ScalingStacks

Aside from its importance in mirror symmetry, the SYZ conjecture is interesting for other diverse fields such as minimal surface theory, Riemannian geometry, Kähler and algebraic geometry.

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  Special Lagrangians are minimal submanifolds, and in fact calibrated submanifolds. Currently, there are few methods for producing special Lagrangians in sufficiently large supply of Calabi-Yau manifolds, although there is a series of conjectures initiated by Thomas-Yau [75][74] and further developed in [43][57].

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  The behaviour of a family of Einstein metrics depends strongly on whether the volume of geodesic balls satisfies the noncollapsing condition for some uniform constant $\kappa>0$.

  $$\text{Vol}_{g}(B_{g}(r))\geq\kappa r^{\dim_{\mathbb{R}}X},\quad\forall 0<r\leq\text{diam}(X).$$

  A good convergence and regularity theory is available in the non-collapsing case [13]. On the other hand, metric degeneration in the collapsing case is largely terra incognita in Riemannian geometry, and the semiflat metric asymptote is a highly nontrivial emergent feature for a collapsing family of Calabi-Yau metrics.

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  A recurring theme of Kähler geometry is the interplay between metric and complex geometry. Käher-Einstein metrics in the non-collapsing case is tied to projective geometry [23]. The large complex structure limit is a very severe kind of polarized degeneration, whose transcendental behaviour (related to exponential and logarithms) is not adequately captured by traditional projective geometry, and instead non-archimedean geometry stands out as a natural framework. One can then ask about the relation between Calabi-Yau metrics and non-archimedean geometry, a problem that turns out to be related to the SYZ conjecture.