I will first overview the classical holomorphic isometry problem between complex manifolds, in particular between bounded symmetric domains. When the source is the unit ball, in general the characterization of holomorphic isometries to bounded symmetric domains is not quite clear. With Shan Tai Chan, we recently characterized the holomorphic isometries from the Poincare disc to the product of the unit disc with the unit ball and it  provided new examples of holomorphic isometries from the Poincare disc into irreducible bounded symmetric domains of rank at least 2.

Under some topological assumption (which gives the boundedness of Sobolev constant), we construct the space of ALE SFK metrics on minimal ALE Kahler surfaces asymptotic to C^2/G, where G is a finite subgroup of U(2). This is a joint work with Jeff Viaclovsky.

In 1934, Wilhelm Blaschke’s attention focused on a recent construction in metric geometry proposed by Dan Barbilian as a generalization of various models of hyperbolic geometry. It was the year when S.-S. Chern started his doctoral program under Blaschke’s supervision in Hamburg and when in several academic centers in Europe scholars were interested in generalizations of Riemannian geometry. Introduced originally in 1934, Barbilian’s metrization procedure induces a distance on a planar domain through a metric formula given by the so-called logarithmic oscillation. In 1959, Barbilian generalized this process to more general domains. In our discussion we plan to show that these spaces are naturally related to Gromov hyperbolic spaces. In several works written with W.G. Boskoff, we explore this connection. We conclude our talk by stating several open problems related to this content.

The Nash embedding theorem states that every Riemannian
manifold can be isometrically embedded into some Euclidean space with
dimension bound. Isometric means preserving the length of every
path. Nash's proof involves sophisticated perturbations of the
initial embedding, so not much is known about the geometry of the
resulted embedding.
     In this talk, using the eigenfunctions of the Laplacian
operator, we construct canonical isometric embeddings of compact
Riemannian manifolds into Euclidean spaces, and study the geometry of
embedded images. They turn out to have large mean curvature
(intuitively, very bumpy), but the extent of oscillation is about the
same at every point. More can be said about global quantities like
the center of mass. This is a joint work with Xiaowei Wang.

We will review some recent work on free boundary minimal
hypersurfaces. In particular, we will explain a geometric classification
of the critical catenoid (joint with Ivaldo Nunes) and discuss what
information about such hypersurfaces in a general ambient manifold one can
extract from the knowledge of their Morse index (joint with Alessandro
Carlotto and Ben Sharp).