PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 102(116) (2017), 85–91 DOI: https://doi.org/10.2298/PIM1716085S

BI-LIPSCHITZITY OF QUASICONFORMAL HARMONIC MAPPINGS IN n-DIMENSIONAL SPACE WITH RESPECT TO k-METRIC

Shadia Shalandi

Abstract. We explore conditions which guarantee bi-Lipschitzity of harmonic quasiconformal maps with respect to k-metric. We prove that harmonic k-quasiconformal maps with nonzero Jacobian between any two domains in ℝ^{n} are bi-Lipschitz with respect to k-metric, and prove the converse too.

1. Introduction

We prove results about bi-Lipschitzity of harmonic k-qc mappings f: D_{1} → D_{2}, where D_{1} and D_{2} are arbitrary proper subdomains of ℝ^{n}, with respect to k-metric.

Similar problems have been studied at the Belgrade Seminar for Complex Analysis. In [1], Mateljević proved such a result in n-dimensional space, but only in the case when both D_{1} and D_{2} are the upper half space in ℝ^{n}. Also, in the same paper, Proposition 5 gives an estimate in dimension 2 for minimal and maximal moduli of directional derivative at a point, in terms of distance to the boundary, for arbitrary codomain. As a corollary, he proved that every harmonic quasiconformal map of the unit disk is a quasi-isometry with respect to hyperbolic distances. He posed a question if analogue of Proposition 5 holds in higher dimensions. In the case n = 2, Manojlović proved in [2] that, when D_{1} and D_{2} are arbitrary domains in the plane, then harmonic quasiconformal maps are bi-Lipschitz with respect to k-metric.

Note that the Lipischitz condition for maps between domains in ℝ^{n} was obtained by Mateljević and Vourinen [6]. Here a different proof, based on results of Božin and Mateljević [3], is given.

Let B^{n}(x, r) = {z ∈ ℝ^{n} : ∥z − x∥ < r}, 𝕊^{n−1} (x, r) = ∂B^{n}(x, r), and let B^{n}, 𝕊^{n−1} stand for the unit ball and the unit sphere in ℝ^{n}, respectively. For a domain G ⊂ ℝ^{n} let ρ: G → (0, ∞) be a continuous function. We say that ρ is a metric density if, for every locally rectifiable curve γ in G, the integral

2010 Mathematics Subject Classification: 30C65; 42B37.

Key words and phrases: harmonic maps, quasi-conformal maps, k-metric, bi-Lipschitz maps.

Communicated by Miodrag Mateljević.

l_{ρ}(γ) = ∫_{γ} ρ(x)ds,

exists. In this case we call l_{ρ}(γ) the ρ-length of γ. A metric density d_{ρ}: G × G → [0, ∞) defines a metric as follows. For a, b ∈ G, let d_{ρ}(a, b) = inf_{γ} l_{ρ}(γ), where the infimum is taken over all locally rectifiable curves in G joining a and b. For a fixed a, b ∈ G, suppose that there exists a d_{ρ}-length minimizing curve γ: [0, 1] → G with γ(0) = a, γ(1) = b such that d_{ρ}(a, b) = l_{ρ}(γ|[0, t]) + l_{ρ}(γ|[t, 1]), for all t ∈ [0, 1]. Then γ is called a geodesic segment joining a and b.

In dimensions n ⩾ 3, we do not have a Riemann mapping theorem, and it is natural to look for counterparts of the hyperbolic metric. So-called hyperbolic type metrics have been the subject of many recent papers. One of the most important of these metrics is the quasihyperbolic metric k_{G} of a domain G ⊂ ℝ. The quasihyperbolic k-metric k = k_{G} of G is a particular case of the geodesic metric d_{ρ} when ρ(x) = 1/d(x, ∂G) [4, 5], where d(x, ∂G) is the distance from point x to the boundary of G.

