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Motivated by the work of Li and Mantoulidis, we study singular metrics which are uniformly Euclidean $(L^infty)$ on a compact manifold $M^n$ ($nge 3$) with negative Yamabe invariant $sigma(M)$. It is well-known that if $g$ is a smooth metric on $M$ with unit volume and with scalar curvature $R(g)ge sigma(M)$, then $g$ is Einstein. We show, in all dimensions, the same is true for metrics with edge singularities with cone angles $leq 2pi$ along codimension-2 submanifolds. We also show in three dimension, if the Yamabe invariant of connected sum of two copies of $M$ attains its minimum, then the same is true for $L^infty$ metrics with isolated point singularities.
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Extending Aubins construction of metrics with constant negative scalar curvature, we prove that every $n$-dimensional closed manifold admits a Riemannian metric with constant negative scalar-Weyl curvature, that is $R+t|W|, tinmathbb{R}$. In particular, there are no topological obstructions for metrics with $varepsilon$-pinched Weyl curvature and negative scalar curvature.
The scalar curvature equation for rotation invariant Kahler metrics on $mathbb{C}^n backslash {0}$ is reduced to a system of ODEs of order 2. By solving the ODEs, we obtain complete lists of rotation invariant zero or positive csck on $mathbb{C}^n backslash {0}$ in lower dimensions. We also prove that there does not exist negative csck on $mathbb{C}^n backslash {0}$ for $n=2,3$.
We classify manifolds of small dimension that admit both, a Riemannian metric of non-negative scalar curvature, and a -- a priori different -- metric for which all wedge products of harmonic forms are harmonic. For manifolds whose first Betti numbers are sufficiently large, this classification extends to higher dimensions.
In this paper, we consider a special class of singular Finsler metrics: $m$-Kropina metrics which are defined by a Riemannian metric and a $1$-form. We show that an $m$-Kropina metric ($m e -1$) of scalar flag curvature must be locally Minkowskian in dimension $nge 3$. We characterize by some PDEs a Kropina metric ($m=-1$) which is respectively of scalar flag curvature and locally projectively flat in dimension $nge 3$, and obtain some principles and approaches of constructing non-trivial examples of Kropina metrics of scalar flag curvature.
In the first part of this paper, we consider the problem of fill-in of nonnegative scalar curvature (NNSC) metrics for a triple of Bartnik data $(Sigma,gamma,H)$. We prove that given a metric $gamma$ on $mathbf{S}^{n-1}$ ($3leq nleq 7$), $(mathbf{S}^{n-1},gamma,H)$ admits no fill-in of NNSC metrics provided the prescribed mean curvature $H$ is large enough (Theorem ref{Thm: no fillin nonnegative scalar 2}). Moreover, we prove that if $gamma$ is a positive scalar curvature (PSC) metric isotopic to the standard metric on $mathbf{S}^{n-1}$, then the much weaker condition that the total mean curvature $int_{mathbf S^{n-1}}H,mathrm dmu_gamma$ is large enough rules out NNSC fill-ins, giving an partially affirmative answer to a conjecture by Gromov (see P.,23 in cite{Gromov4}). In the second part of this paper, we investigate the $theta$-invariant of Bartnik data and obtain some sufficient conditions for the existence of PSC fill-ins.
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