NDLI: The Yamabe problem for almost Hermitian manifolds

Content Provider	Springer Nature Link
Author	Rio, Heberto Simanca, Santiago R.
Copyright Year	2003
Abstract	The conformal class of a Hermitian metric g on a compact almost complex manifold (M$^{2m}$, J) consists entirely of metrics that are Hermitian with respect to J. For each one of these metrics, we may define a J-twisted version of the Ricci curvature, the J-Ricci curvature, and its corresponding trace, the J-scalar curvature s$^{J}$. We ask if the conformal class of g carries a metric with constant s$^{J}$, an almost Hermitian version of the usual Yamabe problem posed for the scalar curvature s. We answer our question in the affirmative. In fact, we show that (2m−1)s$^{J}$−s=2(2m−1)W(ω, ω), where W is the Weyl tensor and ω is the fundamental form of g. Using techniques developed for the solution of the problem for s, we construct an almost Hermitian Yamabe functional and its corresponding conformal invariant. This invariant is bounded from above by a constant that only depends on the dimension of M, and when it is strictly less than the universal bound, the problem has a solution that minimizes the almost complex Yamabe functional. By the relation above, we see that when W (ω, ω) is negative at least one point, or identically zero, our problem has a solution that minimizes the almost Hermitian Yamabe functional, and the universal bound is reached only in the case of the standard 6-sphere $$\mathbb{S}^6 $$ equipped with a suitable almost complex structure. When W(ω, ω) is non-negative and not identically zero, we prove that the conformal invariant is strictly less than the universal bound, thus solving the problem for this type of manifolds as well. We discuss some applications.
Starting Page	185
Ending Page	203
Page Count	19
File Format	PDF
ISSN	10506926
Journal	Journal of Geometric Analysis
Volume Number	13
Issue Number	1
e-ISSN	1559002X
Language	English
Publisher	Springer-Verlag
Publisher Date	2003-01-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	almost Hermitian manifolds J-scalar curvature almost complex Yamabe problem conformal deformation Differential Geometry Convex and Discrete Geometry Fourier Analysis Abstract Harmonic Analysis Dynamical Systems and Ergodic Theory Global Analysis and Analysis on Manifolds
Content Type	Text
Resource Type	Article
Subject	Geometry and Topology

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