gamma(0)=p_{0}$ . Then$t\in(-t_{0}, t_{0})\mapsto\gamma’(t)$ is amap. Let $P(t)\subset R^{N}$ be the hyperplane containing the point. Because is , for asufficiently $smaU$ neighborhood of , every
and perpendicular tobelongs to aunique $P(t)$ . The size of is only dependent on the curvature of .
point
so that is the hyperplane $P(t)$ containing . In other words,
Define amap :. Since is a
is the nearest point projection map, thenif:
. Furthermore,
maps, is also([Si] 2.12.3). Therefore, $as$ acomposition of
curve,is.
at isto the 2-plane tangent to, the map
which takes
since
is
$V$ :
$V(p)$
is
map
so
that
be
the
maps, we can let
and
are
Since
map $H$ :. Now define athe unit vector associated to the line
. Thus and $H$by setting $H(p)$ to be the unit vector perpendicular to $V(p)$ in the plane
be an arclength parametrization of the. Let
are orthonormal vector fields on
is acharacteristici.e.
By
construction
with
curve
as the characteristic curve of the vector field $H$ withcurve of the vector field V. We define
foUows $hom$ the standard ODE. The existence and uniqueness ofcondition$H$
is the characteristic curve ofandvector field on
is atheory because$H$ whose image is. In this way, we have constructed apair of orthogonal foliations on a(with chosen smaller than if necessary) of .
neighborhood, let theand be the curves intersecting
as foUows. For
We define amap :of aneighborhood
aparametrization$\phi(p)=(t,s)$
map
defines. Then theat . Then we set$ts$
of by an open neighborhood of the upper half space of the -plane. Furthermore the pulled-backis representedof the 3-dimensional Euclidean metric , which we denote by
metric$ts$
ison the upper half -plane as adiagonal matrix near the origin, since the surface orthogonally. We are done by choosing $r>0$ sufficiently small and letting
andfoliated by the leaves$u(x,y)=\phi^{-1}(rx, ry)$ .
be defined
Repeat the same argument for $i=2$ .and 3. Q.E.D.
$\Sigma$$\Sigma_{1}$$\Gamma$$\gamma$$C^{1}$$\gamma(t)$$\Gamma$$\gamma’(t)$$C^{2}$$\mathcal{V}$$p_{0}$$\Gamma$$\mathcal{V}$$p\in \mathcal{V}$$\mathcal{P}$$\mathcal{V}\subset R^{N}arrow R^{N}$$p$$\mathcal{P}$$\mathcal{P}(p)=Po\gamma^{-1}\circ\pi_{\Gamma}(p)$$\mathcal{V}arrow\Gamma$$\pi_{\Gamma}$$C^{1}$$C^{1}$$C^{2}$$\Gamma$$C^{1}$$\mathcal{P}$$\pi_{\Gamma}$$C^{2}$$\Sigma$$\mathcal{P}$$C^{1}$$\mathcal{P}(p)\cap T(p)$$V$$\sigma_{t}(s)$$\sigma_{t}’(s)=V(\sigma_{t}(s)),$$\sigma_{t}(0)=\gamma(t).$$(\Sigma\cap\Gamma)\cap P(t)$$C^{1}$$(\Sigma\cup\Gamma)\cap \mathcal{V}arrow R^{N}$$\mathcal{T}(p)$$(\Sigma\cap\Gamma)\cap \mathcal{V}$$p$$C^{1}$$(\Sigma\cup\Gamma)\cap \mathcal{V}arrow R^{N}$$C^{1}$$\mathcal{T}$$\Sigma$$p\in(\Sigma\cup\Gamma)\cap \mathcal{V}$$\mathcal{T}$$\sigma_{t}$$\gamma_{8}(t)$$\gamma_{s}(t)$$\gamma_{s}(0)=\sigma_{0}(s)$$C^{1}$$(\Sigma\cup\Gamma)\cap \mathcal{V}$$\gamma_{0}=\gamma$$\Gamma\cap \mathcal{V}$$\mathcal{U}\subset\Sigma\cup\Gamma$$\phi$$\mathcal{V}$$\mathcal{U}$$\mathcal{U}arrow R^{2}$$p\in \mathcal{U}$$C^{2}$$p$$p_{0}$$\gamma_{s}$$\sigma_{t}$$\phi^{-1}$$p_{0}$$(\phi^{-1})^{*}g_{0}$$G$$g_{0}$$\{\gamma_{s}\}$$\{\sigma_{t}\}$$u:\triangle^{+}\cup Aarrow\Sigma\cup\Gamma$

