Differential Geometry Working Seminar

Daren Cheng, Department of Pure Mathematics, University of Waterloo

"The second variation of area of minimal surfaces in four-manifolds"

I'll talk about the work of Micallef-Wolfson (Math. Ann. '93) on closed, immersed, oriented minimal surfaces in oriented Riemannian 4-manifolds, focusing on the result where they obtain a topological lower bound on the index plus nullity of such minimal surfaces under a condition involving the scalar curvature and Weyl tensor of the ambient space. The main ingredient is an "averaged" version of the second variation formula of the area functional, which I'll derive at the beginning. I'll then explain how the aforementioned curvature condition arises from this formula and mention a few situations where it is known to hold. Finally I'll describe how the index+nullity lower bound follows with the help of the Riemann-Roch theorem, applied to the normal bundle.

Zoom meeting:

• Meeting ID: 958 7361 8652
• Passcode: 577854