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    <h2>Open Problems about Minimal Surfaces</h2>
$$ \def\R{\mathcal R} $$

<p> J.C.C. Nitsche published his book <em> Vorlesungen &uuml;ber Minimalfl&auml;chen</em> in 1975
(although the foreword is dated 1974).  It contained a list of open problems.   The first five chapters
were translated into English and significantly revised and updated, and appeared in 1989 as 
<em> Lectures on Minimal Surfaces, Volume 1 </em>.   The planned second volume never appeared, 
due to the transitory nature of life on Earth. </p>

<p> The purpose of this web page is to review the two lists of open problems in Nitsche's 1975 and 1989
books, and assess the current status of those problems.  The 1975 list contains 95 problems, numbered
from 874 to 968.  The 1989 list updates only the part of the 1975 list pertaining to the first five 
chapters, i.e., to Volume 1 of the English edition.  Another purpose of this web page is to provide
an English translation of the 1975 list.
</p> 

<p>874 asks for a simpler proof of the inequality that the total curvature of an open complete minimal 
surface is less than or equal to $2\pi \chi$, where $\chi$ is the Euler characteristic.  (No change in 1989.)
(The known proofs apply to a more general class of surfaces, so the problem asks if the proof can be 
simplified by using the hypothesis that the surface is minimal.)
</p>
<p>875.  Near each point $p$ of a minimal surface $S$, consider the sphere of radius $r$ centered at $p$.
Suppose that for each $p$ and each sufficiently small $r$,  the surface $S$ divides the surface area of 
this sphere exactly in half, i.e., the area of the sphere on each side of $S$ is $2\pi r^2$.   The problem
is to state the conclusion.  It seems that the implied conjecture is that $S$ must be (part of) a plane.
(No change in 1989.)
</p>
<p>876 concerns one-to-one harmonic maps $\varphi$ from the open unit disk $B$ onto $B$, whose derivatives do not vanish.
Define $\mu$ to be the infimum of $\vert \nabla \varphi \vert^2 (0)$ over all $\varphi$.  The problem is 
to determine the value of $\mu$.  It is conjectured to be $27/(2\pi^2)$.  (Not mentioned in 1989.)
That value is about $1.367$.  The best known estimate is $1.282$, itself the result of a sequence of 
improvements.  
</p>
<p>877.  In the situation of 876, we ask instead for the infimum of $\vert \nabla \varphi \vert^2 (z)$
over all $z$ and all $\varphi$.  Heinz proved it is at least $2/\pi^2$.  No conjecture is offered and,
like 876,  the problem is not mentioned again in 1989.
</p>
<p>878. In the situation of 876, we ask for the function $R$ such that $R(\rho)$ is the smallest value 
such that no $\varphi$ takes the annulus $\rho < \vert z \vert < 1$ into the annulus $R(\rho) < r < 1$.
The conjecture is 
$$ R(\rho) = \frac {2\rho}{1+\rho^2}.$$
(Not mentioned in 1989.)
</p>
<p>879 asks to characterize the harmonic maps $\varphi$ as in 876 that take concentric circles to 
concentric circles. (Not mentioned in 1989.)
</p>
<p> 880 asks about explicit non-parametric representations of "generalized Scherk surfaces", described
on p. 147<em>ff</em>.  (Not mentioned in 1989.)
</p>
<p> 881.  Consider a minimal surface defined in a punctured plane.  The topology of the surface is 
given by the topology of the universal covering surface.  Determine its conformal type.
</p>
<p> 882 asks for a classification of periodic minimal surfaces whose fundamental domain is made 
of pentagons, or arbitrary polygons.
</p>
<p>883.  Consider two "solutions of Plateau's problem" bounded by the same Jordan curve $\Gamma$,
whose Frechet distance is not zero.  Then as point sets the surfaces are not identical.   Here
"solutions of Plateau's problem" probably means $C^2$ immersed surfaces rather than conformal harmonic 
surfaces.  No hypothesis about boundary regularity is mentioned in the problem.
</p>
<p>884.  Let Jordan curves $\Gamma_n$ converge to a Jordan curve $\Gamma$.  Let $X$ be a solution
of Plateau's problem bounded by $\Gamma$.  Is $X$ necessarily the limit of a sequence of solutions
of Plateau's problem bounded by $\Gamma_n$?  (The metrics to be used are not specified.)
