An official website of the United States government
Abstract In a previous paper [7], the first two authors classified complete Ricci-flat ALF Riemannian 4-manifolds that are toric and Hermitian, but non-Kähler. In this article, we consider general Ricci-flat metrics on these spaces that are close to a given such gravitational instanton with respect to a norm that imposes reasonable fall-off conditions at infinity. We prove that any such Ricci-flat perturbation is necessarily Hermitian and carries a bounded, non-trivial Killing vector field. With mild additional hypotheses, we are then able to show that the new Ricci-flat metric must actually be one of the known gravitational instantons classified in [7].
more »
« less
Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher.
Some full text articles may not yet be available without a charge during the embargo (administrative interval).
What is a DOI Number?
Some links on this page may take you to non-federal websites. Their policies may differ from this site.
-
-
The Yamabe invariant is a diffeomorphism invariant of smooth compact manifolds that arises from the normalized Einstein-Hilbert functional. This article highlights the manner in which one compelling open problem regarding the Yamabe invariant appears to be closely tied to static potentials and the first eigenvalue of the Laplacian.more » « less
-
Abstract. For compact complex surfaces (M^4,J) of Kaehler type, it was previously shown (LeBrun 1999) that the sign of the Yamabe invariant Y (M) only depends on the Kodaira dimension Kod(M,J). In this paper, we prove that this pattern in fact extends to all compact complex surfaces except those of class VII. In the process, we also reprove a result from (Albanese 2021) that explains why the exclusion of class VII is essential here.more » « less
-
The fact that every compact oriented 4-manifold admits spinc structures was proved long ago by Hirzebruch and Hopf. However, the usual proof is neither direct nor transparent. This article gives a new proof using twistor spaces that is simpler and more geometric. After using these ideas to clarify various aspects of four-dimensional geometry, we then explain how related ideas can be used to understand both spin and spinc structures in any dimension.more » « less
-
Dearricott, O.; Tuschmann, W.; Nikolayevsky, Y.; Leistner, T.; Crowley, D. (Ed.)The author has elsewhere given a complete classification of the compact oriented Einstein 4-manifolds that satisfy W⁺ (⍵, ⍵) > 0 for some self-dual harmonic 2-form ⍵, where W⁺ denotes the self-dual Weyl curvature. In this article, similar results are obtained when W⁺ (⍵ , ⍵) ≥ 0, provided the self-dual harmonic 2-form ⍵ is transverse to the zero section of Λ⁺→ M. However, this transversality condition plays an essential role in the story; dropping it leads one into wildly different territory where entirely different phenomena predominate.more » « less
