## Inner-variational equations

It’s been a while since the last time I posted a post-colloquium post. This one is based on a colloquium given by me, but don’t worry: none of my results are included here.

As a warm-up, consider the (trivial) problem of finding the shortest path between two points $a,b\in\mathbb R^n$. The naive approach is to minimize the length $L(f)=\int_0^1 |f\,'(t)|\,dt$ among all maps $f\colon [0,1]\to \mathbb R^n$ that are sufficiently smooth and satisfy the boundary conditions $f(0)=a$ and $f(1)=b$. This turns out to be a bad idea: $L(f)$ is neither strictly convex nor differentiable, and the set of minimizing maps is huge, containing some rather nonsmooth specimen.

The right approach is to minimize the energy $E(f)=\int_0^1 |f\,'(t)|^2\,dt$. While this functional is not immediately related to length for general maps, it is not hard to see that for minimizing maps $f$ we have $E(f)=L(f)^2$. Indeed, consider performing the inner variation $\widetilde{f}(t)=f(t+\epsilon \eta(t))$ where $\eta\colon [0,1]\to\mathbb R$ is smooth and vanishes at the endpoints. Expanding the inequality $E(\widetilde{f})\ge E(f)$, we arrive at $\int_0^1 |f\,'(t)|^2\eta'(t)\,dt=0$, which after integration by parts yields $\frac{d}{dt}|f\,'(t)|^2=0$. Thus, minimization of energy enforces constant-speed parametrization, and since for constant-speed maps we have $E(f)=L(f)^2$, the geometric nature of the variational problem has not been lost.

As a side remark, the inner-variational equation $\frac{d}{dt}|f\,'(t)|^2=0$ could be written as $f\,''\cdot f\,'=0$, which is a nonlinear second-order equation.

For comparison, try the first variation $\widetilde{f}(t)=f(t)+\epsilon \eta(t)$ where $\eta\colon [0,1]\to\mathbb R^n$ is smooth and vanishes at the endpoints. Expanding the inequality $E(\widetilde{f})\ge E(f)$, we arrive at $\int_0^1 f\,'(t)\cdot \eta'(t)\,dt=0$, which after integration by parts yields $f\,''\equiv 0$, a linear second-order equation. This Euler-Lagrange equation immediately tells us what the minimizing map $f$ is: there is only one linear map with the given boundary values. Obviously, $f\,''\equiv 0$ is a much stronger statement that $f\,''\cdot f\,'=0$. However, if $f+\epsilon \eta$ is not admissible for some geometric reason (e.g., the curve must avoid an obstacle), the Euler-Lagrange equation may not be available.

Moving one dimension up, consider the problem of parameterizing a given simply-connected domain $\Omega\subset \mathbb C$. Now we are to minimize the energy of diffeomorphisms $f\colon \mathbb D\to\Omega$ which is defined, as before, to be the sum of squares of derivatives. (Here $\mathbb D$ is the unit disk.) In complex notation, $E(f)=\iint_{\mathbb D}(|f_z|^2+|f_{\bar z}|^2)$. For definiteness assume $f$ is sense-preserving, that is $|f_z|\ge |f_{\bar z}|$. Minimizers ought to be conformal maps onto $\Omega$, but how can we see this from variational equations?

The Euler-Lagrange equation that we get from $E(f)\le E(f+\epsilon\eta)$ turns out to be the Laplace equation $\Delta f=0$. This is much weaker than the Cauchy-Riemann equation $f_{\bar z}=0$ that we expect. One problem is that $\eta$ must vanish on the boundary: otherwise $f+\epsilon\eta$ will violate the geometric constraint. We could try to move the values of $f$ in the direction tangent to $\partial\Omega$, but since does not necessarily make sense since the boundary of $\Omega$ could be something like the von Koch snowflake. And of course, it is not at all clear why the minimum of $E$ must be attained by a diffeomorphism. If the class of maps is expanded to include suitable limits of diffeomorphisms, then it’s no longer clear (actually, not true) that $f+\epsilon\eta$ belongs to the same class. All things considered, the approach via the first variation does not appear promising.

Let’s try the inner variation instead. For small $\epsilon$ the map $z\mapsto z+\epsilon\eta$ is a diffeomorphism of $\mathbb D$, hence its composition with $f$ is as good a candidate as $f$ itself. Furthermore, since the inner variation deals with the model domain $\mathbb D$ and not with the generic domain $\Omega$, it is easy to allow modification of boundary values: $\eta$ should be tangent to $\partial \mathbb D$, i.e., $\mathrm{Re}\,\bar z\eta(z)$ should vanish on the boundary. It takes a bit of computation to turn $E(f(z+\epsilon \eta(z)))$ into something manageable, but the final outcome is remarkably simple. The inner-variational equation says that the function $\varphi:=f_z\overline{f_{\bar z}}$ is holomorphic in $\mathbb D$ and $z^2 \varphi$ is real on the boundary $\partial \mathbb D$. (Technically speaking, $\varphi\, dz^2$ must be a real holomorphic quadratic differential.) What can we conclude from this? To begin with, the maximum principle implies that $z^2 \varphi$ is a constant function. And since it vanishes at $z=0$, the inevitable conclusion is $\varphi\equiv 0$. Recalling the sense-preserving constraint $|f_z|\ge |f_{\bar z}|$, we arrive at $f_{\bar z}\equiv 0$, the desired Cauchy-Riemann equation.

Executive summary

• Suppose we are to minimize some quantity (called “energy”) that depends on function (or a map) $f$
• We can consider applying the first variation $f+ \epsilon\eta$ of the inner variation $f\circ (\mathrm{id}+\epsilon \eta)$. Here $\eta$ is a small perturbation which is applied differently: in the first case it changes the values of $f$, in the second it shuffles them around.
• Inner variation applies even in the presence of geometric constraints that make first variation illegal. One example of such constraint is “ $f$ must be injective”.
• Inner-variational equations are quite different from the Euler-Lagrange equations. Even for simple quadratic functionals they are nonlinear.
• Inner-variational equations are useful because they tell us something about the maps of minimal energy.