Surfaces, currents, and varifolds
What is a surface? Different technical definitions serve different purposes. Here we’ll focus on two-dimensional surfaces S in R3 or R4.
(1) One could just define S as a set, or perhaps a set with positive, finite area. The best definition of area is two-dimensional “Hausdorff measure” H2. This definition of surface is too general, because it includes some fractal curves, for example. It is common to restrict to rectifiable sets, which have “approximate tangent planes” at almost all points.
(2) One could define S locally as the graph of say a C1 function f from R2 to R or R2. Then S will have tangent planes everywhere. This definition is fine for smooth surfaces, but it excludes even rather mild singularities. For example, in R4 = C2, {w2-z3 = 0} is not the graph of a function in any neighborhood of (0,0).
(3) Often it is more convenient to define S as the level set {g = c} of say a C1 function. When the gradient of g is nonvanishing, this is equivalent to the graph definition (2). If the gradient of g is allowed to vanish, you can pick up singularities such as {w2-z3 = 0}.
(4) In calculus, you often define a surface S locally as the image of a C1 function from R2 to R3 or R4. Again, when the Jacobian is nonvanishing, this is equivalent to definition (2), and the surface is called a C1 manifold. If the Jacobian vanishes, you can pick up singularities such as {w2-z3 = 0}.
(5) Any measurable set S yields a measure 
One could define a surface as a measure with certain properties. Measures have nice compactness properties, but are much too general.
(6) If a surface S has a tangent plane 
(7) Similarly, if a surface S has a tangent plane at all points, one can integrate a smooth differential form over S and thus view S as a linear functional on differential forms, called a current. Currents have nice compactness properties and include surfaces with multiplicities. Currents and varifolds are the fundamental surfaces of geometric measure theory.
For more, see my Geometric Measure Theory book.
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Sam Ferguson:
In my Topology course, we use a definition somewhat similar to (2), but less restrictive.
I’ll describe it below.
First, we would like to have a concept of “closeness” in our surface, in other words – distance! But it turns out that you really only need “open sets” to describe closeness. This leads to the natural first requirement that our surface be a topological space.
Second, we would like distinct points to be separated! This is something that our (or my) intuition demands.
We can accomplish this by requiring that our surface be “Hausdorff,” which amounts to the above.
Finally, we would like it to look (at least locally!) like two-dimensional real space, but do we really need “smoothness?” It might be more lenient to start off with continuity; that sounds less restrictive. So our last requirement should be that for each point of our surface, there is some open set containing it, which can be sent continuously to and from two-dimensional real space.
It seems to me that, at least initially, these three requirements should be enough to yield a surface, and a similar definition holds for manifolds of an arbitrary ( but finite!) number of dimensions.
Later on, we could certainly add all sorts of structures:
I like tangent planes, linear functionals, and differential forms just as much as the next guy!
Regardless, the graph of {w^2 – z^3 = 0} is a surface (three dimensional!) under the definition which I have described above.
Your Geometric Measure Theory book sounds like a good read – I will request that it be sent to my nearest library!
3 February 2009, 11:00 pmFrank Morgan:
Right Sam, one abstract definition of a surface is a manifold. In applications, one often wants something more specific—a surface in say
with tangent planes—and more general (with some singularities allowed).
FM
4 February 2009, 10:25 am