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.

  • $\begingroup$ 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? $\endgroup$
    – Leo Moos
    Oct 22, 2021 at 10:23
  • $\begingroup$ @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. $\endgroup$ Oct 22, 2021 at 11:12

1 Answer 1


There is a lot of current study of the level sets of harmonic functions in this exact way.

See these papers: https://arxiv.org/abs/1209.4669 https://arxiv.org/abs/2108.08402 and https://arxiv.org/abs/1911.06754 .

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 .

  • $\begingroup$ Great! Thank you for the references, I will take a closer look. $\endgroup$ Oct 23, 2021 at 2:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.