Characterizing Level-set Families of Harmonic Functions

TL;DR

This paper characterizes families of level sets of harmonic functions without critical points using differential geometry and explores how the gradient evolution relates to the mean curvature of these level sets.

Contribution

It provides a local geometric condition for level-set families of harmonic functions and constructs harmonic functions from geometric data, linking gradient flow to mean curvature.

Findings

Level-set families of harmonic functions are characterized by a differential-geometric condition.

Harmonic functions can be constructed from given geometric data.

Gradient evolution along flow is governed by the mean curvature of level sets.

Abstract

Families of hypersurfaces that are level-set families of harmonic functions free of critical points are characterized by a local differential-geometric condition. Harmonic functions with a specified level-set family are constructed from geometric data. As a by-product, it is shown that the evolution of the gradient of a harmonic function along the gradient flow is determined by the mean curvature of the level sets that the flow intersects.

Equations

$$\phi =\frac{\det d\Phi }{\Vert {\gamma }_t^{\prime }\Vert }\text{and}\mathbf{N}=\frac{\left(-\partial y/\partial \sigma \right){\mathbf{e}}_1+\left(\partial x/\partial \sigma \right){\mathbf{e}}_2}{\Vert {\gamma }_t^{\prime }\Vert }\text{ ;}$$

$$\frac{\partial }{\partial \sigma }\left(\frac{\partial \phi }{\partial s}+\kappa \phi \right)\equiv 0\text{ .}$$

$$\frac{\partial \phi }{\partial s}(q)=(\phi \circ {\Phi }^{-1}\circ \alpha {)}^{\prime }(0)={\frac{d}{ds}|}_{s=0}\phi ({\Phi }^{-1}(\alpha (s)))\text{ .}$$

$$\Phi :\mathbb{R}\times (-ϵ,ϵ)\stackrel{\pi \times \mathrm{Id}}{⟶}{S}^1\times (-ϵ,ϵ)\stackrel{\Psi }{\to }{\mathbb{R}}^2\text{ ,}$$

$${\Phi }_i:\mathbb{R}\times (-ϵ,ϵ)\stackrel{{\varphi }_i\times \mathrm{Id}}{⟶}{\mathcal{O}}_i\times (-ϵ,ϵ)\hookrightarrow S^1\times (-ϵ,ϵ)\stackrel{\Psi }{\to }{\mathbb{R}}^2\text{ .}$$

$$\Vert \nabla U(\alpha (s))\Vert =\Vert \nabla U(\alpha (0))\Vert \exp \left((n-1){\int }_0^{s}H(\alpha (\xi ))d\xi \right)\text{ ,}$$

$$\kappa (p)=-\frac{D_{\mathbf{T}}^{2}f(p)}{D_{\mathbf{N}}f(p)}=\frac{D_{\mathbf{N}}^{2}f(p)-\Delta f(p)}{D_{\mathbf{N}}f(p)}\text{ .}$$

$$\kappa \equiv \frac{D_{\mathbf{N}}^{2}f}{D_{\mathbf{N}}f}\text{on }\Omega \text{.}$$

$$⟨\frac{d}{dt}\nabla f(\gamma (t)),{\gamma }^{\prime }(t)⟩+⟨\nabla f(\gamma (t)),{\gamma }^{\prime \prime }(t)⟩=0\text{ .}$$

$$⟨{\frac{d}{dt}|}_{t=0}\nabla f(\gamma (t),{\gamma }^{\prime }(0)⟩=D_{\mathbf{T}}^{2}f(p)\text{ ,}$$

$$⟨\nabla f(\gamma (0)),{\gamma }^{\prime \prime }(0)⟩=\kappa (p)\Vert \nabla f(p)\Vert \text{ .}$$

$$D_{\mathbf{T}}^{2}f(p)+D_{\mathbf{N}}^{2}f(p)=\mathrm{Tr}{Q}_p=\Delta f(p)\text{ .}$$

$$\phi =\frac{\det d\Phi }{\Vert {\gamma }_t^{\prime }\Vert }\text{and}\mathbf{N}=\frac{\left(-\partial y/\partial \sigma \right){\mathbf{e}}_1+\left(\partial x/\partial \sigma \right){\mathbf{e}}_2}{\Vert {\gamma }_t^{\prime }\Vert }\text{ .}$$