</p>
<p> The answer to this question is "No", as a consequence of the cusp normal form theorem of
[Beeson-Tromba 1984]. In that paper, we exhibited a two-parameter family of Jordan curves $\Gamma_{a,b}$
(with $\Gamma_{0,0}$ equal to Enneper's wire with $r=1$) such that the manifold of minimal surfaces near
Enneper's surface is locally given by $(t,x)$ such that $t^3-bt + a = 0$.  Thus for a given $(a,b)$,
there are three solutions when $t \ge 0$ and just one when $t < 0$.  Choose $\Gamma = \Gamma_a,b$
for $(a,b)$ chosen so that there are just two solutions, namely any $(a,b)$ such that $27a^2 = 4b^3$.
Then let $\Gamma_n$ be $\Gamma_{a + 1/n,b}$.  Then $\Gamma_n$ bounds just one minimal surface 
$X_n$ (in some neighborhood of Enneper's surface), and those $X_n$ converge to one of the two minimal 
surfaces bounded by $\Gamma$.  Hence the other minimal surface bounded by $\Gamma$ is not the limit of 
minimal surfaces bounded by $\Gamma_n$.
</p>
<p>885.  Suppose the Jordan curve $\Gamma$ has a symmetry, for example reflection in a plane.  
Does every solution of Plateau's problem, or at least every least-area solution, have the same symmetry?
Does there even exist a symmetric solution?
</p>
<p>886. Does the "method of descent" always lead to an absolute minimum?  or sometimes only to a relative minimum?
</p>
<p>887.  Let $\Gamma_1$ and $\Gamma_2$ be Jordan curves lying on a cylinder over a Jordan curve in the $(x,y)$ plane.
(I believe Nitsche means to assume that the curves each wrap  around the cylinder once, since he uses the 
word "verlaufen".)
Let $S_1$ and $S_2$ be minimal surfaces bounded by $\Gamma_1$ and $\Gamma_2$. 
Suppose $\Gamma_1$ and $\Gamma_2$ do not meet.  Then must $S_1$ and $S_2$ also not meet?  
</p>
<p> This problem is easily solved.  Say $\Gamma_2$ lies below $\Gamma_1$.  Then lower $\Gamma_2$ and $S_2$
until $S_1$ and $S_2$ do not meet (possible by the convex hull property).  Then raise $S_2$ again until the 
first point of contact with $S_1$, or until the original position is reached without contact (in which case we are done).
The first point of contact cannot occur in the interior (of the parameter domain), nor can it occur on the boundary
before the original position is reached.  That completes the proof.  Note that the proof works even if we do not 
assume that the two curves wrap around the cylinder.
</p>
<p>888 asks to sharpen the Gauss-Bonnet inequality for branched minimal surfaces, by estimating the total 
curvature in terms of the geometry of the boundary.  The question is not precise.
</p>
<p>889 asks for better proofs of the absence of (true and false) interior branch points for least-area solutions 
of Plateau's problem.  
</p>
<p> This problem has been solved, by Beeson in 1980 using Fourier series, and again by Tromba in 2012
using Cauchy's theorem.   The results are discussed <a href="index.php?include=regularity"> here </a>.
</p>
<p>890 asks for a proof  of the absence of boundary branch points for least-area solutions 
of Plateau's problem.  
</p>
<p> This problem has been solved for real-analytic boundaries by Brian White in [<a href="http://math.stanford.edu/~white/gull.ps"> White 1997 </a>], and again by Tromba in 2012
using Cauchy's theorem.   The results are discussed <a href="index.php?include=regularity"> here </a>.
The problem is still open for the other two cases Nitsche mentions, namely $C^n$ boundaries,  and 
polygonal boundaries.
</p>
<p> 891.  Let $\Gamma_r$ be <a href="index.php?include=enneper">Enneper's wire </a>.  When $r=1$ does $\Gamma_r$ bound
a unique solution of Plateau's problem?
</p>
<p> This problem was solved (affirmatively) by Ruchert in 1981.
</p>
<p> 892 asks for the number of different solutions of Plateau's problem bounded by Enneper's wire $\Gamma_r$,
and for a description of the bifurcation process.   
</p>
<p> This problem has been completely solved.  The bifurcation process at $r=1$ was described rigorously 
in [Beeson-Tromba 1984], and the number of minimal surfaces (least area or not) bounded by $\Gamma_r$
was shown by Beeson in 2015 to be 3, when $r > 1$, and by Ruchert in 1981 to be 1 when $r \le 1$.
</p>

