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Below is a rather longwinded description of the special case when the singularity is at worst an isolated point. I suspect you know all this already. The magic comes at the very end (see paragraph that starts with "Here's the critical trick"). I don't know if this is the same thing that gives Schoen-Uhlenbeck the extra oomph or not.

There are three applications I know of: Minimal hypersurfaces, self-dual Yang-Mills connections, and Einstein manifolds. The regularity theory described below is used for both a convergence theorem except possibly a finite number of points and a removable singularity theorem. These theorems are then used to establish the so-called bubbling phenomenon. The story below applies to the latter two applications; the details for minimal hypersurfaces are slightly different.

Assume for convenience that we're on a smooth $n$-dimensional complete Riemannian manifold, where $n > 2$. Denote the Laplacian on both functions and tensors by $\Delta = g^{ij}\nabla_i\nabla_j$.

Denote the $L_p$ norm of a function or tensor $u$ with respect to the Riemannian metric by $\|u\|_p$.

Throughout the discussion below we will restrict to a geodesic ball $B(x, r)$ and assume that the following Sobolev inequality holds for a fixed constant $C_S$ and any smooth function $u$ compactly supported in $B$: $$\|\nabla u\|_2 \ge C_S\|u\|_{2n/(n-2)}.$$

First, you consider the scalar elliptic inequality $-\Delta u \le bu$, where $b$ can be viewed as a given potential function. Using Moser iteration, you show that if

$$\|b\|_{q/2}, \|u\|_p < C,$$

where $q > n$, for some $p > 1$ on $B(x,r)$, then there is a bound on $\|u\|_\infty$ on, say, $B(x,r/2)$.

Second, you use Moser iteration to show that if $\|b\|_{n/2}$ is sufficiently small (depends on $C_S$) on $B(x,r)$, then there is a bound for $\|u\|_{q/2}$ for some $q > n$ on $B(x,r/2)$.

Combining the first two shows that if $u$ satisfies $-\Delta u \le cu^2$ and $\|u\|_ {n/2}$ is sufficiently small on $B(x,r)$, then there is a bound on $\|u\|_\infty$ on $B(x,r/2)$.

In each application there is a curvature tensor $F$ that satisfies a PDE of the form $$-\Delta F = Q(F),$$ where $Q$ depends quadratically on $F$. Moreover, there is a convergence theorem when there is a uniform pointwise bound on $F$ (for Einstein manifolds you use the Cheeger-Gromov convergence theorem). Applying the results above to $u = |F|$ using coverings with smaller and smaller balls leads to a convergence theorem when there is a uniform bound on $\|F\|_ {n/2}$ where the convergence can fail at only a finite number of points (where in the limit there is too much of $\|F\|_{n/2}$ for the estimates above to hold).

Now you want to study the limit object near each point singularity. If you keep close track of the dependence on $r$ in the estimates above, the best you can do is a bound on $F$ that blows up like $r^{-2}$, where $r$ is the distance to the singularity. This is not enough to remove the singularity, so you need to use more than the elliptic PDE above.

Here's the critical trick: When doing Moser iteration on $u = |F|$, you use the standard Cauchy-Schwarz inequality to obtain the following pointwise inequality: $$|F\cdot\nabla F| \le |F||\nabla F|$$ But in all of the applications, you have extra information about $F$ and its covariant derivative. In particular, $F$ and/or its covariant derivative have certain symmetries, which allow you to prove a pointwise bound of the form $$|F\cdot\nabla F| \le c|F||\nabla F|,$$ where $c < 1$. This improvement when used with Moser iteration allows you to show that $F$ blows up more slowly than $r^{-2}$. Iterating this improvement leads to a uniform pointwise bound on $F$, which in turn allows the singularity to be removed using a straightforward geometric ODE argument.

The removable singularity theorem allows you to analyze both the limiting object with the bubbles removed as well as the bubbles themselves.