We will consider Euclidean harmonic maps, also called harmonic maps in this paper, i.e., those with zero Laplacian of each coordinate function. Also, we will deal with quasiconformal maps. For a domain D in ℝ^{n}, a map f: D → ℝ^{n} is K-quasiconformal if it is a homeomorphism of D to f(D), and if f belongs to the Sobolev space W_{1,loc}^{n} (D) and there exists K, 1 ⩽ K < ∞, such that ∥Df(x)∥^{n} ⩽ KJ_{f} (x) a.e. on D, where ∥Df(x)∥ denote the operator norm of the Jacobian matrix of f at x.

Our main result is that harmonic k-quasiconformal mappings which do not have zero of Jacobian f: D_{1} → D_{2} are bi-Lipschitz. This result is based on two Theorems from [3]. We also prove that every harmonic mappings f: D_{1} → D_{2} which is bi-Lipschitz with respect to k-metric is quasiconformal, where D_{1} and D_{2} are domains in ℝ^{n}.

2. Background

In this section we give some background results which will be used in our main proofs.

Theorem 2.1. [6] Let D_{1} and D_{2} be two domains in R^{n} and let ρ_{1} and ρ_{2} be two densities, ds = ρ_{1}(z)|dz|, and ds = ρ_{2}(w)|dw| where |dz|, and |dw| stand for Euclidean metric, and Λ_{f} (z), λ_{f} (z) are respectively the maximum and the minimum of modulus of eigenvalues of the Jacobian matrix at z, and suppose that f: D_{1} → D_{2} is a C^{1} quasiconformal mapping

(A) If there is a positive constant c_{1} such that at every point z, ρ_{2}(f(z))Λ_{f} (z)⩽

c_{1}ρ_{1}(z), z ∈ D_{1}, then d_{ρ_{2}} (f(z_{1}), f(z_{2})) ⩽ c_{1}d_{ρ_{1}} (z_{1}, z_{2}). (B) If f(D_{1}) = D_{2}, and there is a positive constant c_{2} such that at every point z, λ_{f} (z)ρ_{2}(f(z)) ⩾ c_{2}ρ_{1}(z), z ∈ D_{1}, then d_{ρ_{2}} (f(z_{1}), f(z_{2})) ⩾ c_{2}d_{ρ_{1}} (z_{1}, z_{2}), z_{1}, z_{2} ∈ D_{1}.

For convenience, we give a proof of this known result.

Proof. Part (A). Suppose that γ is geodesic with parametrization

γ(t) = (γ_{1}(t), γ_{2}(t), γ_{3}(t), …, γ_{n}(t)),

and derivative

γ′(t) = (γ′_{1}(t), γ′_{2}(t), γ′_{3}(t), …, γ′_{n}(t)).

Let γ_{∗}(t) = f(γ(t)); then γ′_{∗}(t) = D_{f} (γ(t))γ′(t), and ∥γ′_{∗}(t)∥ ⩽ Λ_{f} ∥γ′(t))∥. We have

d_{ρ_{1}} (z_{1}, z_{2}) = inf_{γ} ∫_{γ} ρ_{1}(z)|dz| ⩽ ∫_{0}^{1} ρ_{1}(γ(t))∥γ′(t)∥dt.

Letting w_{1} = f(z_{1}) and w_{2} = f(z_{2}), we can write

d_{ρ_{2}} (w_{1}, w_{2}) = inf_{γ_{∗}} ∫_{γ_{∗}} ρ_{2}(w)|dw| ⩽ ∫_{0}^{1} ρ_{2}(γ_{∗}(t))∥γ′_{∗}(t))∥dt.

Using change of variable,

d_{ρ_{2}} (w_{1}, w_{2}) ⩽ ∫_{0}^{1} ρ_{2}(f(γ(t)))Λ_{f} ∥γ′(t))∥dt,

and by (A), we get

d_{ρ_{2}} (w_{1}, w_{2}) ⩽ c_{1} ∫_{0}^{1} ρ_{1}(γ(t))||γ′(t)||dt ⩽ c_{1} ∫_{γ} ρ_{1}(z)|dz| ⩽ c_{1}d_{ρ_{1}} (z_{1}, z_{2}).