<p>893.  The Jordan curve $\Gamma_\delta$ is obtained from a tetrahedron with unit edges
 by splitting two opposite edges
and separating them by $\delta$.  The resulting polygon $\Gamma_\delta$ has eight edges (see illustration
on page 357 of Nitsche 1975). If $\delta$ is small then $\Gamma_\delta$ bounds at least 3 minimal surfaces;
if $\delta$ is large enough, only one.  What is the smallest $\delta$ such that $\Gamma_\delta$ is a 
curve of uniqueness?
</p>

<p>894. If we form a Jordan curve from four circles as on p. 396, where two pairs of circles are 
each separated by $\epsilon$, then when $\epsilon$ is small there are three solutions.  Find the 
upper bound of such $\epsilon$.
</p>

<p>895.  Given $N$, produce an example of a Jordan curve bounding more than $N$ solutions of Plateau's 
problem.
</p>

<p> Many non-rigorous examples have been given, which could be made rigorous by the "bridge theorem".
The bridge theorem was finally proved by Brian White in 1994. Also,  B&ouml;hme proved in 1982 that 
there are curves with curvature $4\pi + \epsilon$  bounding more than $N$ minimal surfaces (but these are 
branched minimal surfaces). So, problem 895 is solved (twice).  
</p>

<p>896.  A polygon of 4 sides bounds only one solution of Plateau's problem.
A polygon of 8 sides can sometimes bound three.  What is the least $m$ such that some $m$-sided polygon 
bounds more than one solution of Plateau's problem?
</p>

<p>897.  Let the Jordan curve $\Gamma$ have total curvature less than $4\pi$.  Then must $\Gamma$ be a 
curve of uniqueness for Plateau's problem?
</p>
<p> This question was answered in the affirmative by Nitsche himself in 1978.  He even obtained uniqueness
if the total curvature is $\le 4\pi$, not just strictly less than.  B&ouml;hme proved in 1982 that 
there are curves with curvature $4\pi + \epsilon$  bounding more than $N$ minimal surfaces (but these are 
branched minimal surfaces).  It might still be open to determine the largest number $\alpha$ such that 
when the total curvature of $\Gamma$ is less than $\alpha$, then $\Gamma$ bounds only one immersed 
minimal surface.  
</p>

<p>898.  Suppose $\Gamma$ has a one-one projection on the $(x,y)$ plane and bounds a non-parametric 
minimal surface $u(x,y)$ defined on $\bar B$,  where $B$ is a domain bounded by the projection of $\Gamma$.
Then (a) does $u$ have the least area among all surfaces bounded by $\Gamma$, and (b) can $\Gamma$ bound 
any other solutions of Plateau's problem?
</p>

<p>899.  Suppose $\Gamma$ has a one-one projection central projection from $(0,0,1)$ to the $(x,y)$ plane, and bounds  
minimal surface $u(x,y)$ defined on $\bar B$,  where $B$ is a domain bounded by the projection of $\Gamma$.
Then (a) does $u$ have the least area among all surfaces bounded by $\Gamma$, and (b) can $\Gamma$ bound 
any other solutions of Plateau's problem?
</p>

<p>900. There are only three known (in 1975) uniqueness theorems for Plateau's problem:  in case $\Gamma$ has a 
convex parallel or central projection on a plane, and in case $\Gamma$ is a quadrilateral. (In 1978, Nitsche
added the $4\pi$ theorem to this list, and in 1981 Ruchert's theorem was added.) Are there any other 
geometric hypotheses on $\Gamma$ 
that guarantee a unique solution of Plateau's problem?
</p>

<p>  There has been at least one new uniqueness theorem since 1975:  [Tromba 2010], page 422,  proves that $\Gamma$ is a curve of uniqueness provided $\Gamma$ is 
sufficiently close to a planar curve $\alpha$. 
</p>

<p>901.  Does there exist a Jordan curve bounding two minimal surfaces of the topological type of the disk,
both surfaces given by explicit equations? 
</p>