ADDED: I can't resist adding an anecdote to this: Right after I learned the trick in the paragraph above from a paper of Schoen-Simon-Yau, I went to a colleague's office to show it to him. As it happens, Eli Stein was there, and he exclaimed, "But it's in my book!" And indeed it is. You will find it presented very nicely in VII.3.1 "A subharmonic property of the gradient" of Stein's 1970 book, "Singular Integrals and Differentiability Properties of Functions". It is obvious that S-S-Y did not know this or forgot, because their proof is much messier than Stein's.

Below is a rather longwinded description of the special case when the singularity is at worst an isolated point. I suspect you know all this already. The magic comes at the very end (see paragraph that starts with "Here's the critical trick"). I don't know if this is the same thing that gives Schoen-Uhlenbeck the extra oomph or not.

There are three applications I know of: Minimal hypersurfaces, self-dual Yang-Mills connections, and Einstein manifolds. The regularity theory described below is used for both a convergence theorem except possibly a finite number of points and a removable singularity theorem. These theorems are then used to establish the so-called bubbling phenomenon. The story below applies to the latter two applications; the details for minimal hypersurfaces are slightly different.

Assume for convenience that we're on a smooth $n$-dimensional complete Riemannian manifold, where $n > 2$. Denote the Laplacian on both functions and tensors by $\Delta = g^{ij}\nabla_i\nabla_j$.

Denote the $L_p$ norm of a function or tensor $u$ with respect to the Riemannian metric by $\|u\|_p$.

Throughout the discussion below we will restrict to a geodesic ball $B(x, r)$ and assume that the following Sobolev inequality holds for a fixed constant $C_S$ and any smooth function $u$ compactly supported in $B$: $$\|\nabla u\|_2 \ge C_S\|u\|_{2n/(n-2)}.$$

First, you consider the scalar elliptic inequality $-\Delta u \le bu$, where $b$ can be viewed as a given potential function. Using Moser iteration, you show that if

$$\|b\|_{q/2}, \|u\|_p < C,$$

where $q > n$, for some $p > 1$ on $B(x,r)$, then there is a bound on $\|u\|_\infty$ on, say, $B(x,r/2)$.

Second, you use Moser iteration to show that if $\|b\|_{n/2}$ is sufficiently small (depends on $C_S$) on $B(x,r)$, then there is a bound for $\|u\|_{q/2}$ for some $q > n$ on $B(x,r/2)$.

Combining the first two shows that if $u$ satisfies $-\Delta u \le cu^2$ and $\|u\|_ {n/2}$ is sufficiently small on $B(x,r)$, then there is a bound on $\|u\|_\infty$ on $B(x,r/2)$.

In each application there is a curvature tensor $F$ that satisfies a PDE of the form $$-\Delta F = Q(F),$$ where $Q$ depends quadratically on $F$. Moreover, there is a convergence theorem when there is a uniform pointwise bound on $F$ (for Einstein manifolds you use the Cheeger-Gromov convergence theorem). Applying the results above to $u = |F|$ using coverings with smaller and smaller balls leads to a convergence theorem when there is a uniform bound on $\|F\|_ {n/2}$ where the convergence can fail at only a finite number of points (where in the limit there is too much of $\|F\|_{n/2}$ for the estimates above to hold).

Now you want to study the limit object near each point singularity. If you keep close track of the dependence on $r$ in the estimates above, the best you can do is a bound on $F$ that blows up like $r^{-2}$, where $r$ is the distance to the singularity. This is not enough to remove the singularity, so you need to use more than the elliptic PDE above.

Here's the critical trick: When doing Moser iteration on $u = |F|$, you use the standard Cauchy-Schwarz inequality to obtain the following pointwise inequality: $$|F\cdot\nabla F| \le |F||\nabla F|$$ But in all of the applications, you have extra information about $F$ and its covariant derivative. In particular, $F$ and/or its covariant derivative have certain symmetries, which allow you to prove a pointwise bound of the form $$|F\cdot\nabla F| \le c|F||\nabla F|,$$ where $c < 1$. This improvement when used with Moser iteration allows you to show that $F$ blows up more slowly than $r^{-2}$. Iterating this improvement leads to a uniform pointwise bound on $F$, which in turn allows the singularity to be removed using a straightforward geometric ODE argument.

