ABSTRACT
This research proposal investigates the interplay between scalar curvature bounds and Seiberg-Witten invariants on 4-manifolds, focusing on deriving new curvature bounds, exploring the Yamabe invariant, and developing families of invariants.
PAPER · PDF
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Key findings
Seiberg-Witten theory constrains scalar curvatures of Riemannian metrics.
Deriving new sharp curvature bounds using refined Weitzenböck formulas.
Investigating the relationship between the Yamabe invariant and Seiberg-Witten basic classes.
Developing families of Seiberg-Witten invariants for parameterized 4-manifolds.
Limitations & open questions
Further research needed to fully understand the implications for the classification of smooth structures and the existence of Einstein metrics.