Beijing Institute of Mathematical Sciences and Applications

In this talk, we will discuss some results on minimal graphs over Riemannian manifolds. On the one hand, we will review some known results on minimal graphs over Euclidean space as well as manifolds. On the other hand, we will talk about a capacity defined by volume functional of graphs over manifolds, and related results on minimal hypersurfaces in product manifolds. Moreover, we get asymptotic estimates for minimal graphic functions on manifolds satisfying volume doubling property and (1,1)-Poincare inequality.

In 1987, Hitchin constructed a complete hyperkähler metric on the moduli space of Higgs bundles, which can be generalized to accommodate singularities. In this talk, we consider Higgs bundles with irregular singularities over the Riemann sphere. We construct a generic ray in the moduli space and study the asymptotic behavior of the Hitchin metric along this ray. Using the techniques developed by Biquard-Boalch and Fredrickson-Mazzeo-Swoboda-Weiss, we show that the Hitchin metric is exponentially close to a simpler semi-flat metric. In dimension four, we obtain an explicit asymptotic formula for the semi-flat metric, which is of type ALG or ALG*. This is a joint work with Nianzi Li.

The Allen-Cahn equation is a semi-linear elliptic equation arising in the van der Waals-Cahn-Hilliard theory of phase transitions. Earlier fundamental work by De Giorgi, Modica, Sternberg etc. revealed intriguing relationship between the Allen-Cahn equation and the theory minimal surfaces. Based on the deep regularity theory by Hutchinson, Tonegawa and Wickramasekera, Guaraco recently introduced a new PDE approach to the existence of minimal surfaces via the Allen-Cahn equation and there have been substantial progress along this direction in the past few years. In this talk, we will consider the Allen-Cahn equation on bounded domains and describe some geometric and analytic aspects of the boundary behaviour of the limit-interfaces. This work is substantially supported by research grants from Hong Kong Research Grants Council and National Science Foundation China.

Volume preserving mean curvature flow was introduced by Huisken in 1987 and it was proved that the flow deforms convex initial hypersurface smoothly to a round sphere. This was generalized later by McCoy in 2005 and 2017 to volume preserving flows driven by a large class of 1-homogeneous symmetric curvature functions. In this talk, we will discuss the flows with higher homogeneity and describe the convergence result for volume preserving curvature flows in Euclidean space by arbitrary positive powers of k-th mean curvature for all k=1,...,n. As key ingredients, the monotonicity of a generalized isoperimetric ratio will be used to control the geometry of the evolving hypersurfaces and the curvature measure theory will be used to prove the Hausdorff convergence to a sphere. We also discuss some generalizations including the flows in the anisotropic setting, and the flows in the hyperbolic setting. The talk is based on joint work with Ben Andrews (ANU), Yitao Lei (ANU), Changwei Xiong (Sichuan Univ.), Bo Yang (CAS) and Tailong Zhou (USTC).

In this talk, I will discuss the unobstructed deformations of d-closed forms, logarithmic differential forms, and $\bar{\partial}$-closed forms on compact complex manifolds satisfying some special conditions. These results are based on several joint works with Professors Kefeng Liu, Sheng Rao, Quanting Zhao, and Wei Xia.

In this talk, I will present recent works on scalar curvature along several Ricci-type flows.

This is a survey talk on differential geometry of Lagrangian surfaces in the complex projective plane. We firstly discuss properties of Lagrangian surfaces with certain geometric properties, Gauss maps of Lagrangian surfaces, and Lagrangian version of Willmore conjecture. Then we shall focus on the construction of minimal Lagrangian surfaces via loop group method.

We study the differences of two consecutive eigenvalues $\lambda_{i}-\lambda_{i-1}$ up to $i=2g-2$ for the Laplacian on hyperbolic surfaces of genus $g$, and show that the supremum of such spectral gaps over the moduli space has infimum limit at least $\frac{1}{4}$ as genus goes to infinity. A min-max principle for eigenvalues on degenerating hyperbolic surfaces is also established. This is a joint work with Haohao Zhang and Xuwen Zhu.

Dirac-harmonic maps with curvature term is a geometric variational model originated in the supersymmetric nonlinear sigma model of quantum field theory. The Euler-Lagrange equation couples a system of second order elliptic PDE/ODEs and a Dirac equation, both of them are nonlinear. In this talk, we will present recent results on the existence and uniqueness of the solutions.

In this talk, the speaker will first survey the past and recent progress on the Kahler-Ricci flow on compact Kahler manifold with semi-ample canonical line bundles. Then, he will discuss his recent work on the higher-order regularity of such a flow when the generic fibers are biholomorphic, which proves that under such an assumption, the flow would converge smoothly to a generalized Kahler-Einstein on any compact subsets away from singular fibers. Part of the talk is based on a joint work with M.C. Lee.

The non-abelian Hodge correspondence is a real analytic map between the moduli space of stable Higgs bundles and the deRham moduli space of irreducible flat connections mediated by solutions of the self-duality equation. In my talk I will explain how to construct such solutions for strongly parabolic $\mathfrak{sl}(2,\mathbb C)$ Higgs fields on a 4-punctured sphere with parabolic weights $t\sim 0$ near the trivial parabolic structure using loop groups methods through an implicit function theorem argument. The methods and computations are based on Deligne's interpretion of the twistor space via $\lambda$-connections. As a first application of this approach, I identify the rescaled limit hyper-Kaehler moduli space at $t = 0$ as the Eguchi-Hanson space, and I show that the hyper-Kaehler metric can be expanded to arbitrary order in terms of multiple-polylogarithms. Finally, the geometric properties lead to some identities involving multiple-polylogarithms. This talk is based on joint work with Lynn Heller and Martin Traizet.

