The Minimal Hypersurface Method to Obstruct Positive Scalar Curvature via the Classification of 3-Manifolds

by Jonathan Groß
Master thesis
Date of Examination:2024-12-05
Date of issue:2025-05-05
Advisor:Prof. Dr. Thomas Schick
Referee:Prof. Dr. Thomas Schick
Referee:Prof. Dr. Max Wardetzky
Persistent Address: http://dx.doi.org/10.53846/goediss-11244

 

 

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Abstract

English

In 1979, Schoen and Yau developed a method using descending chains of minimal hypersurfaces to obstruct psc for manifolds initially up to dimension 7. Recent results on regularity for minimal hypersurfaces extend the applicability of this original technique up to dimension 10. However, the original reduction to dimension 2 requires certain (co-)homological properties, which are absent in topological spaces like T^{k−3} × BZ^3_p. Due to the geometrization of 3-manifolds it is possible to extend the Schoen-Yau method only descending to dimension 3. Analogous to the two-dimensional case which Schick used in a counterexample to the Gromov-Lawson-Rosenberg conjecture, we subsequently exclude pscm in dimension 3 from occurring. Generalizing this technique by terminating the descent in dimension 3 allows us to construct new counterexamples of the unstable GLR-conjecture that were previously inaccessible and prove our main theorem: Certain manifolds of dimension k ≤ 10 with p ≥ 3 prime and fundamental group Z^{k−3} ⊕ Z^3_p do not admit a metric of positive scalar curvature.
Keywords: positive scalar curvature; Gromov-Lawson-Rosenberg conjecture; minimal hypersurfaces; Geometrization of 3-manifolds; Homotopy-theoretic obstructions; conformal Laplacian
 
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