$$\frac{\partial }{\partial \sigma }\left(\frac{\partial \phi }{\partial s}+\kappa \phi \right)\equiv 0\text{ ,}$$

$$\nabla t=\frac{1}{\phi }\mathbf{N}\text{ .}$$

$$\nabla U=u^{\prime }\cdot \nabla t=\frac{u^{\prime }}{\phi }\mathbf{N}\text{ .}$$

$$D_{\mathbf{N}}U=\frac{\partial }{\partial s}(u\circ t)=u^{\prime }(t)\frac{\partial t}{\partial s}=u^{\prime }(t)\Vert \nabla t\Vert =\frac{u^{\prime }(t)}{\phi }$$

$$D_{\mathbf{N}}^{2}U=\frac{{\partial }^2}{\partial s^2}(u\circ t)=\frac{\partial }{\partial s}\frac{u^{\prime }(t)}{\phi }=\frac{u^{\prime \prime }(t)\Vert \nabla t\Vert \phi -u^{\prime }(t)\frac{\partial \phi }{\partial s}}{{\phi }^2}=\frac{u^{\prime \prime }(t)-u^{\prime }(t)\frac{\partial \phi }{\partial s}}{{\phi }^2}\text{ .}$$

$$\frac{\partial \phi }{\partial s}+\kappa \phi =\frac{u^{\prime \prime }(t)}{u^{\prime }(t)}\text{ .}$$

$$\frac{d}{dt}\log u^{\prime }(t)=\frac{\partial \phi }{\partial s}+\kappa \phi \text{ .}$$

$$\mathbf{n}=(-1)^{n-1}\det\left[\begin{array}[c]{cccc}\mathbf{e}_{1}&\frac{\partial y_{1}}{\partial x_{1}}&\cdots&\frac{\partial y_{1}}{\partial x_{n-1}}\\ \mathbf{e}_{2}&\frac{\partial y_{2}}{\partial x_{1}}&\cdots&\frac{\partial y_{2}}{\partial x_{n-1}}\\ \vdots&\vdots&\vdots&\vdots\\ \mathbf{e}_{n}&\frac{\partial y_{n}}{\partial x_{1}}&\cdots&\frac{\partial y_{n}}{\partial x_{n-1}}\end{array}\right]\text{ .}$$

$$\phi =\frac{\det d\Phi }{\Vert \mathbf{n}\Vert }\text{and}\mathbf{N}=\frac{\mathbf{n}}{\Vert \mathbf{n}\Vert }\text{.}$$

$$\frac{\partial }{\partial x_i}\left(\frac{\partial \phi }{\partial s}+(n-1)H\phi \right)=0\text{for }i\in \{1,2,\cdots ,(n-1)\}\text{.}$$

$${\Phi }_i:{\mathbb{R}}^{n-1}\times (-ϵ,ϵ)\stackrel{{\varphi }_i\times \mathrm{Id}}{⟶}{\mathcal{O}}_i\times (-ϵ,ϵ)\hookrightarrow M^{n-1}\times (-ϵ,ϵ)\stackrel{\Psi }{\to }{\mathbb{R}}^n\text{ .}$$

$$H(p)=\frac{1}{n-1}\frac{D_{\mathbf{N}}^{2}f(p)-\Delta f\left(p\right)}{D_{\mathbf{N}}f(p)}\text{ .}$$

$$H\equiv \frac{1}{n-1}\frac{D_{\mathbf{N}}^{2}f}{D_{\mathbf{N}}f}\text{on }\Omega \text{.}$$

$$\frac{\partial \phi }{\partial s}+(n-1)H\phi =\frac{u^{\prime \prime }(t)}{u^{\prime }(t)}\text{ .}$$

$$\alpha (0)=p_0\text{and}{\alpha }^{\prime }(s)=\mathbf{N}(\alpha (s))\text{ .}$$

$$g^{\prime }(s)=D_{\mathbf{N}}U(\alpha (s))=\Vert \nabla U(\alpha (s))\Vert \text{and}{g}^{\prime \prime }(s)=D_{\mathbf{N}}^{2}U(\alpha (s))\text{ .}$$

$$(n-1)H(\alpha (s))=\frac{g^{\prime \prime }(s)}{g^{\prime }(s)}=\frac{d}{ds}\log g^{\prime }(s)=\frac{d}{ds}\log \Vert \nabla U(\alpha (s))\Vert \text{ .}$$