Velazquez constructed a countable collection of mean curvature flow solutions in R^N, N is even and N\geq 8. Each of these solutions becomes singular in finite time at which time the second fundamental form blows up. Guo and Sesum , Stolarski studied the behavior of the mean curvature of Velazquez's solution. In this talk, we will construct the mean curvature flow solutions in R^N which become singular in finite time for any N\geq 8.

In this talk I will report on joint work with Steven Charlton, Sebastian Heller and Martin Traizet about recent progress in constructing higher genus harmonic maps using integrable systems methods. By taking the genus as parameter, we obtain in particular a Taylor expansion for the area of the Lawson surfaces ($\xi_1$,g) with an algorithms to compute all coefficients. Remarkably, these coefficients turns out to be Multiple Zeta Values. Moreover, the Taylor series is shown to converge for genus $>2$ which also yields monotonicity of the area in g for $g>3$. Rather than focussing on the algorithmic details, I aim at explaining the key ideas of the integrable systems approach relating moduli spaces of flats connections to the geometry of the surface.

In this talk, we will report recent progress on the Schwarz symmetrization in Riemannian manifolds and submanifold. As applications, we obtain the estimates of the eigenvalues and eigenfunctions of the Laplacian with Dirichlet and Robin boundary conditions. This is partly joint work with Prof. Haizhong Li and Dr. Yilun Wei.

The Weil-Petersson metric is an important tool in understanding the geometry/topology of the Teichmuller space of Riemann surfaces. To study the Weil-Petersson geometry on the Teichmuller space of Kahler-Einstein manifolds in general, the first step is establishing the deformation theory of such manifold. In this talk, we shall discuss the complex deformation of compact Kahler-Einstein manifolds based on developing the Kodaira-Spencer-Kuranishi's method. We shall also mention applications of such deformation theory.

The curvature behavior is one of the most important aspects of the Kaehler-Ricci flow. In this talk we shall provide an overview on the curvature behavior of long-time solutions to the Kaehler-Ricci flow of collapsing structure, which intimately relate to the underlying complex structure. In particular, some recent criteria for type-IIb singularities basing on certain analytic or algebraic properties of the underlying manifolds will be introduced.

The deformed Donaldson-Thomas (dDT) connection is a Hermitian connection of a Hermitian line bundle over a G2-manifold satisfying certain nonlinear PDEs. This is considered to be the mirror of a calibrated (associative) submanifold via mirror symmetry. As the name indicates, the dDT connection can also be considered as an analogue of the Donaldson-Thomas connection (G2-instanton). In this talk, after reviewing these backgrounds, I will show that dDT connections indeed have properties similar to associative submanifolds and G2-instantons. I would also like to present some related problems. A part of this talk is based on the joint work with Hikaru Yamamoto.

In this talk we introduce sereval new pinching results of Legendrian submanifolds which satisfy certain geometric pdes in unit spheres. In particular, we will focus on a new characterization of the Calabi torus in unit spheres which is the minimal Calabi product Legendrian immersion of the totally geodesic Legendrian sphere and a point. This is based on joint works with Prof. Linlin Sun and Prof. Jiabin Yin.

We derive the first order regularity estimate for any space-time, in terms of the bounds of the Ricci curvature and Lie derivative of the Lorentzian metric relative to an arbitrary timelike vector field.

A complete Riemannian n-manifold M is called epsilon-collapsed, if every unit ball in M has volume less than epsilon (while often a bound on `curvature' must be imposed to prevent a rescaling of metric). In 1978, Gromov classified `almost flat manifolds' (or the `maximally collapsed manifolds' with sectional curvature bounded in absolute value by one and small diameter) ; a bounded normal covering space of M is diffeomorphic to the quotient of a simply connected nilpotent Lie group modulo a manifold up to a co-compact lattice. This result has been a corner stone in the collapsing theory of Cheeger-Fukaya-Gromov in 90's: there is a nilpotent structure on any epsilon-collapsed manifold with bounded sectional curvature, and this theory has found important applications in Metric Riemannian geometry. We will survey some recent development in generalizing the above collapsing theory to epsilon-collapsed manifolds of Ricci curvature bounded below and the (incomplete) universal cover of every unit ball in M is not collapsed. The study of these collapsed manifolds is partially fueled with many constructions of collapsed Calabi-Yau metrics using certain underlying singular nilpotent fibrations.

Isoperimetric inequality is one of the oldest problems in mathematics, which relates with convex geometry, differential geometry and geometric PDEs, etc. Recently, the isoperimetric type inequalities in hyperbolic space have been widely investigated by using the hypersurface curvature flows, including the inverse curvature flows, quermassintegral preserving curvature flows, contracting curvature flows, and locally constrained curvature flows. In this talk, I will survey the recent progress in this direction, which is also based on my joint works with Ben Andrews(ANU), Yong Wei(USTC), Changwei Xiong(SCU), Yingxiang Hu(Beihang U.).