By Nomizu K., Sasaki T.
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Additional resources for Affine differential geometry. Geometry of affine immersions
We shall prove by contradiction that there is no closed minimal surface in R3 . Let M be a closed minimal surface in R3 . Then since M is bounded as a subset of R3 there is a plane that does not intersect M. We move this plane parallel to itself in the direction of M until it first touches M at some point P (we denote the tangent plane by n). Clearly, P is an internal point of M and the plane II (M has no boundary points because it is closed) and M lies on one side of lI . By the maximum principle M is flat in some neighborhood of P.
However, in geometry there is another definition of completeness of a surface, imposing weaker restrictions, which nevertheless extends to a wider class of surfaces and turns out to be quite sufficient to obtain meaningful results. DEFINITION. Let M be an arbitrary immersed connected surface, and p the intrinsic metric defined above. Then M is said to be complete if the metric space (M, p) is complete. If M is an embedded connected surface, then it is easy to see that p(A, B) > IABI for any points A and B of M.
In this coordinate system the surface M in a neighborhood of P can be given by a graph x = x, y = y, z = f (x , y). All such functions f are described by the minimal surface equation (see § 1 of Chapter 2): (I+Jy)fxx-2fxfyfxy+(1+ f2)fyy=0. Let us form three functions: 1+f 1+ f2+fy' 1+f ffy 1+ f2+ fY 1+f2+fy' Henceforth for convenience we put p = fx , q = f, , and w = J1 + f2 + fy . We recall that V l + f2 + f? dx d y is the element of area of the surface M . It turns out that from the minimal surface equation we have the following relations: (I wp2) y- (wq)x' (1 wg2)x= (wq)r In fact, direct calculation shows that Y (I x Ulfy) x - (f fy) y = - wx [(l + Jy )fxx - 2fj fxy + (1 + f) f y].
Affine differential geometry. Geometry of affine immersions by Nomizu K., Sasaki T.