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This is a very naive question, is there a way to geometrically understand Sobolev spaces without going through analysis and PDE's? To my knowledge, Sobolev spaces where created precisely to study PDE's so it might be a bit nonsensical to not want to go through that path, but perhaps one can understand these spaces via some construction on manifolds or something analogous?

In case the answer to the above is "No." then I would ask what you would consider to be the nicest use of Sobolev spaces to geometry (including solving a particular PDE and stuff like that, of course)?

Also, it would be nice to know a bit of the history behind the modern usage of Sobolev spaces...

Thanks!

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A nice way to learn about Sobolev spaces and their contributions in geometry is through Aubin's book Some Nonlinear Problems in Riemannian Geometry. –  Ben McKay Mar 5 at 16:43
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Another nice approach is Hebey's book Sobolev Spaces on Riemannian Manifolds. –  Ben McKay Mar 5 at 16:46
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Approaching Sobolev spaces, I think the first thing to try to feel is the Sobolev norm. When a function, even smooth, is a large or small in this sense. –  Pietro Majer Mar 5 at 16:50
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You can get some feeling for the Sobolev norm by looking at how it reacts to high frequency waves and to small bumps; see my notes euclid.ucc.ie/pages/staff/Mckay/pde/2014/spring/… –  Ben McKay Mar 5 at 18:42
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It might be also good to note that as this is in a sense an opinion question I am a bit reluctant to picking an "accepted answer" so I might just choose the one with the most up votes in a couple of days, apologies! –  Juan OS Mar 5 at 21:42

3 Answers 3

No time to give a complete answer but just a hint to a possible direction. Sobolev spaces in $R^n$ arise as the largest possible spaces on which some functional ('energy') can be defined. So they are the natural domain of some important functionals, the basic example being the Dirichlet functional $\int|\nabla u|^2dx$. This is the most synthetic point of view I can think of, and I wouldn't say it has a geometric nature. However, you have natural generalizations of this kind of functionals on manifolds, so if that is your background, this might give you a better hold on the nature of these spaces.

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That connection with the Dirichlet functional brings one naturally to the calculus of variations, which sort of brings one full circle back to the study of PDEs, or am I babbling? –  Tim Seguine Mar 5 at 21:02
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It's the farthest from PDEs I managed to get :) –  Piero D'Ancona Mar 5 at 21:31
    
Thanks for your answer! So I guess you might say that Sobolev spaces might "measure" how complicated the energy functional is in some manifold? or is that not at all what you mean? –  Juan OS Mar 5 at 21:36
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Sobolev norms measure regularity and size of functions. Regularity is a local thing, so it does not really have much to do with geometry. Size has in part to do with the behaviour of the geometry at infinity, but again, I think the contact with geometry is not really essential. –  Piero D'Ancona Mar 5 at 23:03

Unlike their Holder cousins, most Sobolev spaces are reflexive Banach spaces. Reflexivity is a highly desirable feature for variational problems because it gives a little bit of compactness enough so you can prove the existence of minimizers (or more general critical points) of various energy functionals. Such critical points satisfy the Euler-Lagrange equations and thus you obtain existence of (weak) solutions of many important equations in geometry or physics. Given that their norms have an integral description, the Sobolev spaces are tailor made for energy functionals described by integrals. The famous and for a while controversial Dirichlet principle states that any function $u$ defined on a compact smooth domain $\newcommand{\bR}{\mathbb{R}}$ $\Omega\subset \bR^N$ which is zero on the boundary of $\Omega$ and minimizes the energy functional

$$E(u)=\int_\Omega\left(\frac{1}{2}|\nabla u(x)|^2 -f(x) u(x)\right) dx $$

must be a solution of the Poisson problem $\Delta u=f$ in $\Omega$ satisfying the boundary conditions $u=0$ on $\partial \Omega$.

Weierstrass pointed out a major flaw in the classical understanding of this principle by constructing, in a special case, a minimizer of this energy functional which is not twice differentiable so the Laplacian does not make sense.

That stopped things in their tracks for a while until Hilbert, in his famous 1900 Paris Conference talk included this in the list of his famous 27 problems. In particular, he hinted to a way out by stating that any variational problem has a solution provided that, if need be, we suitably define the concept of solution.

You can read more about this and see many applications of Sobolev spaces to geometry in these lectures.

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Thank you very much! It seems those lectures contain the answers to many of the questions that I come up with! thank you for sharing the link with the community and, of course, typing the manuscript up! –  Juan OS Mar 6 at 8:31

In my view, in decreasing order of importance, Sobolev spaces have a tremendous impact on geometry because:

  1. On $\mathbb R^n$ they have an extremely nice behavior with respect to Fourier transform. This leads to pseudo differential operators and fourier integral operators and give very powerful tools for solving linear PDE's. See [Shubin: Pseudodifferential operators and spectral theory, Springer] for a compact treatment.

  2. A priori estimates for elliptic equations, the Sobolev inequality, and Rellich's lemma have uses in linear theory and nonlinear theory. In particular, they lead to module properties of Sobolev spaces (that they are Banach algebras, etc.)

  3. Using 1 and 2 one can view certain nonlinear PDE's is smooth (Lipschitz) vector fields on suitable Sobolev spaces. This can be used to do geometry on manifolds of mapping, diffeomorphisms, shapes, etc. Now one can argue, that juggling flows of vector fields is a good part part of differential geometry (essentially this part which goes beyond algebraic geometry for smooth spaces). Just an indication: horizontal lifts of flows and their commutation properties leads directly to curvature

I hope that this is helpful.

EDIT: added a reference to 1.

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Thanks for your answer! All points of view are helpful for me as I'm trying to get a general feel for these objects, could you please recommend a reference for the first point you mention? –  Juan OS Mar 5 at 21:38

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