Topic
Scalar curvature
About: Scalar curvature is a research topic. Over the lifetime, 12701 publications have been published within this topic receiving 296040 citations. The topic is also known as: Ricci scalar.
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TL;DR: In this paper, the authors generalize their apriori estimates on cscK(constant scalar curvature K\\\"ahler) metric equation to more general curvature type equations (e.g., twistedcscK metric equation) under the assumption that the automorphism group is discrete.
Abstract: In this paper, we generalize our apriori estimates on cscK(constant scalar curvature K\\\"ahler) metric equation to more general scalar curvature type equations (e.g., twisted cscK metric equation). As applications, under the assumption that the automorphism group is discrete, we prove the celebrated Donaldson's conjecture that the non-existence of cscK metric is equivalent to the existence of a destabilized geodesic ray where the $K$-energy is non-increasing. Moreover, we prove that the properness of $K$-energy in terms of $L^1$ geodesic distance $d_1$ in the space of K\\\"ahler potentials implies the existence of cscK metric. Finally, we prove that weak minimizers of the $K$-energy in $(\\mathcal{E}^1, d_1)$ are smooth.
83 citations
TL;DR: In this article, it was shown that the metric of any compact Yamabe gradient soliton (M, g ) is a metric of constant scalar curvature when the dimension of the manifold n ⩾ 3.
83 citations
83 citations
TL;DR: A conformal invariant in dimension four is 1 2 Q (AR R2+3 IRicI2) as discussed by the authors, where R denotes the scalar curvature and Ric the Ricci tensor.
Abstract: An important problem in conformal geometry is the construction of conformal metrics for which a certain curvature quantity equals a prescribed function, e.g. a constant. In two dimensions, the uniformization theorem assures the existence of a conformal metric with constant Gauss curvature. Moreover, J. Moser [20] proved that for every positive function f on S2 satisfying f (x) = f(-x) for all x E S2 there exists a conformal metric on S2 whose Gauss curvature is equal to f. A natural conformal invariant in dimension four is 1 2 Q (AR R2+ 3 IRicI2), where R denotes the scalar curvature and Ric the Ricci tensor. This formula can also be written in the form
83 citations
TL;DR: In this paper, the effect of quantum perturbations on the energy momentum tensor was analyzed in slow-roll inflation with one-loop quantum corrections to the amplitude of curvature and tensor perturbation, as well as quantum fluctuations from light scalars and light Dirac fermions.
Abstract: We obtain the effective inflaton potential during slow roll inflation by including the one loop quantum corrections to the energy momentum tensor from scalar curvature and tensor perturbations as well as quantum fluctuations from light scalars and light Dirac fermions generically coupled to the inflaton. During slow roll inflation there is a clean and unambiguous separation between superhorizon and subhorizon contributions to the energy momentum tensor. The superhorizon part is determined by the curvature perturbations and scalar field fluctuations: both feature infrared enhancements as the inverse of a combination of slow roll parameters which measure the departure from scale invariance in each case.Fermions and gravitons do not exhibit infrared divergences. The subhorizon part is completely specified by the trace anomaly of the fields with different spins and is solely determined by the space-time geometry. The one-loop quantum corrections to the amplitude of curvature and tensor perturbations are obtained to leading order in slow-roll and in the (H/M_PL)^2 expansion. This study provides a complete assessment of the backreaction problem up to one loop including bosonic and fermionic degrees of freedom. The result validates the effective field theory description of inflation and confirms the robustness of the inflationary paradigm to quantum fluctuations. Quantum corrections to the power spectra are expressed in terms of the CMB observables:n_s, r and dn_s/dln k. Trace anomalies (especially the graviton part) dominate these quantum corrections in a definite direction: they enhance the scalar curvature fluctuations and reduce the tensor fluctuations.
83 citations