PROOF.[Construction 2] We use the so-called hodographic projection of [KNS] to paramand
are locally graphs over the tangent plane
eterize a neighborhood of . The surfaces
be the orthogonal
andbelow. Letlies above the plane,. We suppose that
$p_{0}$$T_{p0}\Sigma_{1}$$\Sigma_{3}$$\Sigma_{2}$$\Sigma_{3}$$\Sigma_{2}$$\Pi_{2}$$\Pi_{3}$111

projection maps fromtoandfor sufficiently small . We canchoose the coordinates so that $p_{0}=(0,0,0)\in R^{3}$ , the tangent line to atis the -axis$\Pi_{2}^{-1}(x_{1\}}x_{2})=(x_{1},x_{2},
u_{2}(x))\in\Sigma_{2}$andis the-plane. Defineby the conditions
and $\Pi_{3}^{-1}(x_{1}, x_{2})=(x_{1)}x_{2}, u_{3}(x))\in\Sigma_{3}$ . Near the origin, the map $h(x_{1}, x_{2}):=(x_{1}, u_{3}(x)-u_{2}(x))$
is of rank two. It sends
to the -axis and its image is contained in the upper
half plane. The map
is the hodographic projection and its inverse map
definedon a sufficiently small half disk centered at the origin defines the real analytic parameterizationofaround , while
can be parameterized using
defines that of . Similarly,
the tangent plane. Denote these three maps parameterizing the neighborhood of by$u=(u_{1}, u_{2}, u_{3})$ :. The real analyticity and the continuity
with :$H$of follows from the construction. Q.E.D.
$\Sigma_{2}\cap B_{\delta}(p_{0})$$\Sigma_{3}\cap B_{\delta}(p_{0})$$\delta$$T_{p0}\Sigma_{1}$$\gamma$$T_{p0}\Sigma_{1}$$x_{1}x_{2}$$x_{1}$$\Pi_{2}^{-1}\circ h^{-1}$$h\circ\Pi_{2}$$\Pi_{3}^{-1}\circ h^{-1}$$q$$x_{1}$$u_{2},$ $u_{3}$$\Pi_{2}(\gamma)=\Pi_{3}(\gamma)$$\Sigma_{2}$$p_{0}$$\Sigma_{1}$$\Sigma_{3}$$T_{p0}(\Sigma_{2})$$p_{0}$$Y_{0} arrow(\bigcup_{i=1}^{3}\Sigma_{i})\cup\gamma$3$u_{i}$$\triangle_{i}^{+}\cup A_{i}arrow\Sigma_{i}$Construction of isothermal coordinates