Then d_{ρ_{2}} (f(z_{1}), f(z_{2})) ⩽ c_{1}d_{ρ_{1}} (z_{1}, z_{2}).

Part (B). Let g be an inverse function of f. We have f(z_{1}) = w_{1} → z_{1} = g(w_{1}) and f(z_{2}) = w_{2} → z_{2} = g(w_{2}).

Let γ(t) = g(γ_{∗}(t)); then γ′(t) = D_{g}(γ_{∗}(t))γ′_{∗}(t), and thus ∥γ′(t)∥ ⩽ Λ_{g}∥γ_{∗}(t)∥. Here Λ_{g} = \frac{1}{λ_{f}} , because D_{g}(w) = [D_{f} (z)]^{−1} . It follows that

d_{ρ_{1}} (g(w_{1}), g(w_{2})) = inf_{γ} ∫_{γ} ρ_{1}(z)|dz| ⩽ ∫_{0}^{1} ρ_{1}(γ(t))∥D_{g}(γ_{∗}(t))γ_{∗}(t))∥dt.

By assumption in (B), we get

(2.1) d_{ρ_{1}} (g(w_{1}), g(w_{2}))⩽ ∫_{0}^{1}ρ_{1}(g(γ_{∗}(t)))\frac{1}{λ_{f}} ∥γ′_{∗}(t)∥dt ⩽ \frac{1}{c_{2}} ∫_{0}^{1}ρ_{2}(γ_{∗}(t))∥γ′_{∗}(t)∥dt

⩽ \frac{1}{c_{2}} ∫_{γ_{∗}} ρ_{2}(w)|dw| ⩽ \frac{1}{c_{2}} d_{ρ_{2}} (w_{1}, w_{2}).

Then c_{2}d_{ρ_{1}} (z_{1}, z_{2}) ⩽ d_{ρ_{2}} (f(z_{1}), f(z_{2})). □

In the following two theorems from [3, Theorems 4.1 and 4.2], nonzero Jacobian families are defined as closed families of harmonic maps with nonzero Jacobians (see [3]).

Theorem 2.2. For every nonzero Jacobian closed family of k-quasiconformal harmonic maps, there is a constant c > 0, such that if f: B^{n} → ℝ^{n} is from the family, d(0, ∂f(B^{n})) ⩾ 1, and f(0) = 0, then J_{f} (0) ⩾ c.

Theorem 2.3. There is a constant c > 0, depending only on k, such that if f: D_{1} → ℝ^{n} is k-quasiconformal harmonic map, d(0, ∂f(B^{n})) ⩽ 1, and f(0) = 0, then J_{f} (0) ⩽ c.

We will also need the following well known theorem for qc maps, called local quasi-symmetry (see, for instance, [4]).

Theorem 2.4. If f: B^{n} → ℝ^{n} is a K-quasiconformal map and f̂: B̄^{n} → ℝ^{n}

its continuous extension, then for any two points a, b ∈ 𝕊^{n−1}

\frac{d(f̂(0), f̂(a))}{d(f̂(0), f̂(b))} ⩽ c(k, n),

for some constant c(K, n) independent of f.

3. Bi-Lipschitzity with respect to k-metric

Theorem 3.1. Suppose that f: D_{1} → D_{2}, where D_{1}, D_{2} ⊊ ℝ^{n}, is a harmonic quasi-conformal mapping, and that f belongs to a nonzero Jacobian family of harmonic maps, then the following holds for some constant C

\frac{1}{C}J^{\frac{1}{n}} (z) ⩽ \frac{d(f(z), ∂D_{2})}{d(z, ∂D_{1})} ⩽ CJ^{\frac{1}{n}} (z).

