Qun Chen (Wuhan University)
Title: Image structures of harmonic maps into metric spaces
Abstract: In this talk, we discuss image structures of harmonic maps from Riemannian manifolds into metric spaces of curvature bounded above in the sense of Alexandrov. When the target spaces are of nonpositive curvature and the domain manifolds have some analytic structures, we present two convex hull properties satisfied by the image sets of harmonic maps. When the target space has curvature bounded above by a positive constant, under certain conditions on curvature and volume comparison of the domain manifolds, we give an asymptotic property for harmonic maps with finite energy.
This is based on joint work with Jie Wang.
Xiaonan Ma (Université de Paris)
Title: Superconnection and family Bergman kernels
Abstract: We establish an asymptotic version of Bismut's local family index theorem for the Bergman kernel associated with a fiberwise positive line bundle when the power tends to infinity. The key idea is to use the superconnection as in the local family index theorem. In particular, we show the curvature operator of the associated direct image is a Toeplitz operator.
Xinan Ma (University of Science and Technology of China)
Title: Jerison-Lee identity and Semi-linear subelliptic equation on CR manifold
Abstract: In the study of the extremal for Sobolev inequality on the Heisenberg group and the Cauchy-Riemann (CR) Yamabe problem, Jerison-Lee found a three-dimensional family of differential identities for critical exponent subelliptic equation on Heisenberg group H^n by using the computer in Jerison-Lee (JAMS, 1988). They wanted to know whether there is a theoretical framework that would predict the existence and the structure of such formulae. With the help of dimensional conservation and invariant tensors, we can answer the above question. For a class of subcritical exponent subelliptic equations on the CR manifold, several new types of differential identities are found. Then we use those identities to get the rigidity result, where rigidity means that subelliptic equations have no other solution than some constant at least when parameters are in a certain range. The rigidity result also deduces the sharp Folland-Stein inequality on closed CR manifolds.
Xinqun Mei (Universität Freiburg)
Title: Alexandrov-Fenchel inequalities for capillary hypersurfaces in hyperbolic space
Abstract: In this talk, we first introduce the quermassintegrals for compact hypersurfaces with capillary boundaries in hyperbolic space from a variational viewpoint, and then we solve an isoperimetric type problem in hyperbolic space. By constructing a new locally constrained inverse curvature flow, we obtain the Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in hyperbolic space. This generalizes a theorem of Brendle-Guan-Li for convex closed hypersurfaces in hyperbolic space.
This is based on joint work with Liangjun Weng.
Wei Wei (Nanjing University)
Title: A new boundary mass for asymptotically flat half-manifolds
Abstract: We introduce a boundary-type mass for asymptotically flat manifolds with non-compact boundary by using the k-th Gauss-Bonnet curvature. We prove that the mass is well defined and state the corresponding mass for graph hypersufaces with a non-compact boundary.
This is a joint work with Prof. Dr. Guofang Wang
Chao Xia (Xiamen University)
Title: Monotonicity along level set flow of p-capacitary functions and its applications
Abstract: In this talk, we present monotone quantities along the level set flow of p-capacitary functions in asymptotically flat 3-manifolds with nonnegative scalar curvature. As applications, we prove geometric inequalities associated with p-capacitary functions, with rigidity on spatial Schwarzschild manifolds outside rotationally symmetric spheres, which generalizes Miao's result from p=2 to 1<p<3. Moreover, we recover mass-to-p-capacity and p-capacity-to-area inequalities due to Bray-Miao and Xiao. Compare to p=2 case, there is no conformal relationship for general p-capacitary functions between Euclidean and Schwarzschild model. The monotonicity property follows from a direct analysis of a system of ODEs arising from p-capacitary functions in Schwarzschild model.
This is joint work with Jiabin Yin and Xingjian Zhou.
Kai Zhang (Universidad de Granada)
Title: Boundary regularity for uniformly elliptic equations
Abstract: In this talk, we introduce some boundary pointwise regularity results for uniformly elliptic equations, including boundary Holder regularity, boundary Lipschitz regularity, boundary C^{1,a} regularity and boundary C^{2,a} regularity etc.
This talk is a combination of our several works in recent years.
Mingwei Zhang (Universität Freiburg)
Title: Some results of Dirac-harmonic maps with curvature term
Abstract: In this talk, we first introduce Dirac-harmonic maps with curvature term, and then establish the long-time existence of heat flow for Dirac-geodesics and its convergence. The difficulty mainly comes from the extra non-homogeneous term in the Dirac equation. When the map is between surfaces, we discuss some structure results.
This is joint work with Qun Chen
Xuwen Zhang (Universität Freiburg)
Title: Willmore inequalities for hypersurfaces with boundary
Abstract: In this talk we discuss Willmore inequalities for immersed/embedded hypersurfaces with boundary. For embedded hypersurfaces, in its most generality we establish the Willmore inequalities in arbitrary unbounded convex domains, with anisotropy taking into account. A direct application is the establishment of Willmore inequalities for embedded capillary hypersurfaces in the half-space. A different Willmore-type inequality is also proven for immersed capillary (hyper)surfaces in the half-space, its counterpart in the Euclidean ball is also obtained due to the conformal invariance of the Willmore functional. As a by-product, we obtain optimal area estimate for immersed capillary minimal surfaces in the Euclidean ball, which is essentially different to another optimal area estimate proven by S. Brendle.
This talk is based on joint works with Xiaohan Jia, Guofang Wang, and Chao Xia.
Qun Chen
Yuanyuan Lian
Jiayu Li
Zheng Lu
Xiaonan Ma
Xinan Ma
Xinqun Mei
Zhe Pu
Christine Schmidt
Guofang Wang
Wei Wei
Chao Xia
Kai Zhang
Mingwei Zhang
Weiping Zhang
Xiqiang Zhang
Xuwen Zhang