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# How rare are simple Steklov eigenvalues?

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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.

# The roles of concavity, symmetry and sub-solutions in geometric PDEs

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In this talk we discuss the roles of concavity, symmetry and subsolutions in the study of fully nonlinear PDEs, especially those on real or complex manifolds with connection to geometric problems. We shall report some of our results along the line, which give the optimal conditions for the existence of classical solutions, either of the Dirichlet problem, or of equations on closed manifolds. If time permits, we shall also discuss the possibility to weaken or extend these conditions, and a class of equations involving differential forms of higher rank, more specifically real (p, p) forms for p > 1 on complex manifolds. Part of the talk is based on joint work with my student Mathew George.

# A free boundary problem in pseudoconvex domains

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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.

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# On complete Calabi-Yau manifolds asymptotic to cones

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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.

# The parabolic U(1)-Higgs equations and codimension-two mean curvature flows

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Mean curvature flow is the negative gradient flow of the area

functional, and it has attracted a lot of interest in the past few years. In

this talk, we will discuss a PDE-based, gauge theoretic, construction of

codimension-two mean curvature flows based on the Yang-Mills-Higgs

functionals, a natural family of energies associated to sections and metric

connections of Hermitian line bundles. The underlying idea is to approximate

the flow by the solution of a parabolic system of equations and study the

corresponding singular limit of these solutions as the scaling parameter

goes to zero. This is based on joint work with A. Pigati and D. Stern.

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Cancelled.

# The transformation theorem for type-changing semi-Riemannian manifolds

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In 1983 Hartle and Hawking put forth that signature type-change may be conceptually interesting, leading to the so-called no-boundary proposal for the initial conditions for the universe, which has no beginning because there is no singularity or boundary to the spacetime. But there is an origin of time. In mathematical terms, we are dealing with signature type-changing manifolds where a positive definite Riemannian region is smoothly joined to a Lorentzian region at the surface of transition where time begins.

We utilize a transformation prescription to transform an arbitrary Lorentzian manifold into a singular signature-type changing manifold. Then we prove the transformation theorem saying that locally the metric \tilde{g} associated with a signature-type changing manifold (M, \tilde{g}) is equivalent to the metric obtained from a Lorentzian metric g via the aforementioned transformation prescription. By augmenting the assumption by certain constraints, mutatis mutandis, the global version of the transformation theorem can be proven as well.

The transformation theorem provides a useful tool to quickly determine whether a singular signature type-changing manifold under consideration belongs to the class of transverse type changing semi-Riemannian manifolds.