Josai Mathematical Monographs vol 3 (2001), pp. 63-71

SOME GEOMETRIC PROPERTIES OF p-HARMONIC MAPS

SHIGEO KAWAI

1. Introduction

In this note we consider geometric properties of p-harmonic maps and p-harmonic functions. Let (M, g) and (N, h) be connected Riemannian manifolds and p ≥ 2. We suppose from a technical reason that (N, h) is isometrically imbedded in a Euclidean space ℝ^{m} and denote by H^{1,p}(M, N) the space

{u ∈ H^{1,p}(M, ℝ^{m}) | u(x) ∈ N a.e. }.

A H^{1,p}-map u from M into N is called a (weakly) p-harmonic map if it is a weak solution of the following equation

Tr ∇(|du|^{p−2} du) = 0 ,

where “Tr” denotes the trace. It is the Euler-Lagrange equation of the p-energy functional

E_{p}(u) = ∫_{M} |du|^{p}.

This definition of p-harmonic maps does not depend on the embedding on N in a Euclidean space and coinside with that of harmonic maps when p = 2.

Because we are now interested only in ”geometric” properties, pharmonic maps are assumed to be of class C^{1} or C^{2} in the sequel. The following are typical examples of p-harmonic maps.

Example 1. (totally geodesic maps) Every totally geodesic map u : M → N is a p-harmonic map for any p. Indeed if ∇du = 0, then we have

Tr∇(|du|^{p−2}du) = |du|^{p−2}Tr(∇du) + ⟨∇|du|^{p−2}, du⟩
= 0 + (p − 2)|du|^{p−4}⟨⟨∇du, du⟩, du⟩
= 0.

If v : N → L is an isometric totally geodesic immersion, then the identity

Tr∇(|d(v ∘ u)|^{p−2}d(v ∘ u)) = Tr∇(|du|^{p−2}d(v ∘ u))
= dv (Tr∇(|du|^{p−2}du))

holds for every map u : M → N. Hence when N is embedded by i in L as a totally geodesic submanifold, a map u : M → N is p-harmonic if and only if i ∘ u : M → L is p-harmonic.

Example 2. (p-harmonic functions) Consider the case M = ℝ^{n} \ {O} and N = ℝ. Then the following functions which depend only on r = |x| are p-harmonic:

u(r) = \begin{cases}r^{(p−n)/(p−1)} & (n ≠ p) \\ log r & (n = p). \end{cases}

This is shown for example from the formula

Tr∇(|∇u|^{p−2}∇u)

= \frac{∂}{∂r}(|∇u|^{p−2} \frac{∂u}{∂r}) + \frac{n − 1}{r}|∇u|^{p−2} \frac{∂u}{∂r} + Tr_{S}∇_{S}(|∇u|^{p−2}∇_{S}u),

where Tr_{S}, ∇_{S} denote the trace and differentiation in the direction of spheres of radius r respectively.

Example 3. (equator maps) For n, m with n ≥ m + 1, consider the unit ball B^{n} in ℝ^{n} = ℝ^{m+1} × ℝ^{n−m−1} and unit sphere S^{m} in ℝ^{m+1}. Let M = B^{n} \ ({O} × ℝ^{n−m−1}) and N = S^{m}. Then a map u : M → N is defined by the equation

u(y, z) = \frac{y}{|y|} (y ∈ ℝ^{m+1}, z ∈ ℝ^{n−m−1}).

If n < m + 1, then u : B^{n} \ {O} → S^{m} ⊂ ℝ^{m+1} is defined by

u(y) = (\frac{y}{|y|}, O) ∈ S^{n−1} × \{O\} ⊂ ℝ^{n} × ℝ^{m−n+1}

These maps are p-harmonic for any p. More strongly some of them are in fact p-energy minimizing for their boundary data ([1]).

Remark. If u is a p-harmonic function, then Cu and u + C are pharmonic function for any constant C. But u + v is not necessarily p-harmonic even if u and v are p-harmonic functions. Also coordinate functions of a p-harmonic map to a Euclidean space are not necessarily p-harmonic functions.

2. Existence of p-harmonic maps from the spheres.

In this section we treat the existence problem.

