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Consider an embedded surface in $\mathbb R^3$ homeomorphic to a disk. If we wanted to find a similar surface that is minimal, i.e. has mean curvature $H=0$ everywhere, we could hold the boundary fixed and perform gradient flow under the obvious energy function $E_H = \frac12\int H^2\,\mathrm dA$. This is known as Willmore flow, and its evolution equation is $$\dot{\mathbf x} = (\Delta H + 2H(H^2-K))\mathbf n,$$ where $K$ is the Gaussian curvature. A minimal surface is clearly a global minimum of $E_H$ (though I don't know if there are any other local minima for a disk topology with fixed boundary).

Suppose we want a developable surface instead, i.e. $K=0$ everywhere. Will the gradient flow of $E_K = \frac12\int K^2\,\mathrm dA$ do the job? Has this kind of surface flow been studied before, and what is its evolution equation?

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    1. It's not clear that every space curve that bounds a disk lies in some flat surface and bounds a disk in that surface, so offhand I'd guess that in $\mathbf{R}^{3}$, the flow (i.e., implementing the flow via a family of embeddings) runs into geometric problems. 2. The flow makes intrinsic sense, and $E_{K}$ is the analogue of the Calabi energy, i.e., it would be the (one-dimensional) Calabi energy on a surface with no boundary and with fixed area (i.e., a choice of Kähler class). You might search for "heat flow" and "extremal Kähler metrics". – 2017-01-01
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    Thanks for the suggestion! I will spend some time looking into the Calabi energy and Kähler metrics (Google turns up a paper by Simanca that's literally titled "[Heat Flows for Extremal Kähler Metrics](https://arxiv.org/abs/math/0310363)", is that what you were thinking of?), though I only know "naive differential geometry" i.e. of curves and surfaces in $\mathbb R^{\le3}$, so I don't know if I'll be able to make much sense of this stuff. – 2017-01-02

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