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Yunhee Euh

Publications and source records attributed to Yunhee Euh.

8 recordsLinked to original sources
arXiv2025

Critical metrics for the quadratic curvature functional on complete four-dimensional manifolds

We study critical metrics of the curvature functional $\A(g)=\int_M |R|^2\, \vol$, on complete four-dimensional Riemannian manifolds $(M,g)$ with finite energy, that is, $\A(g)<\infty$. Under the natural inequality condition on the curvature operator of the second kind associated with the trace-free Ricci tensor, we prove that $(M,g)$ is either Einstein or locally isometric to a Riemannian product of two-dimensional manifolds of constant Gaussian curvatures $c$ and $-c$ $(c\ne 0)$. This extends the compact classification of four-dimensional $\mathcal{A}$-critical metrics obtained in earlier work to the complete setting.

math.DG↗
arXiv2024

On weakly Einstein K\"ahler surfaces

Riemannian four-manifolds in which the triple contraction of the curvature tensor against itself yields a functional multiple of the metric are called weakly Einstein. We focus on weakly Einstein K\"ahler surfaces. We provide several conditions characterizing those K\"ahler surfaces which are weakly Einstein, classify weakly Einstein K\"ahler surfaces having some specific additional properties, and construct new examples.

math.DG↗
arXiv2024

On weakly Einstein Lie groups

A Riemannian manifold is called \emph{weakly Einstein} if the tensor $R_{iabc}R_{j}^{~~abc}$ is a scalar multiple of the metric tensor $g_{ij}$. We consider weakly Einstein Lie groups with a left-invariant metric which are weakly Einstein. We prove that there exist no weakly Einstein non-abelian $2$-step nilpotent Lie groups and no weakly Einstein non-abelian nilpotent Lie groups whose dimension is at most $5$. We also prove that an almost abelian Lie group is weakly Einstein if and only if at the Lie algebra level it is defined by a normal operator whose square is a multiple of the identity.

math.DG↗
arXiv2022

Curvature identities for Einstein manifolds of dimension 5 and 6

Patterson discussed the curvature identities on Riemannian manifolds in [14], and a curvature identity for any 6-dimensional Riemannian manifold was independently derived from the Chern-Gauss-Bonnet Theorem [8]. In this paper, we provide the explicit formulae of Patterson's curvature identity that holds on 5-dimensional and 6-dimensional Einstein manifolds. We confirm that the curvature identities on the Einstein manifold from the previous work [8] are the same as the curvature identities deduced from Patterson's result. We also provide examples that support the theorems.

math.DG↗