Let be a domain. , .
Recall: from last talk, Zhenghe described the Lagrange’s Equation, in this case the equation is written: (we denote as )
Theorem: The graph of is area-minimizing then satisfies Lagrange’s equation.
Proof: Since is area-minimizing,
is minimized by for given boundary values. Hence the variation of due to an infinitesimal of where . i.e.
Apply integration by parts, since vanishes on , the constant term vanishes, we have:
for all with , hence . i.e.
which is the Lagrange’s equation.
We should note that the converse of the theorem is, in general, not true.
Example: two rectangles, star-shaped 4-gon.
Theorem: For convex, any satisfying Lagrange’s equation has area-minimizing graph.
Let be 2-form in s.t. and . i.e. acts on the unit Grassmannian space of oriented planes in .
Definition: An immersed surface is calibrated by if for all in the unit tangent bundle of .
*All calibrated surfaces are automatically area-minimizing.
Let be the two-form
By construction, for all , , we have , when the plane spanned by is tangent to the graph of at .
which is by Lagrange’s equation. Hence is closed.
Let be the graph of , since whenever the plane spanned by is tangent to at , we have
Suppose is not area-minimizing, there exists 2-chain with with smaller area than that of .
Since is convex, any not contained in cannot be area-minimizing (by projecting to the cylinder). Hence we may assume (So that is well-defined on )
Since bounds a 3-chain, is closed, hence
Beacuse hence .
Therefore we have . i.e. is area-minimizing.
Definition: A minimal surface in is a smoothly immersed surface which is locally the graph of a solution to the Lagrange’s equation.
Note that small pieces of minimal surfaces are area-minimizing but lager pieces may not be.
Example: Enneper’s surface
Theorem: Let be a rectifiable Jordan curve in , there is a area-minimizing 2-chain with
Sketch of proof:
There exists rectifiable 2-chain with boundary being . -Take a point in and take the cone of the curve.
Define flat norm on the space of 2-chains in by i.e. if two chains are close together, they would almost bound a 3-chain with small volume, hence the difference has small norm.
Fact: and , for any chain , we may find a chain inside the grid of mesh where (hence the area of is also bounded). Since there are only finitely many such chains, we have:
is totally bounded under the flat norm .
Hence is compact.
Now we choose sequence of rectifiable chains with boundary and area decreasing to
Choose large enough s.t. . Project radially onto the projection does not increase area.
Hence for all . i.e. and .
Since and is compact, there exists subsequence converging to a rectifiable chain .
We can prove that: (continuity of under the flat norm).
(lower-semicontinuity of area under the flat norm).
Therefore is an area-minimizing surface with .