A PARTIAL REGULARITY RESULT FOR HARMONIC MAPS INTO A FINSLER MANIFOLD

ATSUSHI TACHIKAWA

Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba, 278-8510 Japan

               E-mail:tachikawa atsushi@ma.noda.tus.ac.jp

Let N be a n-dimensional C^{∞} manifold and T N the tangent bundle of N. We express each point in T N as (u, X) with u ∈ N and X ∈ T_{x}N. Moreover, 0 denotes the 0−section {(u, 0)} ⊂ T N when we write T N \0. A Finsler structure of N is a function F : T N → [0, ∞) with the following properties:

(i) Regularity: F ∈ C^{∞}(T N \ 0).

(ii) Homogeneity: F (u, λX) = λF (u, X) for all λ ≥ 0.

(iii) Convexity: The Hessian matrix of F^{2} with respect to X

(h_{ij}(u, X)) = (\frac{1}{2} \frac{∂^{2}F^{2}(u, X)}{∂X^{i}∂X^{j}})

is positive definite at every point (u, X) ∈ T N \ 0.

We call the pair (N, F) a Finsler manifold.

According to Centore [1] the energy density of maps between Finsler manifold is defined as follows.

Definition. Let (M, G) and (N, F) be Finsler manifolds, and I_{x}M a unit ball in T_{x}M, namely

I_{x}M = {ξ ∈ T_{x}M; G(X) ≤ 1}.

For a C^{1}-map u : M → N, we define the energy density e(u)(x) of u at x ∈ M by

e(u)(x) = \frac{∫_{I_{x}M}(u^{∗}F)^{2}(X)dX}{∫_{I_{x}M} dX} .  (0.1)

The above definition is consistent with the standard one for Riemannian cases ([1, Lemma 2]) and it can be regarded as a special case of the definition given by Jost [2] for abstract cases of metric measure spaces ([1, Theorem 5]).

In this talk, we consider harmonic maps from ℝ^{m} into Finsler space (N, F) = (ℝ^{n}, F). For such a case, by virtue of the special structure of ℝ^{m} and the homogeneity of F, we can write the energy density defined by (0.1) more simply.

Namely, for a map u : ℝ^{m} → (N, F) we can define the energy density of u at x ∈ ℝ^{m} as

e(u)(x) = \frac{1}{2c_{m}} ∫_{S^{m−1}} F^{2}(u(x), du_{x} · ξ)dξ,  (0.2)

where c_{m} denotes the area of the (m − 1)-dimensional standard sphere S^{m−1}, du_{x} the differential of u at x and

du_{x} · ξ = ((du_{x} · ξ)^{1}, ..., (du_{x} · ξ)^{n}) = (\frac{∂u^{1}}{∂x^{α}} ξ^{α}, ..., \frac{∂u^{n}}{∂x^{α}} ξ^{α}) .

Moreover, as usual, we define the energy of u on Ω ⊂ ℝ^{m}by

E(u, Ω) = ∫_{Ω} e(u)(x)dx  (0.3)

We consider minimizing problems for the functional E(u, Ω) defined by (0.3) in the class

H_{f}^{1,2}(Ω, N) = {u ∈ H^{1,2}(Ω, N) ; u − f ∈ H_{0}^{1,2}(Ω, ℝ^{n})},  (0.4)

where f : ∂Ω → N is a given L^{∞} function. In the following, “a minimizer of E” means “a minimizer of E in the class H_{f}^{1,2}(Ω, N)”.

As in the case that the target manifold is Riemannian, let us call a solution of the Euler-Lagrange equation of (0.3) to be a harmonic map.

We consider the following assumption on h_{ij}(u, X) = \frac{1}{2} \frac{∂^{2}F^{2}(u,X)}{∂X^{i}∂X^{j}}

(h-1) For some concave increasing function ω with ω(0) = 0 we have

|h_{ij}(u, X) − h_{ij}(v, X)| ≤ ω(|u − v|)  (0.5)
for all u, v, X ∈ ℝ^{n}.

(h-2) There exists a positive constant λ_{0} such that

h_{ij}(u, X)ξ^{i}ξ^{j} > λ_{0}∥ξ∥^{2}  (0.6)

for all u, X, ξ ∈ ℝ^{n}.

Theorem.([3]) Let m ≤ 4. Under the above assumptions a minimizer u of E(u; Ω) is in in the class C^{1,α}(Ω_{0}) for some open subset Ω_{0} ⊂ Ω with H^{m−2−ε}(Ω \ Ω_{0}) = 0 for some ε > 0.

References

[1] P.Centore, Finsler Laplacians and minimal-energy maps,Int. J. Math. 11 (2000), 1–13.

[2] J. Jost, “Nonpositive curvature: Geometric and analytic aspects”, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, 1997.

[3] A.Tachikawa, A Partial regularity result for harmonic maps into a Finsler manifold, to appear in Calc. Var.