No escape to spatial infinity [04EW]

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No escape to spatial infinity

Corollary 5.13.

Assume near the infinity of $X$ , the regularity scale grows to infinity. Fix the homology class of the quantitatively almost calibrated Lagrangian $L$ . Then $L$ is contained in a fixed compact subset of $X$ .

Proof.

From Lemma 2.1 there is an a priori volume bound $\text{Vol}(L)\leq C$ . But if $P$ is in the support of $L$ , then the regularity scale of $X$ near $P$ is bounded by

R^{n}\leq C\text{Vol}(L\cap B(R))\leq C,

whence $P$ must remain in a fixed compact subset. ∎