Rate of decay for the mass ratio of pseudo-holomorphic integral 2-cycles

Abstract

We consider any pseudo holomorphic integral cycle in an arbitrary almost complex manifold and perform a blow up analysis at an arbitrary point. Building upon a pseudo algebraic blow up (previously introduced by the author) we prove a geometric rate of decay for the mass ratio towards the limiting density, with an explicit exponent of decay expressed in terms of the density of the current at the point. With a non-explicit exponent this result was proved using different techniques by Pumberger and Rivière (Duke Math J 152(3):441–480, 2010).

Appendix

The decomposition result (19) was given as an easy consequence of the fact that an integral 1-cycle can be represented as a sum of Lipschitz curves. We present in this appendix a self-contained proof of (19) that generalizes to arbitrary dimensions.

1.1 Decomposition of the slice in the style of [7]

Recall from [7, 4.2.25] the following facts. A compactly supported integral current C is called indecomposable if and only if there exists no integral current R of the same dimension that satisfies the following two conditions simultaneously: $R \ne 0 \ne C-R$ and $M(C)+M(\partial C) = M(R) + M(\partial R) + M(C-R) + M(\partial (C-R))$. In particular remark that when C is a cycle then R should also be a cycle for the second condition to hold. Further we have: an arbitrary integral cycle C can be written as a sum of indecomposable integral cycles $\{C_i\}_{i=1}^\infty$

$$\begin{aligned}C= \sum _{i=1}^\infty {C}_i\quad \text { with }\quad \sum _{i=1}^\infty M(C_i) = M(C).\end{aligned}$$

Using this result for a slice $\langle P, |Z|=r \rangle$ we get a countable family of indecomposable integral cycles $\{R_i\}_{i=1}^\infty$ such that

$$\begin{aligned} \langle P, |Z|=r \rangle = \sum _{i=1}^\infty R_i \quad \text { and }\quad \sum _{i=1}^\infty M(R_i) = M(\langle P, |Z|=r \rangle ). \end{aligned}$$

(40)

1.2 Decomposition of the slice taking care of the connectedness

Consider the possible ways of writing the (compact) set $\mathop {\mathcal {P}_{}}\cap \{|Z|=r\}$ as a union of the following form:

$$\begin{aligned}&\mathop {\mathcal {P}_{}}\cap \{|Z|=r\} = (\mathop {\mathcal {P}_{}}\cap \{|Z|=r\} \cap A ) \cup (\mathop {\mathcal {P}_{}}\cap \{|Z|=r\} \cap B ), \nonumber \\&\quad \text { with }A\text { and }B\text { disjoint open sets}. \end{aligned}$$

(41)

Unless $\mathop {\mathcal {P}_{}}\cap \{|Z|=r\}$ is connected, there exists at least a decomposition of $\mathop {\mathcal {P}_{}}\cap \{|Z|=r\}$ of the type (41) into two non-empty closed (and bounded) sets. Observe that, as a compact set, $\mathop {\mathcal {P}_{}}\cap \{|Z|=r\} \cap A$ is a (strictly) positive distance away from the topological boundary $\partial A$.

For any choice of A and B, the support of each $R_i$ is contained either almost completely in A or almost completely in B. Indeed and . Therefore the boundary of the current is supported in the intersection of $\partial A$ with the compact set $\mathop {\mathcal {P}_{}}\cap \{|Z|=r\}$, and this intersection is empty. Analogously for . We thus get that and are integral cycles that add up to $R_i$ (in the sense of [7, 4.2.25]). By the indecomposability of $R_i$ at least one of these two cycles must be zero, which means that one of the two sets , has zero ${\mathop {\mathcal {H}}}^1$-measure.

