From minimal surfaces to Einstein 4-manifolds and back

Minimal surfaces in 3-manifolds, and Einstein 4-manifolds are two typical and important examples of variational objects, which means that they are critical points of some natural geometric functionals. While at first glance they seem to be very different, they actually share many common properties and I will illustrate through this analogy some ubiquitous principles in geometric analysis, like monotonicity, epsilon regularity, compactness, 2-piece decomposition... Both of these objects are often constructed indirectly, so the way their topology and geometry interact remains mysterious. I will discuss old and more recent results attempting to describe them.

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