be a real-analytic curve and $\Sigma_{i}\subset R^{3}(i=1,2,3)$ be a real-analytic
Theorem 2 Let
is a real-analytic surface with boundary . We assume further that the threesurface so that, theredegree angle. Then for every
surfaces are balanced, namely meeting along at
by a conform, $al$ map fromexists an isothermal coordinate system of a neighborhood ofon each face .
whereis the tnplet of standard Euclidean metric$\Gamma\subset R^{3}$$\Gamma$$\Sigma_{i}\cup\Gamma$$\Gamma$$120^{o}$$p_{0}\in\Gamma$$(Y_{0}, G_{0})$$p_{0}$$iG_{0}$$G_{0}$$\triangle_{i}$in the singular
PROOF. By Lemma 1, there exists a parameterization of a neighborhood of. Let $iG$ be the the pull back of the Euclidean metricsurface so that is real analytic in, denote the metric
under the map . With respect to the Euclidean coordinates ofoncomponents of $iG$ by. We now wish to find an isothermal coordinate by solving the Beltrami$p_{0}$$\triangle_{i}^{+}\cup A_{i}$$u_{\dot{\eta}}$$R^{n}$$\triangle_{i}$$u_{i}$$iG_{\alpha\beta}$

equation$w_{\overline{z}}=\mu w_{z}$where Beltrami coefficient$\mu$

is given by$\mu=\frac{iG_{11}-iG_{22}+2\sqrt{-1}^{i}G_{12}}{iG_{11}+iG_{22}+2\sqrt{iG_{11^{i}}G_{22^{-i}}G_{12}^{2}}}$.

Note here that the Beltrami coefficient is represented by the pull-back metric $iG=(u_{i})^{*}G_{0}$ .
The Beltrami coefficient has moduli strictly less than one. Furthermore, the metric components
are given by
$\mu$

$\mu$

$iG_{\alpha\beta}$${}^{t}G_{\alpha\beta}= \langle\frac{\partial u_{i}}{\partial x^{\alpha}},$$\frac{\partial u_{i}}{\partial x^{\beta}}\rangle_{R^{n}}$.

real analytic there.
since the map
Thus, the components of $iG$ are real analytic on
is the pulled-back area form of the immersed surface
Note that the quantity
is non-degenerate by
by a real analytic map . As the differential of the map
inis strictly positive. Then the quantity
construction, the term$\triangle_{i}^{+}\cup A_{i}$$u_{i}1s$$\sqrt{iG_{11^{i}}G_{22^{-i}}G_{12}^{2}}$$u_{i}(\triangle_{i}^{+}\cup A_{i})$$R^{n}$$u_{i}$$iG_{11^{i}}G_{22^{-}}^{i}G_{12}^{2}$$u_{i}$$\sqrt{iG_{11^{i}}G_{22^{-i}}G_{12}^{2}}$112

is real analytic.

$(G_{11}-G_{22}+2\sqrt{-1}G_{12})(x, y)$and $(G_{11}+G_{22}+2\sqrt{G_{11}G_{22}-G_{12}^{2}})(x, y)$ on thenear the origin.
Let $P=u_{i}(p)$ be a point on the free boundary with $p=(x_{0},0)\in R\cap A$ . We set$\alpha(x, y)+i\beta(x, y)$ and $\mu=\eta(x, y)+i\zeta(x, y)$ to rewrite the Beltrami equation defined on the half
as the following system of equations with real analytic coefficients:
disk
open set$R$$w(z,\overline{z})=$$\triangle_{i}^{+}$$(\alpha_{y}\beta_{y})=(\begin{array}{llll}\zeta (1+ \eta)(l- \eta) \zeta \end{array})(\begin{array}{ll}(l- \eta)u_{x}+\zeta v_{x}-(u_{x} \eta-(1+)v_{x}\end{array})$

The inverse matrix on the right hand side exists because $|\mu|^{2}=\eta^{2}+\zeta^{2}<1$ . We also have theCauchy initial data$\alpha(x, 0)=x-x_{0}$ and $\beta(x, 0)=0$