<p>902.  This problem concerns the solution of the non-parametric minimal surface equation over a 
non-convex quadrilateral, shown in Fig. 45.  The boundary values describe a space quadrilateral $\Gamma$,
three sides of which are in the $(x,y)$ plane, and the height of the fourth point, over the non-convex corner,
is $h$.  The unique solution of Plateau's problem for $\Gamma$ is <em> not </em> in non-parametric form.  There is,
however, for $h$ small enough,
 a solution of the non-parametric minimal surface equation taking boundary values on $\Gamma$ except at the 
 non-convex vertex.  This surface can be seen in soap film by 
adding a vertical wire from the $(x,y)$ plane to point $C$.  This solution touches that vertical wire
from $\alpha(h)$ to $h$.  More precisely, as $(x,y)$ approaches the non-convex corner $C^\prime$ in various 
directions,  every possible limit between $\alpha(h)$ and $h$ will be taken on.   The problem is to 
determine the function $\alpha(h)$ explicitly, or at least, find bounds for $\alpha(h)$ in terms of $h$. 
</p>

<p>903. The non-parametric minimal surface equation in 902 is not solvable when $h$ is large enough.  Problem 903
is to give a "non-parametric proof" of that fact, i.e. one not using the uniqueness of the solution to Plateau's problem,
but only arguments about the non-parametric minimal surface equation itself.
</p>

<p>904.  Let $V$ be a region in $\R^3$ whose boundary has non-negative mean curvature.  Then $V$ is 
$H$-convex, which means that for every point $p$ on the boundary of $V$,  there is some neighborhood $K$
of $p$ and a minimal surface $S$ through $p$ such that $V \cap K$ lies entirely on one side of $S$.
</p>

<p>905.  Give an example of a rectifiable Jordan curve bounding two solutions of Plateau's problem,
of different area, but both are relative minima of area.  
</p>

<p> From [Beeson-Tromba 1984],  in which the bifurcation of Enneper's surface is embedded in a two-parameter
family described by the cusp catastrophe, we know that there are perturbations of Enneper's wire $\Gamma_r$
for $r$ slightly greater than 1 for which there are three solutions of Plateau's problem, all of different area 
and two of them relative minima.   However, since we can't give formulas for those two solutions, perhaps this 
doesn't count as "giving an example".     
</p>

<p>906. Is there a "block" of minimal surfaces bounded by the same Jordan curve $\Gamma$,  that contains 
more than a single minimal surface?  Here a "block" is a connected set of minimal surfaces with the 
same value of Dirichlet's energy.  In particular a block would be a one-parameter family.
</p>

<p> After the work of B&ouml;hme and Tomi, this problem is equivalent to the next problem (907).
</p>

<p>907. Is every solution of Plateau's problem for a "sufficiently rectifiable" Jordan curve $\Gamma$
isolated? (i.e., separated in a suitable metric from every other solution?)  Is this true at least for 
every relative  minimum of Dirichlet's energy?   Or at least for every absolute minimum?  Or for 
every solution of Plateau's problem free of boundary branch points?
</p> 

<p> Nitsche lists as possible hypotheses on $\Gamma$: that $\Gamma$ be real-analytic;  that $\Gamma$ be real-analytic
with geometric hypotheses that rule out boundary branch points; that $\Gamma$  be $C^{m,\alpha}$ (possibly
with further hypotheses that rule out boundary branch points);  that $\Gamma$ be a polygon;  that 
$\Gamma$ with some hypotheses as above and also of total curvature less than $4\pi$; that $\Gamma$ be
(only) rectifiable.
</p> 

<p> By compactness, this problem is equivalent to asking whether there can ever be infinitely many 
solutions of Plateau's problem for a given boundary curve $\Gamma$. </p>
<p>[Tomi 1973] solved this problem for absolute minima,  assuming $\Gamma$ is real-analytic.<br>
[Beeson 2015] solved the problem for relative minima, again assuming $\Gamma$ is real-analytic.<br>
[Nitsche 1978] solved the case of total curvature less than $4\pi$. 
</p>

<p>908. Let $B$ be a connected component of the set of minimal surfaces bounded by a given Jordan
curve $\Gamma$ satisfying one of the possible hypotheses mentioned in 907.  Then must all the members of 
$B$ have the same value of Dirichlet's integral?  
</p>