SOME GEOMETRIC PROPERTIES OF p-HARMONIC MAPS 65

Theorem 1. ([8]) Let N be a compact simply connected Riemannian manifold isometrically embedded in a Euclidean space. Then for any C^{1}-map u : S^{n} → N with n ≥ 2, there exist a finite number of n-harmonic maps u_{1}, u_{2}, …, u_{k} : S^{n} → N which satisfy the following:

(1) [u] = [u_{1}] + [u_{2}] + ⋯ + [u_{k}].

(2) inf_{v∈[u]} E_{n}(v) = E_{n}(u_{1}) + ⋯ + E_{n}(u_{k}).

(3) u_{j} is a minimizer of E_{n} in [u_{j}] for j = 1, 2, , … k.

In the above [ ] denotes the free homotopy class of maps from the sphere. If N is not simply connected, the same results holds up to the action of π_{1}(N) on π_{n}(N). This result is shown by a bubbling argument and a generalization of the work in [9], [11] for harmonic maps from S^{2}. The case π_{n}(N) = {0} with arbitrary M was first proved by a different method in an unpublilshed paper [6], and can be deduced from [14] too. As an application of this theorem, we can give an alternative proof of a result concerning manifolds with strongly p-th moment stable stochastic dynamical systems.

Theorem 2. ([4], [8]) If a compact manifold N admit a strongly p-th moment stable stochastic dynamical system, then π_{k}(N) = 0 for k = 1, 2, …, p. In particular for p ≥ dim N/2, a p-th moment stable stochastic dynamical system can only exist on homotopy spheres.

¿From the previous theorem we can show that the elements of pdimensional homotopy groups are represented by p-harmonic maps. The proof of the following theorem is almost the same as that for π_{2}(N) in [9].

Theorem 3. ([8]) Let N be a compact simply connected Riemannian manifold isometriccaly embedded in a Euclidean space. Then for every n ≥ 2 there exist finite number of n-harmonic maps f_{1}, f_{2}, … such that

(1) [f_{1}], [f_{2}], … ≠ 0.

(2) [f_{1}], [f_{2}], … generate π_{n}(N).

(3) E_{n}(f_{1}) = inf{E_{n}(f) | [f] ≠ 0}.

E_{n}(f_{j}) = inf{E_{n}(f) | [f] ∉ ⟨[f_{1}], [f_{2}], …, [f_{j−1}]⟩} where ⟨a, b, c, … ⟩ denotes the subgroup generated by elements a, b, c, …

3. Convex functions and p-harmonic maps.

In this section, we consider geometric properties of p-harmonic maps to manifolds which have convex functions. The next result on harmonic maps is well known.

Theorem 4. ([5]) If φ : M → N is a harmonic map and f : N → ℝ is a C^{2} convex function, then f ∘ φ is a subharmonic function on M. If in addition M is compact and f is strictly convex, then φ is constant.

This is derived by the identity

Tr∇(d(f ∘ φ)) = Tr(∇df)(dφ, dφ)

which is easily verified from the composition law. For p-harmonic maps with p ≥ 2, the following lemma is useful.

Lemma. ([7]) If φ : M → N is a p-harmonic map and f : N → ℝ is a convex function (both of class C^{2}), then we have the following identity:

Tr∇(|dφ|^{p−2}d(f ∘ φ)) = |dφ|^{p−2}Tr(∇df)(dφ, dφ).

This is also a consequence of simple computation including the composition law. If φ is assumed to be only of class C^{1}, then we get an identity “in a weak form”. In the rest of this section, we consider the applications of the above identity. The first is a generalization of the result of W.B Gordon which was obtained in [2] by calculating the first variation directly.

Theorem 5. ([2], [7]) Let φ : M → N be a p-harmonic map of class C^{1}. If M is compact and there exists a strictly convex function f of class C^{2} on N, then φ is constant.

Proof. If ϕ is of class C^{2}, then we have only to integrate both sides of the identity in the above lemma. When ϕ is only of class C^{1}, we use the identity ”in a weak form”.

Theorem 6.([7]) Let M, N be Riemannian manifolds. Suppose that M is complete and noncompact. If N has a strictly convex function f of class C^{2} such that the uniform norm |df| is bounded. Then every p-harmonic map φ : M → N of class C^{1} with

∫_{M} |dφ|^{p−1} < ∞

is a constant map.