for $(x, 0)\in A_{i}$ near $(x_{0},0)\in A_{i}$ . Therefore, we can apply the Cauchy-Kowalewski Theorem andobtain, in some neighborhood of the point , a unique solution to the Beltrami equation. Thisof $(x_{0},0)$ tois a quasiconformal diffeomorphism from a neighborhood
solutionof $(0,0)$ . By construction, the pulled-back metric of the Euclidean
a neighborhood
provides a parameterization ofis conformal to $iG$ . Thus the map
underof
metric$p=(x_{0},0)$
openin. After scaling,
set
by an
in
the neighborhood of
we have constructed an isothermal coordinate system $F=(F_{1}, F_{2}, F_{3})$ : $(Y_{0}, G_{0})arrow(Y_{0}, G)$ of$p\in A$ . The map $f=(f_{1}, f_{2}, f_{3})$ withsatisfies the desired properties of the isothermal
Q.E.D.
coordinate system of$p$$\mathcal{U}_{i}\subset\triangle_{i}^{+}\cup A_{i}$$w_{i}$$\mathcal{V}_{i}\subset\triangle_{i}^{+}\cup A_{i}$$G_{0}$$\triangle_{i}^{+}$$w_{i}^{-1}$$w_{i}$$(\triangle_{i}^{+}\cup A_{i},{}^{t}G)$$\mathcal{V}_{i}$$\triangle_{i}^{+}\cup A_{i}$$f_{i}=u_{i}\circ F_{i}$$P\in\Gamma.$

4

Harmonic functions on simplicial complexes

Let $f=(f_{1}, f_{2}, f_{3}):(Y_{0}, G_{0})arrow R^{3}$ be as in Theorem 2. The equality(1)$f_{i}(x, 0)=f_{j}(x, 0)$

for $i,j=1,2,3$ implies$\frac{\partial f_{i}}{\partial x}(x, 0)=\frac{\partial f_{j}}{\partial x}(x, 0)$Using the conformality of
written as$f_{i}$.

(2), the balancing of the three surfaces along the singular curve$0= \sum_{i=1}^{3}\frac{4\partial\partial y}{|_{\text{\^{o}} y^{i}}^{\lrcorner}\partial|}(x, 0)=\sum_{i=1}^{3}\frac{\lrcorner\partial_{i}\partial y}{|_{\partial x^{i}}^{\partial}\lrcorner|}(x, 0)$$\Gamma$can be.

This combined with (2) implies$0= \sum_{i=1}^{3}\frac{\partial f_{i}}{\partial y}(x, 0)$.

(3)

If we let$\tilde{f}_{1}(x, y)=-f_{1}(x, -y)+\frac{2}{3}\sum_{i=1}^{3}f_{i}(x, -y)$

(4)

113
then (1) and (3) imply that$f_{1}(x, 0)=\tilde{f}_{1}(x, 0)$

We claim (5) shows$U_{1}:\trianglearrow R^{n}$and$\frac{\partial f_{1}}{\partial y}(x, 0)=\frac{\partial\tilde{f}_{1}}{\partial y}(x, 0)$.

(5)defined by setting$U_{1}(x, y)=\{\begin{array}{l}f_{1}(x, y) for y\geq 0\tilde{f}_{1}(x, y) for y<0\end{array}$is harmonic. Indeed, for any smoothgives.

$\xi$:$\trianglearrow R^{n}$with compact support, integration by parts$- \int_{\Delta+}\nabla\xi\cdot\nabla U_{1}dxdy=\int_{\triangle}+\xi\triangle f_{1}dxdy-\int_{I}\xi\frac{\partial f_{1}}{\partial y}(x, 0)dx$

and$- \int_{\triangle}-\nabla\xi\cdot\nabla U_{1}dxdy=\int_{\triangle}-\xi\triangle\tilde{f}_{1}dxdy+\int_{I}\xi\frac{\partial\tilde{f}_{1}}{\partial y}(x, 0)dt$

where $\triangle^{+}=\{(x,y)\in\triangle : y>0\},$ $\triangle^{-}=\{(x, y)\in\triangle : y<0\}$ and $I=\{(x, y)\in\partial\triangle^{+}:y=0\}$ .
Summing up the above two equations and using the harmonicity ofand , we obtain$\tilde{f}_{1}$$f_{1}$$- \int_{\triangle}\nabla\xi\cdot\nabla U_{1}dxdy=0$.

