Infinite energy harmonic maps from quasi-compact Kähler surfaces

2.1 NPC spaces

We refer to [13] for more details.

2.2 Maps into NPC spaces

In this paper, we consider harmonic maps into NPC spaces. Important examples are when the target space X ̃ is a smooth Riemannian manifold of non-positive sectional curvature. In this case, the energy of a smooth map f : Ω → X ̃ is

where (Ω, g) is a Riemannian domain and dvol _g is the volume form of Ω.

In the case when the target is an arbitrary NPC space, we use the following definition of energy due to Korevaar-Schoen. We refer to [14] for more details.

Let (Ω, g) be a bounded Lipschitz Riemannian domain. Let Ω _ϵ be the set of points in Ω at a distance least ϵ from ∂Ω. Let B _ϵ (x) be a geodesic ball centered at x and S _ϵ (x) = ∂B _ϵ (x). We say f: Ω → X is an L ²-map (or that f ∈ L ²(Ω, X)) if

For f ∈ L ²(Ω, X), define

where σ _x,ϵ is the induced measure on S _ϵ (x). We define a family of functionals

We say f has finite energy (or that f ∈ W ^1,2(Ω, X)) if

It is shown in [14] that if f has finite energy, the measures e _ϵ (x)dvol _g converge weakly to a measure which is absolutely continuous with respect to the Lebesgue measure. Therefore, there exists a function e(x), which we call the energy density, so that e _ϵ (x)dvol _g ⇀ e(x)dvol _g . In analogy to the case of smooth targets, we write |∇f|²(x) in place of e(x). In particular, the (Korevaar-Schoen) energy of f in Ω is

For V ∈ ΓΩ where ΓΩ is the set of Lipschitz vector fields on Ω, |f _*(V)|² is similarly defined. The real valued L ¹ function |f _*(V)|² generalizes the norm squared on the directional derivative of f. The generalization of the pull-back metric is the continuous, symmetric, bilinear, non-negative and tensorial operator

where

We refer to [14] for more details.

Let (x ¹, …, x ⁿ ) be local coordinates of (Ω, g) and g = (g _ij ), g ⁻¹ = (g ^ij ) be the local metric expressions. Then energy density function of f can be written (cf. [14, (2.3vi)])

Next assume (Ω, g) is a 2-dimensional Hermitian domain and let (z ¹ = x ¹ + ix ², z ² = x ³ + ix ⁴) be local complex coordinates. We extend π _f linearly cover C and denote

and

Thus,

2.3 Isometries of an NPC space

Throughout this paper, we denote the group of isometries of an NPC space X ̃ by I s o m ( X ̃ ) . Isometries of an NPC space are classified as follows.

2.4 Property (⋆)

Given a homomorphism ρ : π 1 ( M ) → I s o m ( X ̃ ) , we say ρ(π ₁(M)) satisfies Property (⋆) if following holds:

1. Every I ∈ ρ(π ₁(M)) has exponential decay to its translation length. In other words, either

   1. I is semisimple, or

   2. I is parabolic, fixes ξ ∈ ∂ X ̃ and there exists a geodesic ray c : [ 0 , ∞ ) → X ̃ and a, b > 0 such that

      d 2 ( I ( c ( t ) ) , c ( t ) ) ≤ Δ I 2 + b e − a t .

2. For any commuting pair of isometries I ₁, I ₂ ∈ ρ(π ₁(M)), either

   1. I ₁, I ₂ do not fix a common point of ∂ X ̃ , or

   2. I ₁, I ₂ fix a common point ξ ∈ ∂ X ̃ and there exist an arclength parameterized geodesic ray c : [ 0 , ∞ ) → X ̃ in the equivalence class ξ and a, b > 0 such that

      d 2 ( I i ( c ( t ) ) , c ( t ) ) ≤ Δ I i 2 + b e − a t , i = 1,2 .

