A Geometric Approach to Pde Lecture Notes from the 1997 Daewoo Workshop

In this lecture we will see how to study a system of partial differential equations (henceforth denoted pde) from a geometric perspective. Example 1, the minimal surface equation. Given a surface in Euclidean 3-space, M 2 ⊂ E 3 , described locally as a graph z = z(x, y), at a point p = (x, y, z(x, y)), let H(p) = 1 2 (1 + z 2 y)z xx − 2z x z y z xy + (1 + z 2 x)z yy (1 + z 2 x + z 2 y) 3 2. H(p) is called the mean curvature function of the surface M. H(p) has geometric meaning, that is, it is a well defined function on M , independent of coordinates chosen and invariant under the displacement of M by rigid motions. Consider the pde for surfaces in E 3 with the property that H(p) ≡ 0. Geometrically such surfaces are critical for the variation of area and are called minimal surfaces. If M is a minimal surface, then for all x ∈ M , there exists a open neighborhood˜U x ⊂ M such that for all open subsets U ⊂ ˜ U x and patches of surface V with ∂V = ∂U , then area(V) ≥ area(U) (see e.g. [S, III]). We will be studying minimal surfaces and higher dimensional generalizations in the next lecture. One amazing aspect of the pde for minimal surfaces in E 3 is that the solutions are exactly given by solutions to the Cauchy-Riemann equations. More precisely, there is the following theorem: Theorem (Weierstrass [S, IV, p 395]). Minimal surfaces M 2 ⊂ E 3 can be described by holomorphic functions. More precisely, every minimal surface in E 3 is locally of the form: x(w, w) = Re 1 2 f (w)(1 − g(w) 2)dw y(w, w) = Re i 2 f (w)(1 + g(w) 2)dw z(w, w) = Re f (w)g(w)dw