By Weyl’s Lemma,
extensionsofis aharmonic map. Similarly, there exists
and . We call this construction of the real analytic extensionof the multi-sheeted reflection.
By summarizing the argument above, we have$C^{\omega}$$C^{\omega}$$U_{1}$$f_{3}$$U_{i}$$U_{2},$$U_{3}$$f_{2}$$f_{i}$

can be extended real analytand as in Theorem 2. The surface
Theorem 3 Letbythe conformal, harmonic map
cally across the curve . This extended surface is parametrized
via the multi-sheeted reflection.
$\Sigma_{1},$$\Sigma_{2},$$\Sigma_{3}$$\Sigma_{i}$$\gamma$$i$$\gamma$$U_{i}$

across a real analytic boundary curve
We note that the extendability of the minimal surface
follows from a celebrated result of H.Lewy [Le]. On the other hand, Theorem 3 gives a moreofis given inprecise picture of the extension. Indeed, the extension of the parameterization
as deflned in (4).
terms of a linear combination of odd reflections of(i.e. the unit
curve in
When three minimal surfaces are geometrically balanced along aouter normal of the three surfaces sum to zero as in (3) , the entire configuration is completely
determined by one of the three surfaces. This follows from the so-called Bj\"oling’s problem resolved
by H.Schwarz. We will explain below how to use this and arguments in the proof of Theorem 3
to give a construction of Lewy’s extension.
$\Sigma_{i}$$\gamma$$f_{1}$$f_{1},$ $f_{2},$$\Sigma_{1}$$f_{3}$$C^{\omega}$$R^{3}$$)$

with a real analytic boundary can be extended across the
Theorem 4 A minimal surface inboundary by a multi-sheeted refiection.
$R^{3}$

114
PROOF. We start with a surface
with a real analytic boundary curve . Let
be the unit
outer normal to the surface
along . Letandbe the two unit vector fields defined on ,normal to , each making the angle of $\pi/3$ to , Note here $\eta_{1}+\eta_{2}+\eta_{3}=0$ . The solution by
Schwarz of the Bj\"orlng’s problem ([Ni] III \S 149) then provides locally defined, uniquely determined,
minimal surfaces
andalong so thatandare unit outer normals to
and
along
respectively.
Recall that we have the harmonic and conformal parameterization :without branch point. Furthermore, we also have$\Sigma_{1}$$\gamma$$\Sigma_{1}$$\gamma$$\eta_{2}$$\gamma$$\eta_{1}$$\eta_{3}$$\gamma$$\eta_{1}$$\Sigma_{2}$$\Sigma_{3}$$\gamma$$\eta_{2}$$\Sigma_{2}$$\eta_{3}$$\Sigma_{3}$$\gamma$$f_{i}$$\triangle_{i}^{+}\cup A_{i}arrow(\Sigma_{i}\cup\Gamma)\subset R^{3}$$\sum_{i=1}^{3}\frac{\partial f_{i}}{\partial y}=c\sum_{i=1}^{3}\eta_{i}=0$. Now each :can be extended across the real axis
byofparticular, we have a conformal parameterization of the extension ofas in Theorem 3.
$QE.D(4.).In$

where$c=| \frac{\partial}{\text{\^{o}}}xA|=|_{\text{\^{o}} y}^{\partial}A|$$f_{i}$$\triangle_{i}arrow R^{3}$$A_{i}$$\tilde{f}_{1}$$\Sigma_{1}$Recall that on a locally finite simplicial complex of dimension , a function is called harmonic
if for every simplex of dimension $l-1$ , the average value of the function on all maximal
simplices whose closure contain is zero. We demonstrate that the map which provides our local
uniformization byis harmonic in this sense after a normalization. In particular, we show that
$f$