Proof. Let z_{0} be a point in D_{1}, r_{1} = d(z_{0}, ∂D_{1}), r_{2} = d(f(z_{0}), ∂D_{2}).

Let B(z_{0}, r_{1}) be the n dimensional ball centered at z_{0} of radius r_{1} and let D_{3} = f(B(z_{0}, r_{1})). Also assume that f is K-quasiconformal.

Define f̂: B^{n} → ℝ^{n} by f̂(z) = \frac{1}{r_{2}} (f(z_{0}+r_{1}z)−f(z_{0})). Note that since D_{3} ⊆ D_{2}, we have r_{2} = d(f(z_{0}), ∂D_{2}) ⩾ d(f(z_{0}), ∂D_{3}), and hence d(0, ∂f̂(B^{n})) ⩽ 1. We have

J_{f} (z_{0}) = det [ \begin{matrix}∂f_{1}/∂x_{1} & … & ∂f_{1}/∂x_{n} \\ ∂f_{2}/∂x_{1} & … & ∂f_{2}/∂x_{n} \\ ⋮ & ... & ⋮ \\ ∂f_{n}/∂x_{1} & … & ∂f_{n}/∂x_{n} \end{matrix} ]

Jf̂(0) = ( \frac{1}{r_{2}} )^{n} det [ \begin{matrix}r_{1}∂f_{1}/∂x_{1} & … & r_{1}∂f_{1}/∂x_{n} \\ r_{1}∂f_{2}/∂x_{1} & … & r_{1}∂f_{2}/∂x_{n} \\ ⋮ & ... & ⋮ \\ r_{1}∂f_{n}/∂x_{1} & … & r_{1}∂f_{n}/∂x_{n} \end{matrix} ] = ( \frac{r_{1}}{r_{2}} )^{n}J_{f} (z_{0}).

Since, d(0, ∂f̂(B^{n})) ⩽ 1, by Theorem 2.3 Jf̂(0) ⩽ c, so

\frac{r_{1}^{n}}{r_{2}^{n}} J_{f} (z_{0}) ⩽ c

\frac{1}{c_{1}} J_{f} (z_{0})^{\frac{1}{n}} ⩽ \frac{r_{2}}{r_{1}} = \frac{d(f(z), ∂D_{2})}{d(z, ∂D_{1})}

where c_{1} = c^{1/n} .

Note that, by Theorem 2.4, for any point w ∈ ∂D_{3},

d(f(z_{0}), ∂D_{3}) ⩾ \frac{1}{c(K, n)} d(f(z_{0}), w).

So, since by our construction there is a point w which belongs to both ∂D_{2} and ∂D_{3}, we have

d(f(z_{0}), ∂D_{3}) ⩾ \frac{1}{c(K, n)} d(f(z_{0}), ∂D_{2})

and so we have

d(f(z_{0}), ∂D_{3}) ⩾ \frac{r_{2}}{c(K, n)} .

Now again, define f̂: B^{n} → ℝ^{n} by f̂(z) = \frac{c(K,n)}{r_{2}} (f(z_{0} + r_{1}z) − f(z_{0})) for

z ∈ B^{n}. Note that d(0, ∂f̂(B^{n})) ⩾ 1. We have

J_{f} (z_{0}) = det [ \begin{matrix}∂f_{1}/∂x_{1} & … & ∂f_{1}/∂x_{n} \\ ∂f_{2}/∂x_{1} & … & ∂f_{2}/∂x_{n} \\ ⋮ & ... & ⋮ \\ ∂f_{n}/∂x_{1} & … & ∂f_{n}/∂x_{n} \end{matrix} ]

Jf̂(0) = ( \frac{c(K, n)}{r_{2}} )^{n} = det [ \begin{matrix}r_{1}∂f_{1}/∂x_{1} & … & r_{1}∂f_{1}/∂x_{n} \\ r_{1}∂f_{2}/∂x_{1} & … & r_{1}∂f_{2}/∂x_{n} \\ ⋮ & ... & ⋮ \\ r_{1}∂f_{n}/∂x_{1} & … & r_{1}∂f_{n}/∂x_{n} \end{matrix} ]

Jf̂(0) = c(K, n)^{n}( \frac{r_{1}}{r_{2}} )^{n}J_{f} (z_{0}).

