Relations between threshold constants for Yamabe type bordism invariants. (English) Zbl 1360.53039
In this paper, the authors study the relations between threshold constants for Yamabe-type bordism invariants. (The Yamabe invariant on closed manifolds is a bordism invariant below a certain threshold constant.) These threshold constants are characterized through Yamabe-type equations on products of spheres with rescaled hyperbolic spaces. After variational characterizations of these threshold constants they obtain an explicit positive lower bound for the spinorial threshold constants. Through the paper, after recalling existing results, the authors define the model spaces and several versions of the spinorial and non-spinorial Yamabe invariant for non-compact manifolds.
Central results of the article are noncompact versions of the Hijazi inequalities. Some of these slightly modified Hijazi inequalities are proven only for model spaces, some on more general manifolds (manifolds of bounded geometry with uniformly positive scalar curvature). For the investigation they also give regulatory statements for the Euler-Lagrange equation of the spinorial Yamabe functional. They give improvements of regularity for the Dirac Euler-Lagrange equation. The paper also contains an appendix on weak partial differential inequalities.
Central results of the article are noncompact versions of the Hijazi inequalities. Some of these slightly modified Hijazi inequalities are proven only for model spaces, some on more general manifolds (manifolds of bounded geometry with uniformly positive scalar curvature). For the investigation they also give regulatory statements for the Euler-Lagrange equation of the spinorial Yamabe functional. They give improvements of regularity for the Dirac Euler-Lagrange equation. The paper also contains an appendix on weak partial differential inequalities.
Reviewer: Corina Mohorianu (Iaşi)
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 35J60 | Nonlinear elliptic equations |
| 53C27 | Spin and Spin\({}^c\) geometry |
| 57R65 | Surgery and handlebodies |
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