Proof. Fix a point x of M and take a function η on M which satisfies the property

\begin{cases}0 ≤ η ≤ 1, \\ |dη| ≤ C/R, \\ η ≡ 1 & on B_{R}, \\ η ≡ 0 & on M \ B_{2R}. \end{cases}

In the above, B_{R} and C denote the ball of radius R centered at x and a constant which does not depend on R respectively. Making use of Lemma we get the inequlaity

∫_{B_{R}} |dφ|^{p−2}Tr(∇dφ)(dφ, dφ)η ≤ ∫_{B_{2R}} Tr∇ (|dφ|^{p−2}d(f ∘ φ)) η
≤ \frac{C}{R} ∫_{M} |dφ|^{p−1}.

Letting R → ∞, we obtain the desired result. □

In fact we have only to assume that the growth of p − 1 energy of φ on the spheres of radius R is o(R). Relating results including the case of p-subharmonic functions are found in [7].

In [12] J.H.Sampson proved a form of maximum principle for harmonic maps. Let us say, following [12], the second fundamental form ∇di of a hypersurface S in N embedded by a map i is definite at a point y_{0} ∈ N iff ∇di(X, X) is nonzero and points in a given direction for any tangent vector X ∈ T_{y_{0}}S. The concave side of a hypersurface is the side pointed at by ∇di(X, X).

Theorem 7. ([12]) Let φ : M → N be a nonconstant harmonic map Assume that S is a hypersurface of N with definite second fundamental form at a point y_{0} = φ(x_{0}). Then no neighborhood of x_{0} is mapped entirely in the concave side of S in N.

Proof. ([3]) There exists a strictly convex function f defined in a neighborhood V of y_{0} in N which depends only on the distance from S. In addition we can assume that f < 0 on the concave side of S, and f^{−1}(0) ∩ V = S ([3] p.23). ¿From the theorem of W.B. Gordon, the composition f ∘ φ is a subharmonic function. In addition it is nonpositive in a neighborhood of x_{0} and achieve the maximum in the interior. Hence it is constant from the strong maximum principle, and we obtain

Tr(∇df)(dφ, dφ) = △(f ∘ φ) = 0.

Because f is assumed to be strictly convex, the map φ must be a constant. □

Though we cannot completely generalize his result for the case of p-harmonic maps with p ≥ 2, the next weak property can be proved.

Theorem 8. Let φ : M → N be a nonconstant p-harmonic map such that no point in a neighborhood of x_{0} ∈ M is mapped to φ(x_{0}). Assume that S is a hypersurface of N with definite second fundamental form at a point y_{0} = φ(x_{0}). Then no neighborhood of x_{0} is mapped entirely in the concave side of S in N.

Proof. Take a strictly convex function f with the same property as in the previous theorem. To prove by contradiction, we assume that φ maps a neighborhood U of x_{0} in M to the concave side of S. Then the inequality

f(φ(x)) = 0 ≤ f(φ(x_{0}))

holds for all x in U. From Lemma the following one also holds:

Tr∇(|dφ|^{p−2}d(f ∘ φ)) ≥ 0.

We want to claim that

sup_{∂U}(f ∘ φ) = sup_{U}(f ∘ φ) = 0.

For this purpose suppose that

a = sup_{∂U}(f ∘ φ) < sup_{U}(f ∘ φ) = 0.

Then there exist a constand c > 0 and a subdomain U′ in U such that

g = f ∘ φ − a − c > 0

in U′ and g = 0 on ∂U′. Let us consider a function ψ which is equal to g in U′ and vanishes outside U′. Using an previous inequality we obtain

∫_{U} |dφ|^{p−2}⟨d(f ∘ φ), dψ⟩ ≤ 0.

From the definition of ψ,

∫_{U′} |dφ|^{p−2}|d(f ∘ φ)|^{2} ≤ 0

which implies |dφ| = 0 or |d(f ∘ φ)| = 0 in U′. In both cases we get d(f ∘ φ) = 0, and the function g is constant in U′. This contradicts the definition of g and the equality

sup_{∂U}(f ∘ φ) = sup_{U}(f ∘ φ) = 0

holds.

