The family of level sets of a harmonic function
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ORIGINAL RESEARCH PAPER
The family of level sets of a harmonic function Pisheng Ding1 Received: 15 July 2019 / Accepted: 13 December 2019 Ó Forum D’Analystes, Chennai 2019
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. Keywords Harmonic function Level set Curvature
Mathematics Subject Classification Primary 31A05 31B05 Secondary 53A04 53A05 53A07
1 Introduction Harmonic functions on domains in Euclidean spaces 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 critical-point-free harmonic functions 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. & Pisheng Ding [email protected] 1
Illinois State University, Normal, IL, USA
123
P. Ding
In this Introduction, we state our main result in the two-variable case for simplicity. The general case will be treated in Sect. 3. Theorem 1 Let U : R ð; Þ ! R2 be an orientation-preserving C 2 diffeomorphism onto a domain X R2 ; let ð xðr; tÞ; yðr; tÞÞ ¼ Uðr; tÞ. For each t 2 ð; Þ, define ct to be the curve r7!Uðr; tÞ. Let det dU u¼ c0 t
and
N¼
ðoy=orÞe1 þ ð ox=orÞe2 ; c0 t
let s be the arc-length parameter (modulo an additive constant) along each integral curve of N. For p 2 X, let jðpÞ be the signed curvature at p of the curve ct on which p lies (with j signed in accordance with the normal field N). Then, there exists a critical-point-free harmonic function U on X with fct j t 2 ð; Þg being its levelcurve family iff ðou=osÞ þ ju is constant on each curve ct , i.e., o ou þ ju 0: ð1Þ or os
A few words on notation are in order. The quantity ðou=osÞ þ ju can be construed as a function on both X and R ð; Þ, via the mapping U between the two domains. Strictly speaking, for o=or to be meaningful, we should interpret ðou=osÞ þ ju as a function on R ð; Þ. Thus, for q 2 R ð; Þ, ðjuÞðqÞ means jðUðqÞÞuðqÞ, whereas ðou=osÞðqÞ is interpreted as follows: if a : ðd; dÞ ! X is the (unit-speed) inte
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