2.5 Equivariant maps and sections of the associated flat X ̃ -bundle

Following Donaldson [8], we will replace equivariant maps with sections of an associated fiber bundle. Assume we have the following:

1. a complete Riemannian manifold (M, g) with universal covering Π : M ̃ → M

2. an NPC space X ̃

3. an action of π ₁(M) on M ̃ by deck transformations

4. a homomorphism ρ : π 1 ( M ) → I s o m ( X ̃ )

The quotient under the action of π ₁(M) of the product M ̃ × X ̃ is the twisted product

In other words, M ̃ × ρ X ̃ is the set of orbits [(p, x)] of a point ( p , x ) ∈ M ̃ × X ̃ under the action of γ ∈ π ₁(M) via the deck transformation on the first component and the isometry ρ(γ) on the second component. The fiber bundle

is called the flat X ̃ -bundle over M defined by ρ.

There is a one-to-one correspondence between sections of this fibration and ρ-equivariant maps

satisfying the relationship

Since the energy density function | ∇ f ̃ | 2 of f ̃ is a ρ-invariant function, we can define

We can similarly define the pullback inner product and directional energy density functions of f by using the corresponding notions for f ̃ given in Section 2.2. For U ⊂ M, the energy of a section f is

Furthermore, for sections f ₁, f ₂, we define

where f ̃ 1 , f ̃ 2 are the associated ρ-equivariant maps to sections f ₁, f ₂ respectively.

2.6 Harmonic maps from punctured Riemann surfaces

In this paper, many of the constructions will depend on harmonic maps for punctured Riemann surfaces. Below, we summarize the results of the paper [10], [11].

Let R ̄ be a compact Riemann surface and R = R ̄ \ { p 1 , … , p n } a punctured surface. We fix a conformal disk D j ⊂ R ̄ centered at each puncture p ^j such that D i ∩ D j = ∅ for i ≠ j. Furthermore, let D j * = D j \ { 0 } .

Fix P 0 ∈ X ̃ and a fundamental domain F of R ̃ . Let f ₀ be the section of the fiber bundle R ̃ × ρ X ̃ → R such that, for any p ∈ R ∩ Π ( F ) , f 0 ( p ) = [ ( p ̃ , P 0 ) ] where p ̃ = Π − 1 ( p ) ∩ F . (Note that Π(F) is of full measure in R .)

For a given section f : R → R × ρ X ̃ , define

Recall that d(f(z), f ₀(z)) is defined by (2.2).

We record our result in [6], [7].

3.1 Neighborhoods near the divisor

This subsection closely follows [7]. We let M ̄ be a Kähler surface and Σ be a divisor with normal crossings such that

Furthermore, let

where {Σ _j } is the set of irreducible components of Σ. Let σ _j be the canonical section of O ( Σ j ) with zero set Σ _j .

We denote by D the unit disk in the complex plane and let

For clarity, we will also denote the unit disk by D z to indicate that D is being parameterized by the complex variable z = re ^iθ . We use analogous notation D z , r , D z , r 1 , r 2 . We also use the notation S θ 1 to denote the circle S 1 parameterized by the real variable θ and identify S θ 1 as the boundary of D z via the map θ ↦ e^iθ .

To study a neighborhood of the juncture, let P ∈ Σ _i ∩Σ _j for some i, j ∈ {1, …, L} with i ≠ j, and let V _P be a neighborhood of P containing no other crossings. Choose holomorphic trivializations e _i (resp. e _j ) of O ( Σ i ) (resp. O ( Σ j ) ) on V _P and define z ¹ (resp. z ²) by setting

For each j = 1, …, L, let h _j be a Hermitian metric on O ( Σ j ) such that | e j | h j = 1 in V _P for any crossing P. Let h be a Hermitian metric on M ̄ , not necessarily Kähler, such that the following holds:

1. The metric h is the Euclidean metric in a neighborhood V _P of every crossing P, i.e.

   (3.2) h | V P = d z 1 d z ̄ 1 + d z 2 d z ̄ 2 .

   By rescaling σ ₁ and σ ₂ if necessary, we can assume without the loss of generality that

   (3.3) D ̄ z 1 × D ̄ z 2 ⊂ V P .