Full text

Characterizing Level-Set Families of Harmonic Functions111MSC2010: Primary 31A05, 31B05; Secondary 53A04, 53A05, 53A07. Key words: harmonic function, level set, curvature.

Pisheng Ding

Abstract. Families of hypersurfaces that are level-set families of harmonic functions free of critical points are characterized by a local differential-geometric condition. Harmonic functions with a specified level-set family are constructed from geometric data. As a by-product, it is shown that the evolution of the gradient of a harmonic function along the gradient flow is determined by the mean curvature of the level sets that the flow intersects.

1 Introduction

Harmonic functions on $\mathbb{R}^{n}$ are those whose Laplacian vanishes identically. In this note, by analyzing certain differential-geometric properties of their level sets, we give a local characterization of their level-set families.

For harmonic functions of two variables, it is already quite difficult to characterize their level curves; see, e.g., [1]. Level hypersurfaces of harmonic functions of more than two variables are even more intractable (especially without the complex-analytic tools available in the two-variable case). This note shows that families of level-sets of harmonic functions free of critical points are somehow easier to characterize. The difference between an individual curve or hypersurface and a family of them is that the former is “static” whereas the latter contains “kinematic” information that is more readily relatable to harmonicity.

In this Introduction, we state our main result in the two-variable case for simplicity. The general case will be treated in §3.

Theorem 1

Let $\Phi:\mathbb{R}\times(-\epsilon,\epsilon)\rightarrow\mathbb{R}^{2}$ be an orientation-preserving $C^{2}$ diffeomorphism onto a domain $\Omega\subset\mathbb{R}^{2}$; let $\left(x(\sigma;t),\,y(\sigma;t)\right)=\Phi(\sigma;t)$. For each $t\in(-\epsilon,\epsilon)$, define $\gamma_{t}$ to be the curve $\sigma\mapsto\Phi(\sigma;t)$. Let

[TABLE]

let $s$ be the arc-length parameter (modulo an additive constant) along each integral curve of $\mathbf{N}$. For $p\in\Omega$, let $\kappa(p)$ be the signed curvature at $p$ of the curve $\gamma_{t}$ on which $p$ lies (with $\kappa$ signed in accordance with the normal field $\mathbf{N}$). Then, there exists a critical-point-free harmonic function $U$ on $\Omega$ with $\left\{\gamma_{t}\mid t\in(-\epsilon,\epsilon)\right\}$ being its level-curve family iff $(\partial\varphi/\partial s)+\kappa\varphi$ is constant on each curve $\gamma_{t}$, i.e.,

[TABLE]

A few words on notation are in order. The quantity $(\partial\varphi/\partial s)+\kappa\varphi$ can be construed as a function on both $\Omega$ and $\mathbb{R}\times(-\epsilon,\epsilon)$, via the mapping $\Phi$ between the two domains. Strictly speaking, for $\partial/\partial\sigma$ to be meaningful, we should interpret $(\partial\varphi/\partial s)+\kappa\varphi$ as a function on $\mathbb{R}\times(-\epsilon,\epsilon)$. Thus, for $q\in\mathbb{R}\times(-\epsilon,\epsilon)$, $(\kappa\varphi)(q)$ means $\kappa(\Phi(q))\varphi(q)$, whereas $(\partial\varphi/\partial s)(q)$ is interpreted as follows: if $\alpha:\left(-\delta,\delta\right)\rightarrow\Omega$ is the (unit-speed) integral curve of $\mathbf{N}$ with $\alpha(0)=\Phi(q)$, then

[TABLE]

Remark. To a $C^{2}$ family of simple closed curves $\Psi:S^{1}\times(-\epsilon,\epsilon)\rightarrow\mathbb{R}^{2}$, Theorem 1 can be applied in either of the following two ways.

