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Register a new userAn existence theorem on the isoperimetric ratio over scalar-flat conformal classes
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Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary $partial M$, admitting a scalar-flat conformal metric. We prove that the supremum of the isoperimetric ratio over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric inequality in the Euclidean space, and consequently is achieved, if either (i) $9le nle 11$ and $partial M$ has a nonumbilic point; or (ii) $7le nle 9$, $partial M$ is umbilic and the Weyl tensor does not vanish identically on the boundary. This is a continuation of the work cite{Jin-Xiong} by the second named author and Xiong.
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Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary $partial M$. Suppose that $(M,g)$ admits a scalar-flat conformal metric. We prove that the supremum of the isoperimetric quotient over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric inequality in the Euclidean space, and consequently is achieved, if either (i) $nge 12$ and $partial M$ has a nonumbilic point; or (ii) $nge 10$, $partial M$ is umbilic and the Weyl tensor does not vanish at some boundary point.
We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension $ngeq 3$. We prove the existence of such conformal metrics in the cases of $n=6,7$ or the manifold is spin and some other remaining ones left by Escobar. Furthermore, in the positive Yamabe constant case, by normalizing the scalar curvature to be $1$, there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to $+infty$.
In this paper, we consider the indefinite scalar curvature problem on $R^n$. We propose new conditions on the prescribing scalar curvature function such that the scalar curvature problem on $R^n$ (similarly, on $S^n$) has at least one solution. The key observation in our proof is that we use the bifurcation method to get a large solution and then after establishing the Harnack inequality for solutions near the critical points of the prescribed scalar curvature and taking limit, we find the nontrivial positive solution to the indefinite scalar curvature problem.
We consider conformal deformations within a class of incomplete Riemannian metrics which generalize conic orbifold singularities by allowing both warping and any compact manifold (not just quotients of the sphere) to be the link of the singular set. Within this class of conic metrics, we determine obstructions to the existence of conformal deformations to constant scalar curvature of any sign (positive, negative, or zero). For conic metrics with negative scalar curvature, we determine sufficient conditions for the existence of a conformal deformation to a conic metric with constant scalar curvature -1; moreover, we show that this metric is unique within its conformal class of conic metrics. Our work is in dimensions three and higher.
Let $G$ be a simply-connected semisimple compact Lie group, $X$ a compact Kahler manifold homogeneous under $G$, and $L$ a negative $G$-equivariant holomorphic line bundle over $X$. We prove that all $G$-invariant Kahler metrics on the total space of $L$ arise from the Calabi ansatz. Using this, we then show that there exists a unique $G$-invariant scalar-flat Kahler metric in each Kahler class of $L$.
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