Page 245 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes t = 0, its maximum at t = t where 0 < t < L, that (0) and (t ) lie on the x-axis, and that  enters


0
0
0
the upper-half plane in the interval [0, t ]. All these properties can be achieved by reparametrizing
0
and rotating .
We now claim that p   (0) and q   (t ) are the only points of  on the x-axis. Indeed, suppose
0
that there is another point r   (t ) on the x-axis, then one of these points lies between the other
1
two, and the tangent at that point must, by convexity, contain the other two. Thus, by the
argument used in the proof of Theorem 10 the segment between the outer two is contained in ,
and in particular pq is contained in . If follows that k  0 at p and q where k has its minimum
and maximum, hence k  0, a contradiction since then  is a line and cannot be closed. We
conclude that  remains in the upper half-plane in the interval [0, t ] and remains in the lower
0
half-plane in the interval [t , L].
0
Suppose now by contradiction that (0) and (t ) are the only vertices of . Then it follows that:


0
k  0 on [0, t ], k0 on [t , L].
0
0
Thus, if we write  = (x, y), then we have k y  0 on [0, L], and x = –ky, hence:
L L L


0   0 x ds    0  ky ds   0 k y ds.


Since the integrand in the last integral is non-negative, we conclude that ky  0, hence y  0,
again a contradiction.
It follows that k has another point where k changes sign, i.e., an extremum.
Since extrema come in pairs, k has at least four extrema.

20.4 Fenchel’s Theorem

We will use without proof the fact that the shortest path between two points on a sphere is
always an arc of a great circle. We also use the notation  +  to denote the curve  followed by
1
2
1
the curve  .
2
n
Definition 11. Let  :[0,L]  be a regular curve parametrized by arclength. The spherical
n


image of  is the curve  :[0,L]  n 1 . The total curvature of :[0,L]  is:
 
K  I  ds.
We note that the total curvature is simply the length of the spherical image.
Theorem 9. Let  be a regular simple closed curve in  parametrized by arclength. Then the total
n
curvature of  is at least 2:
K  2,

with equality if and only if  is planar and convex.
The proof will follow from two lemmata which are interesting in their own right.

n
Lemma 2. Let  :[0,L]  be a regular closed curve parametrized by arclength. Then the
spherical image of  cannot map into an open hemisphere. If  maps into a closed hemisphere,
then  maps into an equator.

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