Page 332 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 26: Lines of Curvature

Proof. Let Y be the generator of F(u; t). We will show that there is a smooth family of Notes
diffeomorphisms : U    ,   U such that Y is also the generator of the variation G = X  f.
This proves the proposition since Proposition 5 gives that A (U) is constant. Since Y is tangential,
G
we can write Y = y X . Consider the initial value problem:
i
i
dv i y (v), v (0) u . i
i
i
dt  
Since the y ’s are compactly supported, a solution v = v(u; t) exists for all t. Defining (u; t) =
i
v(u; t), then an application of the inverse function theorem shows that (u; t) is a diffeomorphism
for t in some small interval (–, ). Finally, we see that:

dX   X i dv i  X y  Y.

i
dt dt i
Our next theorem gives an interpretation of the mean curvature as a measure of surface area
variation under normal perturbations.
Theorem 4. Let X : U   be a parametric surface, and let F(u; t) be a compactly supported
3
variation with generator Y . If V  U is open with V compact in U, and the support of F
contained in V , then

dAF(V) 2 (Y N)H dA.
dt    V  ...(10)

Proof. By Propositions 6 and 7, it suffices to consider normal variations with generator Y = fN.
i
ij
ij
In that case, we find that Y = fN + fN, so that g X  Y = fg X  N   fk = –2fH. The theorem
j
j
j
j
i
i
j
i
follows by substituting into (6).
Definition 8. A parametric surface X is area minimizing if A (U) A X(U) for any parametric
X

3


surface X such that X  X on the boundary of U. A parametric surface X : U   is locally area
minimizing if for any compactly supported variation F(u; t), the area A (U) has a local minimum
F
at t = 0.
Clearly, an area-minimizing surface is locally area-minimizing. The following theorem is an
immediate corollary of Theorem 4.
Theorem 5. A locally area minimizing surface is a minimal surface.
Note that in general a minimal surface is only a stationary point of the area functional.
26.4 Bernstein’s Theorem
In this section, we prove Bernstein’s Theorem: A minimal surface which is a graph over an entire
2 3
plane must itself be a plane. We say that a surface X is a graph over a plane Y :    , where
2
Y is linear, if there is a function f :    such that X = Y + fN where N is the unit normal of Y.
Theorem 6 (Bernstein’s Theorem). Let X be a minimal surface which is a graph over an entire
plane. Then X is a plane.

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