I hope not to be too simplistic.


I read about this monotonicity formula [A question on the monotonicity formula for minimal submanifolds](https://mathoverflow.net/questions/397505/a-question-on-the-monotonicity-formula-for-minimal-submanifolds)


I noticed that the monotonicity formula is often used in regularity theory for surfaces, declined in different versions, like for varifold (Allard 1972) or for currents or finite perimeter sets.


My question is what is the main utility of the monotonicity formula? I noticed the results showing that the density exists at every point and it is greater or equal than 1. But how does this help as well? Thanks!



Main utility of the monotonicity formula for generalized surfaces
A basic answer is that "the monotonicity formula places constraints on the shape of a minimal surface" e.g., you cannot have a lot of area concentrated in a ball if then later there is a (relatively) small amount of area. This, along with the convex hull property, already tells you a lot about the possible shape of a minimal surface.


However, I think you are looking for a concrete application. Here is one sample application that is somewhat different from what I have described above:


Suppose that $\Sigma\_i,\Sigma \subset B\_2(0) \subset \mathbb{R}^3$ are smooth embedded minimal surfaces and for any $f \in C^0\_c(B\_2(0))$ it holds that
$$
\tag{\*} \int\_{\Sigma\_i} f|\_{\Sigma\_i} \to \int\_\Sigma f|\_\Sigma
$$
as $i\to\infty$ (this is a very weak notion of convergence of surfaces). Then, if $x\_i \in \Sigma\_i\cap B\_1(0)$ has $x\_i\to x$ then $x\in\Sigma$.


Proof: Assume $x\not \in \Sigma$. Then, there is $B\_{3\epsilon}(x)$ disjoint from $\Sigma$. Choose a bump function $f=1$ on $B\_{2\epsilon}(x)$ and $f$ vanishing outside of $B\_{3\epsilon}(x)$. Now, by (\*) we have that
$$
\int\_{\Sigma\_i} f|\_{\Sigma\_i}\to 0.
$$
Thus, for $i$ large, $B\_{\epsilon}(x\_i)\subset B\_{2\epsilon}(x)$, so we have arranged that
$$
|\Sigma\_i\cap B\_\epsilon(x\_i)|\leq \int\_{\Sigma\_i} f|\_{\Sigma\_i} \to 0. 
$$
On the other hand **the monotonicity formula** implies that
$$
\frac{|\Sigma\_i\cap B\_\epsilon(x\_i)|}{\pi \epsilon^2} \geq \lim\_{r\to0}\frac{|\Sigma\_i\cap B\_r(x\_i)|}{\pi r^2} = 1
$$
since $\Sigma\_i$ is smooth (and thus nearly flat on small scales). This is a contradiction.




---


Note that if the $\Sigma\_i$ are not minimal (i.e., if they don't satisfy the monotonicity formula), it is easy to find a counterexample to the above result by taking a flat disk $\Sigma$ and forming $\Sigma\_i$ by gluing on a lot of "tentacles" which all have small area (and thus small contribution to the integral as in (\*)). Thus, this is a nontrivial result.


(Note that I have not attempted to prove the most general version of this result, if you want you might see Simon's GMT book or other sources on minimal surfaces.)


There are many related applications of the monotonicity formula. A simple yet powerful one consists of White's proof of the Allard regularity theorem. See Section 1.1 here <https://annals.math.princeton.edu/wp-content/uploads/annals-v161-n3-p07.pdf> or Theorem 7.8 here [http://web.stanford.edu/~ochodosh/MinSurfNotes.pdf](http://web.stanford.edu/%7Eochodosh/MinSurfNotes.pdf)).


