| Abstract: |
| It is well-known that any simple closed curve in $\mathbb{R}^3$ bounds at least one minimal disk, which was independently proved by Douglas and Rad\`{o}. However, for any given two disjoint simple closed curves, we cannot guarantee existence of a compact connected minimal surface spanning such boundary curves in general. From this point of view, it is interesting to give a quantitative description for necessary conditions on the boundary of compact connected minimal surfaces. In this talk, we give various necessary conditions and nonexistence results for compact connected minimal submanifolds, Bryant surfaces, and surfaces with small $L^2$ norm of the mean curvature vector in a Riemannian manifold. |
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