In this survey, we discuss some recent results on free boundary minimal surfaces in the Euclidean unit-ball. The subject has been a very active field of research in the past few years due to the seminal work of Fraser and Schoen on the extremal Steklov eigenvalue problem. We review several different techniques of constructing examples of embedded free boundary minimal surfaces in the unit ball. Next, we discuss some uniqueness results for free boundary minimal disks and the conjecture about the uniqueness of critical catenoid. We also discuss several Morse index estimates for free boundary minimal surfaces. Moreover, we describe estimates for the first Steklov eigenvalue on such free boundary minimal surfaces and various smooth compactness results. Finally, we mention some sharp area bounds for free boundary minimal submanifolds and related questions.

Inspired by the Gromov-Hausdorff distance, we define a new notion called the intrinsic flat distance between oriented m dimensional Riemannian manifolds with boundary by isometrically embedding the manifolds into a common metric space, measuring the flat distance between them and taking an infimum over all isometric embeddings and all common metric spaces. This is made rigorous by applying Ambrosio-Kirchheim’s extension of FedererFleming’s notion of integral currents to arbitrary metric spaces. We prove the intrinsic flat distance between two compact oriented Riemannian manifolds is zero iff they have an orientation preserving isometry between them. Using the theory of AmbrosioKirchheim, we study converging sequences of manifolds and their limits, which are in a class of metric spaces that we call integral current spaces. We describe the properties of such spaces including the fact that they are countably Hm rectifiable spaces and present numerous examples.

We develop a Floer theoretical gluing technique and apply it to deal with the most generic singular ber in the SYZ program, namely the product of a torus with the immersed two-sphere with a single nodal self-intersection. As an application, we construct immersed Lagrangians in Gr(2;Cn) and OG(1;C5) and derive their SYZ mirrors. It recovers the Lie theoretical mirrors constructed by Rietsch. It also gives an eective way to compute stable disks (with non-trivial obstructions) bounded by immersed Lagrangians.

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Let F be a foliation in a closed 3-manifold with negatively curved fundamental group and suppose that F is almost trans- verse to a quasigeodesic pseudo-Anosov °ow. We show that the leaves of the foliation in the universal cover extend continuously to the sphere at in¯nity; therefore the limit sets of the leaves are continuous images of the circle. One important corollary is that if F is a Reebless, ¯nite depth foliation in a hyperbolic 3-manifold, then it has the continuous extension property. Such ¯nite depth foliations exist whenever the second Betti number is non zero. The result also applies to other classes of foliations, including a large class of foliations where all leaves are dense, and in¯nitely many examples with one sided branching. One extremely useful tool is a detailed understanding of the topological structure and asymptotic properties of the 1-dimensional foliations in the leaves of e F induced by the stable and unstable foliations of the °ow.