Page 339 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 6. Let X : U   be a parametric surface, and let V  U be open. Let N : U   be the Gauss
2
3
map of X, then .................
7. ................. Let X be a minimal surface which is a graph over an entire plane. Then X is a
plane.

26.9 Review Questions

1. Prove that setting     1, g     1/ in the Weierstrass representation, we get the catenoid.
f
Find the conjugate harmonic surface of the catenoid.
2. Let U   2 , let f : U   be a smooth function, and let X : U   be given by (u, v, f(u,
3
v)), where (u, v) denote the variables in U. Show that X is a minimal surface if and only if
it satisfies the non-parametric minimal surface equation:

1 q p 2  u  2pqp  1 p q 2  v  0,
u
where we have used the classical notation: p = f , q = f . Show that if f satisfies the equation
u
v
above then the following equations are also satisfied:

   1 q 2       pq  , 
2
2
 u  1 p  q 2   v  1 p  q 2 


   

   pq       1 p 2  . 
2
2
 u   1 p  q 2    v   1 p  q 2  


2
3. Let f C (U) be a convex function defined on a convex open set U, and let

2
f   (p,q): U   denote the gradient of f. Prove that for any u , u  U the following
1
2
inequality holds:
  0.
u  u 1   f u      f u 1
2
2
4. Let U   be open. A map ': U ! Rn is expanding if x y   (x)   (y) for all x, y  U. Let
n
n
 : U   be an open expanding map. Show that the image of the ball B (x ) of radius R
R
0
x
B
centered at x  U contains the disk    of radius R centered at  . x 0 Conclude that
R
0
0
if U =  , then  is onto  .
n
n
Answers: Self Assessment
1. line of curvature 2.     0.


3. parametric surface 4. asymptotic line
ˆ
5. curvature H = 0. 6. K (V)  A (V).
N
X
7. Bernstein’s Theorem
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