# Geometric flow by the level sets of a harmonic function

Let $$u$$ be an harmonic function in a cylindrical domain $$B_2^{n-1}\times(-1,1)\subset\mathbb{R}^n$$, and suppose its level sets $$\Gamma_t=\{u=t\}$$ are graphs of functions on $$B_2^{n-1}$$. Consider a linear parametrization of $$u$$: $$u_t:=u-t.$$ Then the nodal set of $$u_t$$ is $$t$$-level set of $$u$$: $$\{u_t=0\}=\{u=t\}.$$ We can regard this $$u_t$$ to be the level set function of evolving surface $$\Gamma_t=\{u=t\}$$. If we denote $$v$$ as the vector field indicating the speed of the flow of $$\Gamma_t$$ follows, then by level set equation $$|v|=\frac{1}{|\nabla u|}.$$ Since we know that $$v$$ is normal to $$\Gamma_t$$, $$v$$ is parallel to $$\nabla u$$, hence $$v=\frac{\nabla u}{|\nabla u|^2}.$$ On the other hand, since $$u$$ is constant on its level surfaces $$\Gamma_t$$, its Laplace-Beltrami operators are identically zero: $$0=\Delta_{\Gamma_t}u=\Delta u+H_{\Gamma_t}\partial_{\nu}u-\partial_{\nu\nu}u=H_{\Gamma_t}\nabla u-\partial_{\nu\nu}u.$$ Therefore with the above expression, we obtain $$v=\frac{|H_{\Gamma_t}|}{|\partial_{\nu\nu}u|}\nu,$$ where $$\nu:=\nabla u/|\nabla u|$$ is a unit normal vector to $$\Gamma_t$$.

So we have that, the surface $$\Gamma_t$$ follows some geometric flow somehow related to mean curvature flow (if $$|\partial_{\nu\nu} u|=1$$ then it is a mean curvature flow). My questions are:

1. Is there any name or related category of this flow?
2. I am interested in the $$C^{2,\alpha}$$ regularity of $$\Gamma_t$$, if $$\Gamma_t$$ was a graph of $$C^{2,\alpha}$$ function, then will $$\Gamma_s$$ be $$C^{2,\alpha}$$ in some neighborhood of $$t$$? If so, can we control $$C^{2,\alpha}$$ norm of $$\Gamma_s$$ by $$\Gamma_t$$?

Just to note, different formulation of the curvature equation above ($$0=H_{\Gamma_t}\nabla u-\partial_{\nu\nu}u$$) is that $$|\nabla u|\Delta_1 u=\Delta_\infty u,$$ where $$\Delta_p$$ is $$p$$-Laplacian. It is interesting for me that two operators with duality have some relation. Unfortunately I don't have much knowledge in this direction ($$p$$-laplacian) so I could not have any meaningful result.

• Although studying the MCF via level sets has a rich history, the approach chosen there---as far as I know---basically goes in the opposite direction of what you're doing here. I'm not an expert in these questions, but I'm a bit skeptical about the point of view you've chosen. For example, as $\partial_{\nu \nu} u \neq 0$ is needed to define your equation, wouldn't you get the answer to the second question from the implicit function theorem anyway? Oct 22, 2021 at 10:23
• @LeoMoos As you can see, if $\partial_{\nu\nu}u=0$ then at the point $H_{\Gamma_t}=0$ as well (here let's assume $\nabla u\neq 0$). In that case I can simply write $v=\nabla u/|\nabla u|^2$. Obtaining $C^{2,\alpha}$ regularity (or more) is easy, but I am interested in getting $C^{2,\alpha}$ estimation of $\Gamma_s$ in terms of $\Gamma_t$, e.g. $\|f_s\|_{C^{2,\alpha}}\lesssim \|f_t\|_{C^{2,\alpha}}$. Thank you for the comment. Oct 22, 2021 at 11:12

Some may not look directly related to what you're asking, but note that anytime you are using the co-area formula, it's like integrating $$\frac{d}{dt} \int_{u=t} F$$ with respect to $$t$$, so it ends up being related to what you're asking.
In terms of the $$p$$-Laplacian, there is some relationship between your flow and inverse mean curvature flow. See https://arxiv.org/abs/1812.05022 .