Let $\{A_\alpha \}$ and $\{B_\alpha \}$ be all the possible choices of open sets in (41) with the condition that $A_\alpha$ is always the open set containing the support of $R_1$. Then consider the sets

$$\begin{aligned}&\cap _\alpha ( \mathop {\mathcal {P}_{}}\cap \{|Z|=r\} \cap A_\alpha ), \\&\mathop {\mathcal {P}_{}}\cap \{|Z|=r\} \cap (\cup _\alpha B_\alpha ). \end{aligned}$$

These two sets are disjoint and their union is $\mathop {\mathcal {P}_{}}\cap \{|Z|=r\}$. The first set is closed bounded and connected by construction and it contains the support of $R_1$. For each $R_i$ ($i \ne 1$) we have the following dicothomy: either $R_i$ is supported completely in all of the $A_\alpha$’s or there exists an index $\overline{\alpha }$ such that the support of $R_i$ is disjoint from $A_{\overline{\alpha }}$ and lies completely in $B_{\overline{\alpha }}$. Thus each $R_i$ is supported either completely in the first set $\cap _\alpha ( \mathop {\mathcal {P}_{}}\cap \{|Z|=r\} \cap A_\alpha )$ or completely in the second set $\mathop {\mathcal {P}_{}}\cap \{|Z|=r\} \cap (\cup _\alpha B_\alpha )$.

This means that

$$\begin{aligned}&\cap _\alpha ( \mathop {\mathcal {P}_{}}\cap \{|Z|=r\} \cap A_\alpha ) = \cup _{i \in I_1}\text {supp}R_i\\&\mathop {\mathcal {P}_{}}\cap \{|Z|=r\} \cap (\cup _\alpha B_\alpha )= \cup _{i \in \mathbb {N}\setminus I_1}\text {supp}R_i \end{aligned}$$

with $1 \in I_1$. So we have the decomposition (as cycles)

$$\begin{aligned}\langle P, |Z|=r \rangle = \sum _{i \in I_1} R_i + \sum _{i \in \mathbb {N}\setminus I_1} R_i,\end{aligned}$$

where the first cycle on the right hand side has for support the connected compact set $\cap _\alpha ( \mathop {\mathcal {P}_{}}\cap \{|Z|=r\} \cap A_\alpha )$ (which contains the support of $R_1$).

By doing the same with any other $R_j$ instead of $R_1$, we decompose

$$\begin{aligned} \langle P, |Z|=r \rangle = \sum _{j=1}^\infty N_j\quad \, \text { with }\quad \,\, \sum _{j=1}^\infty M(N_j) = M(\langle P, |Z|=r \rangle ), \end{aligned}$$

(42)

where each $N_j$ is an integer cycle supported in a 1-rectifiable set that is compact and connected. Write each $N_i$ as the current of integration on a 1-rectifiable set ${\mathop {\mathcal {N}_{}}}_i$ with multiplicity $f_i \in L^1({\mathop {\mathcal {N}_{}}}_i, \mathbb {N})$. The decomposition (19) implies that $\cup {\mathop {\mathcal {N}_{}}}_i$ equals $\mathop {\mathcal {P}_{}}\cap \{|Z|=r\}$ a.e. and $\sum _{i=1}^\infty f_i = f$ a.e.

Remark however that by construction the supports of the $N_i$’s are disjoint, so we have the disjoint unions

$$\begin{aligned}\sqcup _{i=1}^\infty \text {supp}{N_i} = \mathop {\mathcal {P}_{}}\cap \{|Z|=r\} = \sqcup _{i=1}^\infty {\mathop {\mathcal {N}_{}}}_i,\end{aligned}$$

with the second equality to be understood in the a.e. sense. This means that for each i the set ${\mathop {\mathcal {N}_{}}}_i$ agrees ${\mathop {\mathcal {H}}}^1$-a.e. with the support of $N_i$, in particular we can choose a good representative ${\mathop {\mathcal {N}_{}}}_i$ for the underlying set to the current $N_i$, namely we can choose the ${\mathop {\mathcal {N}_{}}}_i$’s to be compact, disjoint and connected.