$l$

$f$

$\sigma$$\sigma$$Y_{0}$

Theorem 5 The coordinate functions of the mapsatisfy the mean value equality:$f$: $Y_{0}arrow R^{3}$ of the singular minimal surface$\int_{B_{\epsilon}(po)}\sum_{i=1}^{3}[f_{i}(x)-f(p_{0})]dx=0$

where $p_{0}=(x_{0},0)$ in , andis a ball of radius, namely the set $\{y\in$inwhere the distance function is with respect to the Euclidean
on each$f_{i}(p_{0})=f(p_{0})$
.
Here
we
note
.
$i=1,2,3$
face
for
$A$

$\epsilon>0$$B_{\epsilon}(p_{0})$$Y_{0}|d(y,p_{0})<\epsilon\}$$(Y_{0}, G_{0})$$d$$metr\dot{\tau}c^{i}G_{0}$$\triangle_{i}^{+}$PROOF. We have shown above that each map defined on
can be canonically extended
across the edge to the disc . The resulting harmonic function
satisfies the mean value$\triangle_{i}^{+}$$f_{i}$$A$$\triangle_{i}$$U_{i}$

equality$\frac{1}{\pi\epsilon^{2}}\int_{B_{\iota}\phi_{0})}U_{i}(x)dx=f_{i}(p_{0})$.

Rewriting the integral as a sum of integrals over the upper disc
, we get$B_{\epsilon}^{+}(p_{0})\subset\triangle_{i}^{+}$and the lower disc

$B_{\overline{\epsilon}}(p_{0})\subset\triangle_{i}^{-}$$\frac{1}{\pi\epsilon^{2}/2}\int_{B_{*}^{+}(po)}f_{i}(x)dx=-\frac{1}{\pi\epsilon^{2}/2}\int_{B^{-}(po)}\tilde{f_{i}}(x)dx+2f_{1}(p_{0})$By rewriting
we get$\tilde{f_{i}}(x, y)$.

as $-f_{i}(x, -y)+ \frac{2}{3}\sum_{j=1}^{3}f_{j}(x, -y)$ for $y<0$ , and taking a sum over $i=1,2,3$ ,$\sum_{i=1}^{3}\frac{1}{\pi\epsilon^{2}/2}\int_{B_{\epsilon}^{+}(po)}f_{i}(x)dx=3f(p_{0})$,115which is the mean value equality. Q.E.D.

References[Alm]

F.J. Almgren. Existence and $Regulari\cdot ty$ Almost Everywhere of Solutions to. Elliptic
ational Problems with Constraints. Memoirs A.M.S., 165, Providence, R.I.: AMS, 1976

[C]

J. Chen. On energy minimizing mappings between and into singular spaces. Duke Math.
J. 79 (1995), no. 1, 77-99.

[KNS]

D. Kinderlerer, L. Nirenberg and J. Spruck. Regularity in elliptic free boundary problems.
I. J. Anal. Math. 34 (1978) 86-119.

[Le]

H. Lewy. On the boundary behavior of minimal surfaces. Proc. Nat. Acad. Sci. U. S. A.
37, (1951). 103-110.

[MY]

C. Mese and S. Yamada. The parameterized Steiner problem and the singular Plateau
problem via energy. Trans.A.M.S. 358 (2006) 2875-2895.

[MY2]

C. Mese and S. Yamada. Local uniformaization and free boundary regularity of singularminimal surfaces preprint (2008).

[Ni]

J.C.C.Nitsche Lectures on Minimal Surfaces Cambridge Univ. Press, Cambridge, 1989.

[Si]

L. Simon. On regularity and singularity of energy minimizing maps. Lectures in Mathematics ETH Z\"urich. Birkh\"auser Verlag, Basel, 1996.

[T]

J. Taylor. The structure of singularities in soap-bubble-like and soap-film-like minimal
surfaces. Ann. of Math. 103 (1976) 89-539.

$Va\uparrow\dot{\eta}-$