Hence there exists some point x_{1} ∈ ∂U with φ(x_{1}) ∈ S \ x_{0}. The hypersurface S is locally a graph of some convex function defined on a neighorhood of x_{0} in the tangent plane at x_{0}. We can take a hypersurface S′ with definite second fundamental form whose intersection with S is only the point x_{0} and the mapping φ maps U entirely in the concave side of S′. Repeating the same reasoning for this hypersurface S′, we get a contradiction. □

4. Inequality between |∇|dφ|| and |∇dφ|

For harmonic maps the following inequality is known.

Theorem 9. ([10], [13]) Let φ : M^{n} → N be a harmonic map. Then on the set {x | dφ(x) ≠ 0} we have the inequality

\frac{n}{n − 1}|∇|dφ||^{2} ≤ |∇dφ|^{2}.

For a tensor T the inequality

|∇|T || ≤ |∇T |

holds generally. The above theorem states that it can be improved in the case T = ∇dφ for some harmonic map φ. Such kind of inequalities appear in many geometric prpblems. The case of harmonic map was proved in [13] with a different constant, and is used in the regularity problem of harmonic maps to spheres. Later [10] proved the above form and improved some of regularity results in [13]. It is not known whether this type of inequality is true for p-harmonic maps or not. But at least we can prove an inequlaity for p-harmonic functions.

Theorem 10. Let u be a p-harmonic function on a Riemannian manifold. Then the following inequality holds on the set {x | |∇u|(x) ≠ 0}:

min \{2, 1 + \frac{(p − 1)^{2}}{n − 1} \} |∇|∇u||^{2} ≤ |∇∇u|^{2}.

Proof. For x with ∇u(x) ≠ 0, let us take a normal coordinate around x such that

u_{1}(x) = |∇u|(x), u_{2}(x) = ⋯ = u_{n}(x) = 0.

Then we have at x

∇_{j}|∇u| = u_{1j}, |∇|∇u||^{2} = ∑_{j} u_{1j}^{2}.

On the other hand, since u is p-harmonic

|∇u|^{p−2}△u + ⟨∇|∇u|^{p−2}, ∇u⟩ = 0.

Consequently we get

|∇u|^{2(p−2)}|∇∇u|^{2} − |∇u|^{2(p−2)}|∇|∇u||^{2}

= |∇u|^{2(p−2)} ∑_{i,j} u_{i,j}^{2} − |∇u|^{2(p−2)} ∑_{j} u_{1j}^{2}

≥ |∇u|^{2(p−2)} ∑_{i≥2} u_{i1}^{2} + |∇u|^{2(p−2)} ∑_{i≥2} u_{ii}^{2}

≥ |∇u|^{2(p−2)} ∑_{i≥2} u_{i1}^{2} + \frac{1}{n − 1}|∇u|^{2(p−2)} (∑_{i≥2} u_{ii})^{2} .

The second term on the right hand side is equal to

\frac{1}{n − 1} (|∇u|^{p−2}u_{11} + ⟨∇|∇u|^{p−2}, ∇u⟩)^{2}

= \frac{1}{n − 1} (|∇u|^{p−2}u_{11} + (p − 2)|∇u|^{p−3}u_{11}u_{1})^{2}

= \frac{(p − 1)^{2}}{n − 1} |∇u|^{2(p−2)}u_{11}^{2}.

Hence we obtain

|∇u|^{2(p−2)}|∇∇u|^{2} − |∇u|^{2(p−2)}|∇|∇u||^{2}

≥ |∇u|^{2(p−2)} ∑_{i≥2} u_{i1}^{2} + \frac{(p − 1)^{2}}{n − 1} |∇u|^{2(p−2)}u_{11}^{2}

≥ min \{1, \frac{(p − 1)^{2}}{n − 1} \} |∇u|^{2(p−2)}|∇|∇u||^{2}

which implies the desired inequlaity. □

Remark. For the equator map φ : B^{n} \ ({O} × ℝ^{n−m−1}) → S^{m} defined in Example 3,

\sqrt{2} |∇|dφ|| = |∇dφ|.

Remark. We can also show the next inequality for a p-harmonic function u with a similar calculation:

(1 + min \{\frac{1}{(p − 1)^{2}} , \frac{1}{(n − 1)}\}) |∇|∇u|^{p−1}|^{2} ≤ |∇ (|∇u|^{p−2}∇u)|^{2} .

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Department of Mathematics Faculty of Calture and Education Saga University, Saga 840-8502 Japan