By Theorem 2.2, since d(0, ∂f̂(B^{n})) ⩾ 1, we have Jf̂(0) ⩾ c, so

\frac{r_{1}^{n}}{r_{2}^{n}} J_{f} (z_{0}) ⩾ \frac{c}{c(K, n)^{n}} .

Then

c_{2}J_{f} (z_{0})^{\frac{1}{n}} ⩾ \frac{r_{2}}{r_{1}} = \frac{d(f(z_{0}), ∂D_{2})}{d(z_{0}, ∂D_{1})}

where c_{2} = \frac{c(K,n)}{c^{1/n}} . Finally, set C = max(c_{1}, c_{2}). □

A consequence of Theorem 3.1 is the following:

Theorem 3.2. Suppose that f: D_{1} → D_{2}, where D_{1}, D_{2} ⊊ ℝ^{n}, is a harmonic K-quasiconformal mapping, and that f belongs to a nonzero Jacobian family of harmonic maps. Then f is bi-Lipschitz with respect to k-metric.

Proof. From the quasiconformality condition and using that our map is C^{1}, we have a constant k such that at every point z

Λ_{f} (z) ⩽ kJ^{\frac{1}{n}} (z), λ_{f} (z) ⩾ \frac{1}{k}J^{\frac{1}{n}} (z)

where Λ_{f} and λ_{f} are the greatest and smallest moduli of eigenvalues of the Jacobian matrix. By Theorem 3.1, there is a constant C such that

\frac{1}{C}J^{\frac{1}{n}} (z) ⩽ \frac{d(f(z), ∂D_{2})}{d(z, ∂D_{1})} ⩽ CJ^{\frac{1}{n}} (z).

The metric densities for k metrics are

ρ_{1}(z) = \frac{1}{d(z, ∂D_{1})}, ρ_{2}(z) = \frac{1}{d(w, ∂D_{2})},

and so we have

ρ_{2}(f(z))Λ_{f} (z) = \frac{1}{d(f(z), ∂D_{2})} Λ_{f} (z) ⩽ \frac{1}{d(f(z), ∂D_{2})} kJ^{\frac{1}{n}} (z)

⩽ \frac{1}{d(f(z), ∂D_{2})} kC \frac{d(f(z), ∂D_{2})}{d(z, ∂D_{1})} = kC \frac{1}{d(z, ∂D_{1})} = kCρ_{1}(z),

ρ_{2}(f(z))λ_{f} (z) = \frac{1}{d(f(z), ∂D_{2})} λ_{f} (z) ⩾ \frac{1}{d(f(z), ∂D_{2})} \frac{1}{k}J^{\frac{1}{n}} (z)

⩾ \frac{1}{d(f(z), ∂D_{2})} \frac{1}{kC} \frac{d(f(z), ∂D_{2})}{d(z, ∂D_{1})} = \frac{1}{kC} \frac{1}{d(z, ∂D_{1})} = \frac{1}{kC}ρ_{1}(z).

Then by Theorem 2.1 the map f is bi-Lipschitz with respect to k-metric. □

Theorem 3.3. If a bijective harmonic map f: D_{1} → D_{2}, where D_{1}, D_{2} ⊂ ℝ^{n},

is bi-Lipschitz with respect to k-metric, then it is a quasiconformal mapping.

Proof. Note that, by elliptic regularity, f is a C^{1} map. Let x, x + Δx be two points in D_{1} where ∥Δx∥ → 0, and suppose that the Jacobian matrix D_{f} (x) maps unit sphere to ellipsoid with minimal and maximal axes equal to λ_{f} and Λ_{f} respectively, and let ρ_{1}, and ρ_{2} be metric density functions in D_{1} and D_{2} respectively. Assume that

\frac{1}{c}d_{ρ_{2}} (f(x), f(y)) ⩽ d_{ρ_{1}} (x, y) ⩽ cd_{ρ_{2}} (f(x), f(y)).