First, we may use the covering map $\pi:\mathbb{R}\rightarrow S^{1}$, $\sigma\mapsto(\cos\sigma,\sin\sigma)$, to define

[TABLE]

which is a local diffeomorphism. With this $\Phi$, Theorem 1 applies.

In the other approach, which would prove suitable for the more general case treated in §3, we cover $S^{1}$ with an atlas of two local charts $\phi_{i}:\mathbb{R}\rightarrow\mathcal{O}_{i}\hookrightarrow S^{1}$ and then define

[TABLE]

Theorem 1 then implies that the given family of simple closed curves is the level-curve family of a harmonic function iff both $\Phi_{i}$ meet Condition (1).

Theorem 1 will be proved in §2.2, following an exposition of some preliminaries in §2.1. In §3, we generalize this result to the many-variable case, wherein the mean curvature $H$ of level hypersurfaces figures in place of $\kappa$ in a condition parallel to (1) that characterizes level-set families of harmonic functions. When the condition is met by a given family, we construct in §4 a harmonic function $U$ whose level-set family is the given one.

Related to the theme of this note is another fundamental observation concerning how the geometry of the level sets of a harmonic function determine its gradient. We now state this result.

Theorem 2

Let $U$ be a critical-point-free harmonic function on a domain $\Omega\subset\mathbb{R}^{n}$. Let $\alpha:\left(-\delta,\delta\right)\rightarrow\Omega$ be an integral curve of the vector field $\nabla U/\left\|\nabla U\right\|$. Then, for $s\in\left(-\delta,\delta\right)$,

[TABLE]

where, for any $p\in\Omega$, $H(p)$ is the mean curvature at $p$ of the level-$U(p)$ hypersurface oriented by the normal field $\nabla U/\left\|\nabla U\right\|$.

Note that $\alpha$ by definition has unit speed and therefore is the arc-length parametrization of a gradient flow.

This theorem will be proved in §3.

2 The Two-variable Case

We base our proof for Theorem 1 on a characterization of harmonicity of a $C^{2}$ function in terms of curvature of its level sets and its directional derivatives, which we first turn to.

2.1 Characterizing Harmonicity in Terms of Curvature of Level Curves

Let $f$ be a $C^{2}$ function on a domain $\Omega\subset\mathbb{R}^{2}$ with no critical points. For $p\in\Omega$ and a unit vector $\mathbf{v}\in T_{p}\mathbb{R}^{2}$, denote by $D_{\mathbf{v}}f(p)$ and $D_{\mathbf{v}}^{2}f(p)$ the first and second directional derivatives of $f$ at $p$ along $\mathbf{v}$. Denote by $Q_{p}:T_{p}\mathbb{R}^{2}\times T_{p}\mathbb{R}^{2}\rightarrow\mathbb{R}$ the Hessian quadratic form of $f$ at $p$; recall that $D_{\mathbf{v}}^{2}f(p)=Q_{p}(\mathbf{v},\mathbf{v})$.

Let $C$ be a level curve of $f$. Install on $C$ the unit normal field $\mathbf{N}:=\nabla f/\left\|\nabla f\right\|$; for the unit tangent field $\mathbf{T}$, let it be such that the frame $(\mathbf{T},\mathbf{N})$ is positively-oriented (but the opposite choice will do as well). The signed curvature $\kappa$ of the curve $C$ at each point thereon is defined by the equation $d\mathbf{T}/d\sigma=\kappa\mathbf{N}$, where $\sigma$ is the arc-length parameter along $C$ (with its increasing direction induced by $\mathbf{T}$); note that the sign of $\kappa$ depends on the choice we make of $\mathbf{N}$, but not of $\mathbf{T}$. So defined, $\kappa$ is a scalar field on $\Omega$.

