Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0705.0710

On Conformally Kaehler, Einstein Manifolds

Xiuxiong Chen·Claude LeBrun·Brian Weber
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove that any compact complex surface with positive first Chern class admits an Einstein metric which is conformally related to a Kaehler metric. The key new ingredient is the existence of such a metric on the blow-up of the complex projective plane at two distinct points.

Explore related subjects

math.DGgr-qcmath.AGmath.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Xiuxiong Chen, Claude LeBrun, Brian Weber. 2007-05-30. On Conformally Kaehler, Einstein Manifolds. https://arxiv.org/abs/0705.0710

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Rigidity of Immersed Gauss Curvature Shrinkers in Euclidean Space

For every $m\geq2$, we prove that any connected closed oriented smooth immersion $X:M^m\to\mathbb{R}^{m+1}$ satisfying \[ K=\left\langle X,ν\right\rangle \] is an embedding onto the unit sphere. No convexity is assumed. At a point maximizing $|X|$, we find a strictly convex component of $\{K>0\}$. If this component does not coincide with $M$, its polar immersion bounds an unbounded convex set. A finite area lemma then extends the local surface area formula for the polar body to a global measure identity. Using the vanishing first moment of surface area measure, we show that the origin lies in the interior of the polar body, contradicting the existence of recession directions. Thus the hypersurface is strictly convex, and the classification of strictly convex shrinkers completes the proof.

math.DG↗
arXiv2026

Orbit stratifications and smooth topology of positively curved six-manifolds with rank-one nonabelian symmetry

We study closed simply connected six-manifolds with positive sectional curvature and an effective isometric action of $SO(3)$ or $SU(2)$. Starting from Liu's corrected orbit-space analysis, we refine the stratification and recover smooth topology in several previously unresolved cases. For a boundaryless $SO(3)$ quotient, singular isotropy is connected, the weighted nonprincipal graph belongs to nine necessary families, and every action with only cyclic exceptional isotropy is classified equivariantly. The remaining branching families admit explicit rational intersection and Pontryagin formulas. For a three-ball $SO(3)$ quotient, the uniform-boundary configurations give sphere bundles over $S^4$ and a completely distinguished odd-integer family of non-spin six-manifolds. A doubled-quotient argument excludes the configuration with two boundary fixed points and an interior singular orbit, while the configuration with one fixed point and an exceptional interval has underlying manifold $\mathbb{CP}^3$. For a boundaryless $SU(2)$ quotient, an adjoint-weight matching argument reduces the exceptional three-orbit case to a unique action on the Eschenburg flag and identifies every two-orbit candidate with a weighted action on $S^2\times S^4$. These classifications imply rational ellipticity throughout the stated symmetry class. We also establish restricted curvature obstructions for the two diagonal product actions, without asserting nonexistence for arbitrary invariant metrics. Abstract graph restrictions, smooth realizability, and positive-curvature realizability are kept distinct. Some of the results in this paper are discovered by ChatGPT-6-astra and verified by the authors.

math.DG↗
arXiv2026

Lojasiewicz inequalities for maps of the 2-sphere

We prove a Lojasiewicz-Simon inequality $$ \left| E(u) - 4πn \right| \leq C \| \mathcal{T}(u) \|^α$$ for maps $u \in W^{2,2}\left( S^2, S^2 \right).$ The inequality holds with $α= 1$ in general and with $α> 1$ unless $u$ is nearly constant on an open set. We obtain polynomial convergence of weak solutions of harmonic map flow $u(t) : S^2 \to S^2$ as $t \to \infty$ on compact domains away from the singular set, assuming that the body map is nonconstant. The proof uses Topping's repulsion estimates together with polynomial lower bounds on the energy density coming from a bubble-tree induction argument.

math.DG↗