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I noticed this question posted on MO, hence I estimated that this may be an acceptable question even in MO (and not for MSE). I studied the notion of current and in a nutshell I understood "varifolds are weaker objects than currents."

My question is what kind of prerequisites one needs to have in order to study varifold theory and varifold geometry. For example, as far as currents are concerned, one needs to have grasped the notions of geometric measure theory and a bit of multilinear algebra and the introduction to distribution theory in functional analysis. I would like to ask the principal prerequisites one needs to know to fully grasp varifold theory. Geometric measure theory, of course, but how much Riemannian geometry? How much partial differential equation theory for the theory of regularity as well? Thank you very much in advance!

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    $\begingroup$ In my opinion you have it slightly backwards. Varifolds - as other low-regularity objects in analysis - are usually considered to compensate for a lack of compactness in the smooth category. That's where prerequisites tend to come form: for example parabolic PDE for Brakke flow, differential geometry for stationary varifolds etc. As far as the varifold aspect is concerned, I think there are few formal prerequisites - some measure theory and functional analysis, perhaps. $\endgroup$ – Leo Moos May 24 at 12:04
  • $\begingroup$ I would suggest that you ask a new question instead of deleting the old one. If you do this, you could also clarify your question and add more detail. For example, why does the postscript in the answer below not address your new question? $\endgroup$ – Leo Moos Jul 13 at 16:35
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    $\begingroup$ @WholeFood, there is a rule here that we don't (significantly) change the question after an answer has been posted. The reason is that someone has worked hard to compose that answer, and a change in the question might render that work useless, which would not be polite. Another reason is that a future reader who hasn't noticed that the question was changed might not understand how the answer addresses it. $\endgroup$ – Alex M. Jul 13 at 17:31
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The general prerequisites are almost the same as for currents, mainly a strong understanding of measure theory and a bit of geometrical intuition.

There is an aspect of multilinear algebra and some functional analysis involved as well, but a lot of that can be studied at the same time. The need for Riemannian geometry depends mostly on what you want to study. For varifolds on Riemannian manifolds, the need is kind of obvious, but for varifolds on $\mathbb{R}^n$, you can do completely without. The notion of mean curvature is involved in many problems involving varifolds, but not much more, as they simply lack the regularity needed for some of the more fancy ideas from differential geometry. Anything else, such as knowledge of PDEs and Calculus of variations is nice to have with regards to context, but the same is true in reverse and you have to start somewhere.

Fundamentally, varifolds in $\mathbb{R}^n$ are just Radon measures on $\mathbb{R}^n \times Gr(n,m)$, where $Gr(n,m)$ is the space of $m$-dimensional linear subspaces of $\mathbb{R}^n$. If you understand all the words in that sentence and know a bit about rectifiability, then you have the prerequisites to study varifolds.

P.S. I wouldn't say that they are either weaker or stronger than currents though. In the technical sense, there are currents that cannot be reasonably expressed as varifolds and varifolds that cannot be expressed as currents (even ignoring the issue of orientation). Similarly, in the practical sense, each has its advantages and its downsides and which one is better is entirely problem-dependent.

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  • $\begingroup$ Thank you for the nice answer! After a general introduction to Varifolds, would you add something more on the prerequisites to jump straight to Allard regularity paper (1972) or not? $\endgroup$ – Whole Food May 24 at 13:06
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    $\begingroup$ @LorenzoLotto If you want to go in that direction I would tend towards Simon's "Introduction to geometric measure theory", which I think covers Allard's results, but is also a self-contained introduction into the topic. $\endgroup$ – mlk May 24 at 13:31

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