Lemma 3

Assume the preceding hypothesis. For $p\in\Omega$,

[TABLE]

Consequently, $f$ is harmonic iff

[TABLE]

Proof. Let $\gamma$ be the unit-speed parametrization of an arc on the level-$f(p)$ curve, with $\gamma(0)=p$ and $\gamma^{\prime}(0)=\mathbf{T}(p)$. By definition, $\gamma^{\prime\prime}(0)=\kappa(p)\mathbf{N}(p)$. For all $t$, $\left\langle\nabla f(\gamma(t)),\,\gamma^{\prime}(t)\right\rangle=0$; hence

[TABLE]

Note that

[TABLE]

whereas

[TABLE]

Now (2) follows by noting that $D_{\mathbf{N}(p)}f(p)=\left\|\nabla f(p)\right\|$ and that

[TABLE]

We are now ready to establish Theorem 1.

2.2 Proof of Theorem 1

Let $J$ denote the interval $(-\epsilon,\epsilon)$. Recall from §1 that $\Phi:\mathbb{R}\times J\rightarrow\mathbb{R}^{2}$, with $\Phi(\sigma;t)=\left(x(\sigma;t),\,y(\sigma;t)\right)$, is an orientation-preserving $C^{2}$ diffeomorphism onto a domain $\Omega$, that $\gamma_{t}$ is the curve $\sigma\mapsto\Phi(\sigma,t)$, and that

[TABLE]

(Note that $\varphi>0$.) Theorem 1 asserts that there exists a harmonic function $U$ on $\Omega$ with $\left\{\gamma_{t}\right\}_{t\in J}$ being its level-curve family iff

[TABLE]

where $s$ is the arc-length parameter along each integral curve of $\mathbf{N}$ and $\kappa$ is the signed curvature of $\gamma_{t}$ (signed in accordance with the normal field $\mathbf{N}$).

We now set out to prove Theorem 1.

Let $t:\Omega\rightarrow J$ denote the second component of $\Phi^{-1}$. (For $p\in\Omega$, $t(p)$ is characterized by the condition that $p$ is on the curve $\gamma_{t(p)}$.) By the inverse function theorem,

[TABLE]

Any function having the $\gamma_{t}$’s as its level curves evidently depends solely on $t$. Therefore, consider a function $U:\Omega\rightarrow\mathbb{R}$ of the form $u\circ t$ where $u:J\rightarrow\mathbb{R}$ is a $C^{2}$ function free of critical points. Then,

[TABLE]

Clearly, $U$ is free of critical point on $\Omega$. Note that $u^{\prime}$ on $J$ is either strictly positive or strictly negative (by Darboux’s theorem). Without loss of generality, assume that $u^{\prime}>0$, in which case $\nabla U/\left\|\nabla U\right\|=\mathbf{N}$ and the curvature of level curves of $U$ as defined in §2.1 has the same sign as the curvature $\kappa$ on $\Omega$ introduced in §1.

Now that $U=u\circ t$ is a critical-point-free $C^{2}$ function on $\Omega$ with $\left\{\gamma_{t}\right\}_{t\in J}$ being its level-curve family, we seek a necessary and sufficient condition for $U$ to be harmonic.

With $s$ denoting the arc-length parameter along the integral curves of $\mathbf{N}$ (which are also gradient flows of $t$), we compute $D_{\mathbf{N}}U$ and $D_{\mathbf{N}}^{2}U$. First,

[TABLE]

(or more simply: $D_{\mathbf{N}}U=\left\langle\nabla U,\mathbf{N}\right\rangle=u^{\prime}(t)/\varphi$ ); also

[TABLE]

Hence, the condition that $D_{\mathbf{N}}^{2}U=\kappa D_{\mathbf{N}}U$ is equivalent to the condition that $\left(u^{\prime\prime}(t)-u^{\prime}(t)\frac{\partial\varphi}{\partial s}\right)/\varphi^{2}=\kappa u^{\prime}(t)/\varphi$, i.e., the condition that

[TABLE]

If $U$ is harmonic, then, by Lemma 3, $D_{\mathbf{N}}^{2}U=\kappa D_{\mathbf{N}}U$ and hence (6) holds. In equation (6), the right side of the equality is constant on each curve $\gamma_{t}$ and hence independent of the parameter $\sigma$ that parametrizes $\gamma_{t}$ ; so must the left side! This proves the necessity of Condition (5) for $U$ to be harmonic.