2. The metric h induces the orthogonal decomposition T M ̄ | Σ j = T Σ j ⊕ N Σ j and under the natural isomorphism

   (3.4) N Σ j ≃ O ( Σ j ) | Σ j ,

   the restriction of h to NΣ _j is same as h _j .

For r ∈ (0, 1], we set

There exists r > 0 such that the restriction of the exponential map

defines diffeomorphism of D j , r to a neighborhood of Σ _j in M ̄ . By rescaling σ _j if necessary, we may assume r > 1. In particular, we identify (3.5) for sufficiently small r > 0 as an open subset of M via the exponential map; i.e.

Denote

The restriction of NΣ _j → Σ _j to D ̄ j defines a disk bundle

We also identity D ̄ j as a subset of M ̄ ; i.e.

We denote by J M ̄ the holomorphic structure on D ̄ j defined by pulling back the complex structure on M ̄ via the exponential map.

We now consider a finite collection of sets near the divisor of the following two types:

1. A set of type (A) admits a local unitary trivialization

   (3.9) π j − 1 ( Ω ) ≃ Ω × D ̄ z 2 ,

   of π j : D ̄ j → Σ j where Ω ⊂ Σ _j is a contractible open subset of Σ _j containing no crossings. We will use coordinates (z ¹, z ²) where z ¹ is a holomorphic local coordinate in Ω and z ² is the standard coordinate of D ̄ . Although the coordinates (z ¹, z ²) are holomorphic with respect to the product complex structure J _prod of Ω × D ̄ z 2 , they are not holomorphic with respect to the complex structure J M ̄ . However, by construction, we have that

   J M ̄ = J prod on T ( z 1 , 0 ) Ω × D ̄ z 2 .

2. A set of type (B) is as in (3.3); i.e.

   (3.10) D ̄ z 1 × D ̄ z 2 ⊂ V P

   where V _P be an open set containing a single crossing P ∈ Σ _i ∩Σ _j (i ≠ j). By the property (i) of the hermitian metric h, (z ¹, z ²) are holomorphic coordinates with respect to J M ̄ . Furthermore, with the identification D ̄ z 1 ≃ D ̄ z 1 × { 0 } ⊂ Σ 1 (resp. D ̄ z 2 ≃ { 0 } × D ̄ z 2 ⊂ Σ 2 ),

   (3.11) π j − 1 D ̄ z 1 ≃ D ̄ z 1 × D ̄ z 2 ( resp.  π i − 1 D ̄ z 2 ≃ D ̄ z 1 × D ̄ z 2 )

   is a local unitary trivialization of π j : D ̄ j → Σ j (resp. π i : D ̄ i → Σ i ).

3.2 Poincaré-type metric

Recall the Poincaré metric

Using the canonical section σ j ∈ O ( Σ j ) and the Hermitian metric h _j on O ( Σ j ) given in Section 3.1, we define the Poincare-type metric on M as follows:

Below we derive some estimates for the metric g in a set of type (A) and of type (B) (See (3.9) and (3.10) for definitions of a set of type (A) and (B)). These are an expanded form of the estimates derived by Mochizuki [7].

3.3 Metric estimates in set of type (A)

Let π j − 1 ( Ω ) ≃ Ω × D ̄ be a set of type (A) with the standard product coordinates (z ¹, z ²). (Recall that Ω ⊂ Σ _j does not intersect Σ _i for i ≠ j.) We will write

Fix a local trivialization e of O ( Σ j ) , holomorphic with respect to the complex structure J M ̄ . Define

With σ _j the canonical section of O ( Σ j ) as before, define a function ζ on Ω × D ̄ by

Thus, ζ is holomorphic with respect to J M ̄ .

Since z ² = 0 = ζ on Ω × {0} and z ² ≠ 0, ζ ≠ 0 on Ω × D ̄ * ,

Taking real and imaginary parts,

Let

be a smooth function bounded above and bounded away from 0 satisfying

Thus,

Plugging in ∂ ∂ ζ in the above equation, we obtain

From this, we immediately obtain

This implies

where