金沙赌城

魏国栋、江瑞奇副教授学术报告(2020.12.10)

In this talk,we first prove the extensibility of an arbitrary boundary metric to a positive scalar curvature metric inside for a compact manifold with boundary, which solves an open problem due to Gromov. Then we introduce a fill-in invariant and discuss its relationship with the positive mass theorems for asymptotically flat (AF) and asymptotically hyperbolic (AH) manifolds. In particular, we prove that the positive mass theorem for AH manifolds implies that for AF manifolds. In the end, we give some estimates for the fill-in invariant, which provide some partially affirmative answers to two conjectures by Gromov. This is a joint work with Prof. Yuguang Shi and Dr. Wenlong Wang.

In this talk, we show that for any two closed Riemannian manifolds M^m and N, there

exists a minimizing (extrinsic) m-polyharmonic map for every free homotopy class in

[M^{2m},N], provided that the homotopy group \pi_{2m}(N) is trivial. This generalizes the celebrated existence results for harmonic maps and biharmonic maps. We also prove that there exists a non-constant smooth polyharmonic map from R^{2m} to N by a blowup analysis at an energy-concentration point for an energy-minimizing sequence if the convergence fails to be strong.

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