<p> For $C^(n,\alpha)$  curves $\Gamma$, with $n \ge 6$,  this follows (I believe) from the work of 
Tromba, specifically Theorem 2 on page 438 of [Tromba 2010].
</p>

<p>909. Suppose it is known that every solution of Plateau's problem for $\Gamma$ is isolated.
It follows that the number of solutions cannot be infinite.   In that case,  the problem is to 
estimate the number of solutions by means of the geometric properties of $\Gamma$.
</p>

<p>  The only such estimate known, other than the uniqueness theorems discussed above, is 
for one specific boundary curve, namely Enneper's wire $\Gamma_r$ with $1 < r < \sqrt 3$. 
[Beeson 2015b] shows that this curve bounds exactly three minimal surfaces.
</p>

<p>910.  Let the Jordan curve $\Gamma$ lie on the boundary of a sphere, or more generally on 
the boundary of a (bounded or unbounded) convex body.  Is it true that (a) every solution of 
Plateau's problem for $\Gamma$ has least area?  (b) every solution of Plateau's problem 
bounded by $\Gamma$ is embedded (has no self-intersections)?  
</p>

<p> This problem has been completely solved.  The final result is stated on page 282 of
[Dierkes <em> et. al. </em> 1992]:  Let $K$ be a strictly convex body in $\R^3$ whose boundary
$\partial K$ is of class $C^2$, and suppose $\Gamma$ is a closed rectifiable Jordan curve 
contained in $\partial K$.  Then there exists an embedded minimal surface of the type of the disk
bounded by $\Gamma$.</p>

<p> 
This result was reached in a series of papers:

We say $\Gamma$ is an <em> extreme curve</em> if $\Gamma$ lies on the boundary of a convex body.<br>
[Gulliver-Spruck 1976] proved that if $\Gamma$ is extreme, and has total curvature less than $4\pi$,
then the unique minimal surface bounded by $\Gamma$ is embedded. [Tomi-Tromba 1978] proved that 
if $\Gamma$ is extreme, then it bounds an embedded minimal surface, but their solution might 
be unstable. <br>
[Almgren-Simon 1979] proved that an extreme curve bounds a stable embedded minimal disk.<br>
 [Meeks-Yau 1982] proved that an extreme curve either bounds two distinct embedded
minimal disks, or else it a curve of uniqueness in the strong sense that any
 minimal surface it bounds (of any genus) is an embedded minimal disk.
</p>

<p>911.  Let three rectifiable Jordan arcs $\Gamma_1$, $\Gamma_2$, and $\Gamma_3$  each 
connect two points $A$ and $B$.  Under which general geometric conditions on the configuration 
of these curves does there exist a three-part surface system of least area bounded by these curves,
in which the three surfaces meet along a Jordan arc ("dividing line")?  Under which conditions 
will the dividing line have only the points $A$ and $B$ in common with any $\Gamma_i$?
</p>

<p>912. Given a system as in 911, prove that (a) the dividing line is real-analytic (also up to the endpoints);
(b) the length of the dividing line is finite,  and less than the sum of the lengths of $\Gamma_1$, $\Gamma_2$,
and $\Gamma_3$; and (c) each of the three surfaces is real-analytic up to the dividing line and has 
no branch points on the dividing line.  Any two of the three surfaces meet at an angle of 120 degrees.
</p>

<p>913. With $\Gamma_1$,  $\Gamma_2$,
and $\Gamma_3$ as in 911, suppose $\Gamma_1$ lies in the $(x,y)$ plane, and $\Gamma_2$ is the reflection 
of $\Gamma_3$ in that plane.  Then must any solution be symmetric under that reflection?
</p>

