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.

Let

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 .