Conversely, if Condition (5) holds, then, $\left(\partial\varphi/\partial s\right)+\kappa\varphi$ depends only on $t$. Viewing $\left(\partial\varphi/\partial s\right)+\kappa\varphi$ as a function of $t$, we can find a function $u$ of $t$ such that

[TABLE]

For any such $u$, (6) holds and hence the composite function $U:=u\circ t$ satisfies the condition $D_{\mathbf{N}}^{2}U=\kappa D_{\mathbf{N}}U$ on $\Omega$, which, by Lemma 3, implies that $U$ is harmonic.

3 Extension to Higher Dimensions

Turning to the $n$-variable case, we give an immediate extension of Theorem 1. We begin with a family of hypersurfaces in $\mathbb{R}^{n}$ and seek a condition that is necessary and sufficient for the existence of a harmonic function whose level-set family is the given one. For simplicity, suppose that the hypersurfaces in the family are all diffeomorphic to $\mathbb{R}^{n-1}$. As we shall remark following Theorem 4, this is not a severe restriction.

Let $J$ denote the interval $(-\epsilon,\epsilon)$. Suppose that $\Phi:\mathbb{R}^{n-1}\times J\rightarrow\mathbb{R}^{n}$, with $\Phi(x;t)=\left(y_{1}(x;t),\,y_{2}(x;t),\cdots,y_{n}(x;t)\right)$, is an orientation-preserving $C^{2}$ diffeomorphism onto a domain $\Omega$ in $\mathbb{R}^{n}$. For each $t\in J$, define $\Gamma_{t}:\mathbb{R}^{n-1}\rightarrow\mathbb{R}^{n}$ to be the parametrized hypersurface $x\mapsto\Phi(x;t)$. Let $\mathbf{n}$ be the Hodge dual of $(\partial\Phi/\partial x_{1})\wedge(\partial\Phi/\partial x_{2})\wedge\cdots\wedge(\partial\Phi/\partial x_{n-1})$, i.e.,

[TABLE]

Let

[TABLE]

Orient each $\Gamma_{t}$ by the normal field $\mathbf{N}$. For $p\in\Omega$, let $H(p)$ be the mean curvature at $p$ of the oriented hypersurface $\Gamma_{t}$ on which $p$ lies. Let $s$ denote the arc-length parameter (modulo an additive constant) along the integral curves of $\mathbf{N}$.

Theorem 4

Assume the preceding hypotheses and notations. There exists a critical-point-free harmonic function $U$ whose family of level sets is $\left\{\Gamma_{t}\right\}$ iff $(\partial\varphi/\partial s)+(n-1)H\varphi$ is constant on each hypersurface $\Gamma_{t}$, i.e.,

[TABLE]

Remark. To a $C^{2}$ family of hypersurfaces $\Psi:M^{n-1}\times(-\epsilon,\epsilon)\rightarrow\mathbb{R}^{n}$, where $M^{n-1}$ is a connected $C^{2}$ $(n-1)$-manifold, Theorem 4 can be applied as follows.

Take a $C^{2}$ atlas of local charts $\phi_{i}:\mathbb{R}^{n-1}\rightarrow\mathcal{O}_{i}\hookrightarrow M^{n-1}$. Define

[TABLE]

Theorem 4 then implies that the given family of hypersurfaces is the level-set family of a critical-point-free harmonic function iff $\Phi_{i}$ meet Condition (7) for all $i$.

All that is needed for the proof is a characterization of harmonicity that extends Lemma 3. We briefly recall the mean curvature of a one-codimensional $C^{2}$ orientable submanifold $M$ of $\mathbb{R}^{n}$. Let $M$ be oriented by a unit normal field $\mathbf{N}$. At a point $p\in M$, any unit tangent vector $\mathbf{v}\in T_{p}M$ and the unit normal $\mathbf{N}(p)$ span a plane $\Pi_{\mathbf{v}}$; the intersection curve $M\cap\Pi_{\mathbf{v}}$ is a so-called normal section at $p$ and has signed curvature $\kappa_{p}(\mathbf{v})=-D_{\mathbf{v}}^{2}f(p)/\left\|\nabla f(p)\right\|$, as in (2). The mean curvature $H(p)$ of $M$ at $p$ is simply the average of $\kappa_{p}(\mathbf{v})$ as $\mathbf{v}$ ranges over the unit $(n-2)$-sphere in the $(n-1)$-dimensional $T_{p}M$. The following result, like Lemma 3, plays a key role in establishing Theorem 4.

