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   <h2> Branched Minimal Surfaces</h2>
    <p>  Much of my work 
    on minimal surfaces has involved  <em>branched minimal surfaces</em>, i.e., minimal surfaces with <em>branch points</em>. This page is intended to provide pictures of branch points, without going deeply into the mathematical formulas, although the most elementary formulas will be mentioned here. </p>
    <p> A <em>branch point</em> is a point where the surface is not parametrized in a one-to-one fashion. At that point, it doesn't meet the classical definition of a geometric surface, which requires that the surface look, in the vicinity of each point, like a curved disk. However, one is forced to consider branch points because the keep on arising in proofs as limits of surfaces without branch points. There are nice formulas that describe the appearance of branch points, so one can draw pictures of them and be assured that every branch point looks like one of these pictures. The pictures are described by two integers: the <em>order</em> and the <em>index</em>. Branch points can be <em>interior </em>branch points or <em>boundary</em> branch points. We will start with pictures of boundary branch points:<em> [Note: in the Safari browser, if the pictures do not rotate after they have finished loading, use the back button and then click the link again. They should then start rotating immediately. Other browsers do not seem to have this problem.]</em></p>
    <p> <a href="BranchPointAnimations/Order2Index1.gif">Order 2, Index 1</a>. </p>
	   <p> <a href="BranchPointAnimations/Order4Index2.gif">Order 4, Index 2</a>. </p>
	   <p>One important thing that is not obvious from the pictures is this: the unit normal extends continuously to the branch point. That means that all the sheets of the surface flatten out near the center and become horizontal. In my work I am mostly concerned with a very small neighborhood of the branch point. Then the picture is much flatter. Here is a picture that shows a smaller neighborhood of the order 2, index 1 branch point, in which you can see how the normal tends to vertical at the origin:</p>
	   <p><a href="BranchPointAnimations/BoundaryOrder2Index1R2.gif">Order 2, Index 1, r=0.2</a></p>
	   <p>These are called <em>boundary</em> branch points because they could occur at the boundary of a surface. In these examples, you see only a neighborhood of the boundary, and the boundary in question is a straight line. The pictures show parts of the <em>x</em>, <em>y</em>, and <em>z</em> axes as well. The next picture shows an <em>interior</em> branch point. Essentially, the first picture above is half of this picture. </p>
	   <p><a href="BranchPointAnimations/InteriorOrder2Index1.gif">Interior branch point, order 2, index 1</a>. </p>
	   <p>  A branch point of order <i>M</i> and index <i>k</i> goes "around" <i>M</i> + 1 times while going 
	      "up and down"  <i>M</i> + <i>k</i> + 1 times.   When the branch point is on the boundary, only half of 
		  this surface is seen.   The order of a boundary branch point must be even if the surface is 
		  to touch the boundary monotonically--with odd order, it would double back on itself.   Thus order 2
		  is the simplest case.   Index 0  would give a piece of a plane, so index 1 is the simplest three-dimensional
		  case.  Thus order 2, index 1 goes around (as an interior branch point) 3 times while going up and down 4 times;
		  and as a boundary branch point, it goes around one and a half times while going up and down twice.  An 
		  order 4, index 2 boundary branch point goes around two and a half times while going up and down three and 
		  a half times.  These are the cases illustrated at the links above.  		  
	</p>   
	<p> The surfaces illustrated have simple formulas using a complex variable <i>z</i>.  Namely, if 
	   the surface is given by three functions <em>X</em>(<em>z</em>), <em>Y</em>(<em>z</em>), and <em>Z</em>(<em>z</em>), 
	    then we have </p>
		<p><em>X</em>+<em>iY</em> = <em>z</em><sup><em>M</em> + 1</sup>/(<em>M</em>+1)  </p>
		<p> Z = Re[ <em>z</em><sup><em>M</em> + <em>k</em> + 1</sup>/(<em>M</em>+ <em>k</em>1)]</p>
		<p>Here Re means "the real part of".  The first formula, which does not mention the real part explicitly,
		   is just a way of saying that X and Y are the real and imaginary parts, respectively, of a constant
		   times <em>z</em><sup><em>m</em></sup>,  so they are (up to a constant) <em>r</em><sup><em>M</em>+1</sup> cos(<em>(M+1)&theta;</em>) and
		   <em>r</em><sup><em>M</em>+1</sup> sin(<em>(M+1) &theta;</em>), where <em>r</em> and <em>&theta;</em> are the 
		   polar coordinates of <em>z</em>.   These formulas corroborate the English descriptions above using the 
		   phrase "goes around" so many times.   The pictures above show the part of the surface defined over the disk |<em>z</em>| 
		   &le; 0.6.</p>
		
        <p>   Here is a good problem for the beginner at this point:  How many lines of intersection of the surface with
        itself will there be,  in a branch point of order <em>M</em> and index <em>k</em>?  
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	<p>  Any branch point must be given by formulas that start out like those above, but there may also be terms with 
	    higher powers of <em>z</em>.   If <i>M+1</i> exactly divides <i>M+k+1</i> then to the first approximation, the surface will trace over 
	      itself exactly. (You will have noticed that if you tried to solve the exercise above.)
		   If there is to be any separation of the different "sheets" of the surface, that must 
		  arise from higher-order terms in the formula for the surface.   You can see that in the following
		  picture, which shows a boundary branch point of order 2 and index 3, so <em>M</em> + 1 is 3 and 
		  <em>M</em> + <em>k</em> + 1 is 6. Notice that at first glance the surface appears to be like a 
		  distorted disk--but looking carefully, you can see from the coordinate markings on the surface 
		  that it actually does go around one and half times, with the last half time exactly overlapping 
		  the first half. </p>
	<p><a href="BranchPointAnimations/Order2Index3.gif">Order 2, Index 3</a><em> A false branch point </em></p>
	<p>If the overlap is exact, as in this picture, the branch point is called a <em> false branch point.</em> But bear in mind that the picture could look <em>almost</em> like this, with the sheets being separated by a very tiny amount (that might not even be visible in a computer-graphic picture) due to a term in a high power of <em>z</em>. A branch point that is not false is called, naturally, a <em>true</em> branch point. </p>
	<p>Here's one last, more complicated, picture: </p>
	<p> <a href="BranchPointAnimations/Order4Index2.gif">Order 4, Index 2</a>. </p>
		<p>&nbsp; </p>
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