Rigidity of Critical Metrics for Quadratic Curvature Functionals
In this paper we prove new rigidity results for complete, possibly non-compact, critical metrics of the quadratic curvature functionals $\mathfrak{F}^{2}_t = \int |\operatorname{Ric}_g|^{2} dV_g + t \int R^{2}_g dV_g$, $t\in\mathbb{R}$, and $\mathfrak{S}^2 = \int R_g^{2} dV_g$. We show that (i) flat...
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Main Authors | , , |
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Format | Journal Article |
Language | English |
Published |
06.10.2021
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Subjects | |
Online Access | Get full text |
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Summary: | In this paper we prove new rigidity results for complete, possibly
non-compact, critical metrics of the quadratic curvature functionals
$\mathfrak{F}^{2}_t = \int |\operatorname{Ric}_g|^{2} dV_g + t \int R^{2}_g
dV_g$, $t\in\mathbb{R}$, and $\mathfrak{S}^2 = \int R_g^{2} dV_g$. We show that
(i) flat surfaces are the only critical points of $\mathfrak{S}^2$, (ii) flat
three-dimensional manifolds are the only critical points of
$\mathfrak{F}^{2}_t$ for every $t>-\frac{1}{3}$, (iii) three-dimensional scalar
flat manifolds are the only critical points of $\mathfrak{S}^2$ with finite
energy and (iv) $n$-dimensional, $n>4$, scalar flat manifolds are the only
critical points of $\mathfrak{S}^2$ with finite energy and scalar curvature
bounded below. In case (i), our proof relies on rigidity results for conformal
vector fields and an ODE argument; in case (ii) we draw upon some ideas of M.
T. Anderson concerning regularity, convergence and rigidity of critical
metrics; in cases (iii) and (iv) the proofs are self-contained and depend on
new pointwise and integral estimates. |
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DOI: | 10.48550/arxiv.2110.02683 |