It is well known that the semi-linear elliptic Allen-Cahn equation arising in phase transition theory is closely related to the theory of minimal surfaces. Earlier works of Modica and Sternberg et. al in the 1970’s studied minimizing solutions in the framework of De Giorgi’s Gamma-convergence theory. The more profound regularity theory for stationary and stable solutions were obtained by the deep work of Tonegawa and Wickramasekera, building upon the celebrated Schoen-Simon regularity theory for stable minimal hypersurfaces. This is recently used by Guaraco to develop a new approach to min-max constructions of minimal hypersurfaces via the Allen-Cahn equation. In this talk, we will discuss about the boundary behaviour for limit interfaces arising in the Allen-Cahn equation on bounded domains (or, more generally, on compact manifolds with boundary). In particular, we show that, under uniform energy bounds, any such limit interface is a free boundary minimal hypersurface in the generalised sense of varifolds. Moreover, we establish the up-to-the-boundary integer rectifiability of the limit varifold. If time permits, we will also discuss what we expect in the case of stable solutions. This is on-going joint work with Davide Parise (UCSD) and Lorenzo Sarnataro (Princeton). This work is substantially supported by research grants from Hong Kong Research Grants Council and National Science Foundation China. 

Steklov eigenvalues are eigenvalues of the Dirichlet-to-Neumann operator which are introduced by Steklov in 1902 motivated by physics. And there is a deep connection between the extremal Steklov eigenvalue problems and the free boundary minimal surface theory in the unit Euclidean ball as revealed by Fraser and Schoen in 2016. In the talk, we will discuss the question of how rare simple Steklov eigenvalues are on manifolds and its applications in nodal sets and critical points of eigenfunctions.

A domain with C^2 boundary in complex space is called pseudoconvex if it has a C^2 defining function with positive complex hessian on its boundary. Pseudoconvexity is a generalization of convexity. It can be realised as a domain with geometric condition on the boundary and its topology can be studied by Morse theory. In this talk, we will discuss the Morse index theorem for free boundary minimal disks for partial energy in strictly pseudoconvex domain and the relation between holomorphicity and stability of the free boundary minimal disk. We will also give an example to illustrate the necessity of strict pseudoconvexity in our index estimate.

We proved a ``no semistability at infinity" result for complete Calabi-Yau metrics asymptotic to cones, by eliminating the possible appearance of an intermediate K-semistable cone in the 2-step degeneration theory developed by Donaldson-Sun. As a consequence, a classification result for complete Calabi-Yau manifolds with Euclidean volume growth and quadratic curvature decay is given. Moreover a byproduct of the proof is a polynomial convergence rate  to the asymptotic cone for such manifolds. Joint work with Song Sun.

Cancelled.