5.2.1 Intrinsic distance bound and potential clustering [04F2]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

5.2.1 Intrinsic distance bound and potential clustering

Suppose $L$ is a smooth immersed Lagrangian, then it inherits a Riemannian metric from the restriction of the Calabi-Yau metric, so we can speak of the intrinsic distance function on $L$ . Instead of extrinsic balls such as $B(P,r)$ , we can talk about intrinsic balls $\hat{B}(P,r)$ . A key distinction is that intrinsic distance does not need to extend to a continuous function on $X\times X$ , the prototypical example being the union of two embedded Lagrangians, whose domains are disjoint, but whose images in $X$ intersect. Clearly, the intrinsic distance bounds extrinsic geodesic distance, so $\hat{B}(P,r)$ is always contained in an extrinsic geodesic ball of radius $r$ , but the converse is far from true. The intrinsic distance between two distinct connected components would simply be infinity. One can think of intrinsic distance as a quantitative measurement of connectedness.

As usual, the regularity scale on $X$ grows to infinity asymptotically by assumption, so the regularity scale has a global lower bound. The following lemma has the same proof as Cor. 5.12. (The essence of this argument also appears in Neves [63, Lem 3.9]).

Lemma 5.15.

(Intrinsic ball volume lower bound) Let $L$ be a smooth immersed compact Lagrangian in $X$ , satisfying the quantitative almost calibrated condition. For any $P$ in the support of $L$ , there is a uniform bound

\text{Vol}(\hat{B}(P,r)\cap L)\geq C^{-1}r^{n},\quad r\leq 1.

Corollary 5.16.

(Intrinsic diameter bound) Assume further that the smooth, quantitatively almost calibrated compact Lagrangian $L$ has connected domain. Then within a fixed homology class, the intrinsic distance of $L$ has a uniform upper bound.

Proof.

Let $d=dist(P,P^{\prime})\geq N=[d]$ be the intrinsic diameter of $L$ . By connectedness, we can find $P_{1},\ldots P_{N}$ with $dist(P,P_{i})=i$ . Now the intrinsic balls $\hat{B}(P_{i},\frac{1}{2})$ are disjoint, but each takes up a nontrivial amount of volume $\geq C^{-1}$ . Thus

C^{-1}N\leq\text{Vol}(L)\leq\frac{1}{\sin\epsilon}\int_{L}\text{Re}(\Omega)\leq C,

so there is an a priori bound on $N$ , hence on $d$ . ∎

Corollary 5.17.

(Potential oscillation bound) Assume $L$ is an exact, immersed, compact Lagrangian, satisfying the almost calibrated condition, such that the Lagrangian potential $f_{L}$ has connected range. Then the potential $f_{L}$ has an a priori bound

\sup_{L}f_{L}-\inf_{L}f_{L}\leq C.

Proof.

Given any $P,P^{\prime}$ on $L$ , we write the potential as a line integral of the Liouville 1-form

f_{L}(P)-f_{L}(P^{\prime})=\int_{P^{\prime}}^{P}\lambda\leq\left\lVert\lambda\right\rVert_{C^{0}}dist(P,P^{\prime}).

Thus the intrinsic ball volume lower bound implies that if $a$ lies in the range of $f_{L}$ , then

\text{Vol}(\{|f_{L}-a|<\frac{1}{2}\})\geq C^{-1}.

The range of $f_{L}$ is by assumption a closed interval. If the interval has length $\geq N$ , then we can find $N$ distinct values of $a$ with disjoint $\{|f_{L}-a|<\frac{1}{2}\}$ , so

C^{-1}N\leq\text{Vol}(L)\leq\frac{1}{\sin\epsilon}\int_{L}\text{Re}(\Omega)\leq C,

This provides an a priori bound on $N$ , hence on the potential oscillation. ∎

Corollary 5.18.

Assume $L$ is an exact, immersed, compact Lagrangian, satisfying the almost calibrated condition, then it satisfies $(N,A)$ -potential clustering (cf. section 5.1.2) for some $N,A$ with uniform bounds. Morever, if $L$ carries an unobstructed brane structure, then the potential clustering property is consistent with the twisted complex interpretation.

Proof.

By Lemma 6.3, a general immersed Lagrangian can be decomposed into a union of Lagrangians $L_{i}$ such that the ranges of the potentials $f_{L_{i}}$ are connected, and any bounding cochain structure naturally produces a twisted complex. The oscillation of each $L_{i}$ is bounded in terms of the quantitative almost calibration condition, and the ambient features of $X$ . Morever, the number of Lagrangian components is also a priori bounded. ∎

As a notable consequence, we obtain the uniform energy bound on the holomorphic curves (cf. Prop. 3.41).

Remark 5.13.

Although it is unclear how to make sense of the intrinsic distance on a general Lagrangian integral current, mildly singular Lagrangians (for instance with local conical singularities) do have a sensible notion of intrinsic distance, and the arguments in this section extend practically to all non-pathological examples, covering all Lagrangians that appear in Joyce’s LMCF program. Furthermore, the potential clustering is robust under limits (cf. Cor. 5.9). As such, we believe it holds for all Lagrangians relevant to our variational program (cf. the class $\mathcal{L}$ in section 5.3 below).