We prove that \frac{Λ_{f}}{λ_{f}} ⩽ c^{2}, wherefrom quasiconformality follows. As Δx → 0, we have

d_{ρ_{1}} (x, x + Δx) = ρ_{1}(x)∥Δx∥(1 + o(1)),
d_{ρ_{2}} (f(x), f(x + Δx)) = ρ_{2}(x)∥DfΔx∥(1 + o(1)).

Note that

Λ_{f} = sup_{e,∥e∥=1} ∥D_{f} (x)e∥ and λ_{f} = inf_{e,∥e∥=1} ∥D_{f} (x)e∥.

Suppose supremum is achieved for vector e_{1}, and infimum is achieved for e_{2} (since matrix multiplication is continuous, and unit sphere is compact, there have to be such vectors e_{1} and e_{2}).

We are going to consider Δx = te_{1}, t → 0 and Δx = te_{2}, t → 0. Putting Δx = te_{1}, t → 0 we have

d_{ρ_{1}} (x, x + te_{1}) = ρ_{1}(x)t(1 + o(1)) as t → 0
d_{ρ_{2}} (f(x), f(x + te_{1})) = ρ_{2}(f(x))Λ_{f} t(1 + o(1)) as t → 0.

Putting Δx = te_{2}, t → 0, we have

d_{ρ_{1}} (x, x + te_{2}) = ρ_{1}(x)t(1 + o(1)) as t → 0
d_{ρ_{2}} (f(x), f(x + te_{2})) = ρ_{2}(f(x))λ_{f} t(1 + o(1)) as t → 0.

Using the bi-Lipschitz condition, we get

\frac{1}{c}ρ_{2}(f(x))tλ_{f} (1 + o(1)) ⩽ ρ_{1}(x)t(1 + o(1)) ⩽ cρ_{2}(f(x))tλ_{f} (1 + o(1)),

\frac{1}{c}ρ_{2}(f(x))tΛ_{f} (1 + o(1)) ⩽ ρ_{1}(x)t(1 + o(1)) ⩽ cρ_{2}(f(x))tΛ_{f} (1 + o(1)).

Letting t tend to zero and dividing by t, we get

\frac{1}{c}ρ_{2}(f(x))λ_{f} ⩽ ρ_{1}(x) ⩽ cρ_{2}(f(x))λ_{f} ,
\frac{1}{c}ρ_{2}(f(x))Λ_{f} ⩽ ρ_{1}(x) ⩽ cρ_{2}(f(x))Λ_{f} .

So \frac{1}{c}ρ_{2}(f(x))Λ_{f} ⩽ cρ_{2}(f(x))λ_{f} , wherefrom \frac{Λ_{f}}{λ_{f}} ⩽ c^{2}. □

Note that the proof of previous theorem assumes only that ρ_{1} and ρ_{2} are positive continous and that f is C^{1}. So in fact we have proved

Theorem 3.4. Suppose that ρ_{1}, ρ_{2} are positive continuous metric densities defined in ℝ^{n} domains D_{1} and D_{2} respectively, and f: D_{1} → D_{2} is C^{1} bijection which is bi-Lipschitz with respect metrics d_{ρ_{1}} and d_{ρ_{2}} . Then f is a quasiconformal mapping.

Acknowledgments. I wish to thank my advisor, Vladimir Božin, and Miša Arsenović for suggestions and discussion regarding this problem. I also thank the referee for pointing out the paper [7], that appeared while this paper was under review.

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Department of Mathematics (Received 29 07 2016) University of Belgrade (Revised 10 05 2017 and 21 06 2017) Belgrade, Serbia

    Shado.shadia@yahoo.com