Lemma 5

For a $C^{2}$ real-valued function $f$ on an open subset $\Omega$ of $\mathbb{R}^{n}$ without critical points, let $\mathbf{N}=\nabla f/\left\|\nabla f\right\|$ and let $H(p)$ be the mean curvature at $p\in\Omega$ of the level-$f(p)$ hypersurface of $f$. Then,

[TABLE]

Consequently, $f$ is harmonic iff

[TABLE]

We omit the simple proof of (8). The proof of Theorem 4 parallels that of Theorem 1; it suffices to note that equation (6), key to Theorem 1, now takes the form

[TABLE]

From Lemma 5, we deduce Theorem 2 stated in §1.

Proof of Theorem 2. Let $U$ be a critical-point-free harmonic function on a domain $\Omega\subset\mathbb{R}^{n}$. Let $\mathbf{N}=\nabla U/\left\|\nabla U\right\|$. Let $s\mapsto\alpha(s)$ be the unit-speed gradient flow originating from $p_{0}$; $\alpha$ is such that

[TABLE]

Consider $g:=U\circ\alpha$. It is elementary that

[TABLE]

By (9),

[TABLE]

Integrating both sides yields that

[TABLE]

proving Theorem 2.    $\blacksquare$

Remark. When $n=2$ in Theorem 2, the mean curvature $H$ of level hypersurfaces is replaced by the signed curvature $\kappa$ of level curves, resulting in

[TABLE]

This formula is suggested in [2] by a purely complex-analytic argument, as a harmonic function on a planar domain is locally the real part of a complex analytic function. However, that argument does not apply when $n>2$.

4 Constructing a Harmonic Function from Its Family of Level Sets

Suppose that the hypersurface family $\left\{\Gamma_{t}\right\}$ defined in Theorem 4 satisfies (7) and therefore is the level-set family of a harmonic function $U$. We seek to determine $U$ from $\left\{\Gamma_{t}\right\}$. As we shall see, the final outcome will provide a link between Theorems 4 and 2, the two main results of this note.

As in §2.2, for $y\in\Omega$, suppose that $U(y)=u(t(y))$ with $u^{\prime}>0$ on $(-\epsilon,\epsilon)$. Recall equation (10), which is equivalent to $U$ being harmonic:

[TABLE]

Integrating both sides yields

[TABLE]

where $x(\tau)\in\mathbb{R}^{n-1}$ is so chosen that the curve

[TABLE]

is a parametrization of the integral curve of $\mathbf{N}$ originating at $\Phi(\mathbf{0};0)$. (Due to (7), the expression $(\partial\varphi/\partial s)+(n-1)H\varphi$ is constant in $x$; therefore, for its evaluation at $\left(x;\tau\right)$, we are free to choose $x$. Our particular choice facilitates further evaluation of the integral in (11).) As $\nabla t=(1/\varphi)\mathbf{N}$, the curve $\ell$ is also a parametrization of the gradient flow of $t$ originating at $\Phi(\mathbf{0};0)$; with $s$ being the arc-length parameter along $\ell$, $dt/ds=\left\|\nabla t\right\|=1/\varphi$. Thus,

[TABLE]

which allows us to conclude from (11) that

[TABLE]

Integrating, we have

[TABLE]

Substituting $ds$ for $\varphi d\tau$ and $\varphi dt$ in the two integrals, the above formula can written in terms of line integrals along $\ell$ with respect to arc length $s$:

[TABLE]

where

[TABLE]

Remark. Formula (12), derived from Theorem 4, provides a satisfying and corroborating link between Theorems 4 and 2. By Theorem 2,

[TABLE]

with this, Formula (12) then reads

[TABLE]

which is a statement of the fundamental theorem of calculus.