The removable singularity theorem allows you to analyze both the limiting object with the bubbles removed as well as the bubbles themselves.

3 added 2 characters in body

Below is a rather longwinded description of the special case when the singularity is at worst an isolated point. I suspect you know all this already. The magic comes at the very end (see paragraph that starts with "Here's the critical trick"). I don't know if this is the same thing that gives Schoen-Uhlenbeck the extra oomph or not.

There are three applications I know of: Minimal hypersurfaces, self-dual Yang-Mills connections, and Einstein manifolds. The regularity theory described below is used for both a convergence theorem except possibly a finite number of points and a removable singularity theorem. These theorems are then used to establish the so-called bubbling phenomenon. The story below applies to the latter two applications; the details for minimal hypersurfaces are slightly different.

Assume for convenience that we're on a smooth $n$-dimensional complete Riemannian manifold, where $n > 2$. Denote the Laplacian on both functions and tensors by $\Delta = g^{ij}\nabla_i\nabla_j$.

Denote the $L_p$ norm of a function or tensor $u$ with respect to the Riemannian metric by $\|u\|_p$.

Throughout the discussion below we will restrict to a geodesic ball $B(x, r)$ and assume that the following Sobolev inequality holds for a fixed constant $C_S$ and any smooth function $u$ compactly supported in $B$: $$\|\nabla u\|_2 \ge C_S\|u\|_{2n/(n-2)}.$$

First, you consider the scalar elliptic inequality $-\Delta u \le bu$, where $b$ can be viewed as a given potential function. Using Moser iteration, you show that if

$$\|b\|_{q/2}, \|u\|_p < C$$

where $q > n$ for some $p > 1$ on $B(x,r)$, then there is a bound on $\|u\|_\infty$ on, say, $B(x,r/2)$.

Second, you use Moser iteration to show that if $\|b\|_{n/2}$ is sufficiently small (depends on $C_S$) on $B(x,r)$, then there is a bound for $\|u\|_{q/2}$ for some $q > n$ on $B(x,r/2)$.

Combining the first two shows that if $u$ satisfies $-\Delta u \le cu^2$ and $\|u\||u\|_ {n/2}$ is sufficiently small on $B(x,r)$, then there is a bound on $\|u\|\infty |u\|_\infty$ on $B(x,r/2)$.

In each application there is a curvature tensor $F$ that satisfies a PDE of the form $$-\Delta F = Q(F),$$ where $Q$ depends quadratically on $F$. Moreover, there is a convergence theorem when there is a uniform pointwise bound on $F$ (for Einstein manifolds you use the Cheeger-Gromov convergence theorem). Applying the results above to $u = |F|$ using coverings with smaller and smaller balls leads to a convergence theorem when there is a uniform bound on $\|F\||F\|_ {n/2}$ where the convergence can fail at only a finite number of points (where in the limit there is too much of $\|F\|{n/2} |F\|_{n/2}$ for the estimates above to hold).

Now you want to study the limit object near each point singularity. If you keep close track of the dependence on $r$ in the estimates above, the best you can do is a bound on $F$ that blows up like $r^{-2}$, where $r$ is the distance to the singularity. This is not enough to remove the singularity, so you need to use more than the elliptic PDE above.

Here's the critical trick: When doing Moser iteration on $u = |F|$, you use the standard Cauchy-Schwarz inequality to obtain the following pointwise inequality: $$|F\cdot\nabla F| \le |F||\nabla F|$$ But in all of the applications, you have extra information about $F$ and its covariant derivative. In particular, $F$ and/or its covariant derivative have certain symmetries, which allow you to prove a pointwise bound of the form $$|F\cdot\nabla F| \le c|F||\nabla F|,$$ where $c < 1$. This improvement when used with Moser iteration allows you to show that $F$ blows up more slowly than $r^{-2}$. Iterating this improvement leads to a uniform pointwise bound on $F$, which in turn allows the singularity to be removed using a straightforward geometric ODE argument.

The removable singularity theorem allows you to analyze both the limiting object with the bubbles removed as well as the bubbles themselves.

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