<p>914. This problem concerns minimal surfaces bounded by a "fixed" Jordan arc $\Gamma_1$  and a 
"movable" Jordan arc $\Gamma_2$,  whose length is constrained to be less than the length of $\Gamma_1$.
We seek a solution of least area, allowing $\Gamma_2$ to vary.   The problem is to show that there are 
no branch points on the arc $\Gamma_2$.  (It then follows from work of Nitsche that $\Gamma_2$,  or at 
least the parts of $\Gamma_2$ that do not touch $\Gamma_1$, are real-analytic.)
</p>
<p>The next three problems concern the non-parametric minimal surface equation ``with obstacle.''  Let $P$ be a 
bounded convex domain in the $(x,y)$ plane.  Let $Q$ be a compact subset of $P$.  Let $f$ be defined
on $Q$;  $f$ is called the ``obstacle.''   We ask for a surface $z = \varphi(x,y)$,  zero on $\partial P$
and lying above the obstacle, i.e. $\varphi(x,y) \ge f(x,y)$ for $(x,y)$ in $Q$. The set of support
of $\phi$ is the set of $(x,y,\varph(x,y))$ with $(x,y)$ in $Q$ such that $f(x,y) = \varphi(x,y)$.  
</p>
<p> 915 
concerns the case when the obstacle defines a convex polyhedron.  Then must the set of support be 
connected?  Must the interior of the set of support also be connected?  Nitsche proved that each open
triangle of the polygonal obstacle either belongs entirely to the set of support,  or entirely not.
</p>
<p> 916.  Now suppose that $\partial P$ meets the $x$-axis in exactly two points $x_1$ and $x_2$,
and the set $Q$ is a closed interval $x_1,x_2]$ on the $x$-axis.
The obstacle is a real-analytic function $f$ defined on $Q$, but negative
outside an interval $[a,b]$ with 
 $x_1 \le a < b \le x_2$.  On $[a,b]$ nothing is assumed about the sign of $f$.  The problem is to 
 show that the projection of the set of support on the $(x,y)$ plane 
 consists of finitely many intervals and isolated points.
</p>
<p>917.  This problem points out that the non-parametric problem with obstacle could be generalized
by not requiring the boundary values to be prescribed on the entire boundary of $P$, but only 
on certain arcs,  and allowing the domain of the obstacle to contain parts of the boundary of $P$.
The problem does not actually ask a question, but presumably the problem is to prove existence and 
describe the set of support.
</p>
<p>The next few problems (918 to 924) concern the ``free boundary problem'', in which we are given a surface $T$
and a Jordan arc $\Gamma$ with distinct endpoints on $T$,  and we ask for a minimal surface $X$ defined
in the unit disk, taking the upper semicircle $\partial^1 P$ onto $\Gamma$ monotonically, and 
the lower semicircle $\partial^2 P$ onto $T$ in the weak sense that for every sequence $z_n$
approaching a point $z$ on $\partial^2 P$, the distance from $X(z_n)$ to $T$ approaches zero. 
It is also required that $X(z_n)$ be continuous at $(0,-1)$ and $(0,1)$. 
</p>
<p>918. This problem asks for sufficient conditions on $\Gamma$ and $T$ that would guarantee that
the free boundary problem has a unique solution.
</p>
<p>919.  Give conditions on the regularity of $\Gamma$ and $T$, and if necessary, on the manner
of approach of $\Gamma$ to $T$ at the endpoints, that suffice to prove that $X$ extends to 
a rectifiable curve on $\partial^2 P$.
</p>
 <p>920.  Give (under appropriate assumptions about the regularity of $\Gamma$ and $T$) an 
 estimate of the length of the trace of $X$ on $T$, i.e. the image of $\partial^2 P$.
 (If $T$ is a plane, this length is bounded by the length of $\Gamma$.)
 </p>
 <p>921. If $T$ is $C^{2,\alpha}$ (or maybe just $C^{1,\alpha}$), show that $X$ meets $T$
 orthogonally.
 </p>
 <p>922.  Boundary regularity for the free boundary problem, when $X$ is not necessarily a
 relative minimum, but only a stationary value for Dirichlet's energy.
 </p>
 <p>923.  Estimate the number of branch points possible in the free boundary value problem
 in terms of geometric properties of $\Gamma$ and $T$.
 </p>
 <p>924. Show that the least-area solutions of the free boundary problem have no (interior or boundary)
 branch points.
 </p>
 <p>925.  Let $\Gamma_1$ and $\Gamma_2$ be two Jordan curves lying in parallel planes $E_1$ and $E_2$.
 Suppose $\Gamma_1$ lies in the interior of a convex curve $C_1$ in $E_1$, and 
 $\Gamma_2$ lies in the interior of a convex curve $C_2$ in $E_2$.  If $\Gamma_1$ and $\Gamma_2$ bound
 a minimal surface of the type of the annulus,  so do $C_1$ and $C_2$.  (Nitsche proved this in case
 $C_1$ and $C_2$ are circles.)
 </p>
 <p>926. 
 </p>
 
</div>
</div>