Page 333 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes We may without loss of generality assume that X is a graph over the plane Y (u, v) = (u, v, 0), i.e.

X(u,v)  u,v,f(u,v) . It is then straightforward to check that X is a minimal surface if and only
if f satisfies the non-parametric minimal surface equation:

1 q p 2  u  2pqp  1 p q 2  v  0, ...(11)
v
where we have used the classical notation: p = f , q = f . We say that a solution of a partial
u
v
differential equation defined on the whole (u, v)-plane is entire. Thus, to prove Bernstein’s
Theorem, it suffices to prove that any entire solution of (11) is linear.
Proposition 8. Let f be an entire solution of (11). Then f is a linear function.
If f satisfies (11), then p and q satisfy the following equations:


   1 q 2       pq  , 
 u  1 p  q 2   v  1 p  q 2  ...(12)
2
2


   

   pq       1 p 2  . 
2
2
 u  1 p  q 2   v  1 p  q 2  ...(13)


   
Since the entire plane is simply connected, Equation (13) implies that there exists a function 
satisfying:
1 p 2 pq

  ,   ,
v
u
2
2

1 p  q 2 1 p  q 2

and Equation (12) implies that there exists a function satisfying:
pq 1 q 2

  ,   .
v
u
2
2


1 p  q 2 1 p  q 2
Furthermore,  =  , hence there is a function h so that h = , h = . The Hessian of the function
v
v
u
u
h is:
h
      h uu h uv         u u   v   ,
h
ij
h
v 
vv 
vu
hence, h satisfies the Monge-Ampère equation:
det h ij 1. ...(14)
  
In addition, h > 0, thus   is positive definite, and we say that h is convex. Proposition 15
h
ij
11
now follows from the following result due to Nitsche.
2
Proposition 9. Let h C 2   be an entire convex solution of the Monge-Ampère Equation (14).

Then h is a quadratic function.
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