Related papers: Bach-flat asymptotically locally Euclidean metrics

We study sequences of conformal deformations of a smooth closed Riemannian manifold of dimension $n$, assuming uniform volume bounds and $L^{n/2}$ bounds on their scalar curvatures. Singularities may appear in the limit. Nevertheless, we…

We study geometry of complete Riemannian manifolds endowed with a weighted measure, where the weight function is of quadratic growth. Assuming the associated Bakry-Emery curvature is bounded from below, we derive a new Laplacian comparison…

Four dimensional simply connected Lie groups admitting a pseudo K\"ahler metric are determined. The corresponding Lie algebras are modelized and the compatible pairs $(J,\omega)$ are parametrized up to complex isomorphism (where $J$ is a…

We obtain an Euclidean volume growth results for complete Riemannian manifolds satisfying a Euclidean Sobolev inequality and a spectral type condition on the Ricci curvature. We also obtain eigenvalue estimates, heat kernel estimates, Betti…

We observe inequalities involving the Herzlich volume of a 4-dimensional asymptotically complex hyperbolic Einstein manifold and its Euler characteristic provided the metrics is either Kaehler or selfdual. In the selfdual case we have to…

In this paper, we establish compactness results of some class of conformally compact Einstein 4-manifolds. In the first part of the paper, we improve the earlier results obtained by Chang-Ge. In the second part of the paper, as…

We construct smooth Riemannian metrics with constant scalar curvature on each Hirzebruch surface. These metrics respect the complex structures, fiber bundle structures, and Lie group actions of cohomogeneity one on these manifolds. Our…

Let (M,J) be a compact complex 2-manifold which which admits a Kaehler metric for which the integral of the scalar curvature is non-negative. Also suppose that M does not admit a Ricci-flat K\"ahler metric. Then if M is blown up at…

We prove that the problem of constructing biharmonic conformal maps on a $4$-dimensional Einstein manifold reduces to a Yamabe-type equation. This allows us to construct an infinite family of examples on the Euclidean 4-sphere. In addition,…

The equations of motion of four-dimensional conformal gravity, whose Lagrangian is the square of the Weyl tensor, require that the Bach tensor $E_{\mu\nu}= (\nabla^\rho\nabla^\sigma + \ft12 R^{\rho\sigma})C_{\mu\rho\nu\sigma}$ vanishes.…

In this paper, we study 3-dimensional complete non-compact Riemannian manifolds with asymptotically nonnegative Ricci curvature and a uniformly positive scalar curvature lower bound. Our main result is that, if this manifold has $k$ ends…

In this paper we prove that every Riemannian metric on a locally conformally flat manifold with umbilic boundary can be conformally deformed to a scalar flat metric having constant mean curvature. This result can be seen as a generalization…

In a series of papers, Bamler [Bam20a,Bam20b,Bam20c] further developed the high-dimensional theory of Hamilton's Ricci flow to include new monotonicity formulas, a completely general compactness theorem, and a long-sought partial regularity…

Bianchi-IX four metrics are $SU(2)$ invariant solutions of vacuum Einstein equation, for which the connection-wise self-dual case describes the Euler Top, while the curvature-wise self-dual case yields the Ricci flat classical…

Given a compact constant scalar curvature Kaehler orbifold, with nontrivial holomorphic vector fields, whose singularities admit a local ALE Kaehler Ricci-flat resolution, we find sufficient conditions on the position of the singular points…

We consider four-dimensional, Riemannian, Ricci-flat metrics for which one or other of the self-dual or anti-self-dual Weyl tensors is type-D. Such metrics always have a valence-2 Killing spinor, and therefore a Hermitian structure and at…

We produce solutions to the K\"ahler-Ricci flow emerging from complete initial metrics $g_0$ which are $C^0$ Hermitian limits of K\"ahler metrics. Of particular interest is when $g_0$ is K\"ahler with unbounded curvature. We provide such…

In this paper we obtain three results concerning the geometry of complete noncompact positively curved K\"{a}hler manifolds at infinity. The first one states that the order of volume growth of a complete noncompact K\"{a}hler manifold with…

We construct self-dual(SD) but not locally conformally flat(LCF) metrics on families of non-simply connected 4-manifolds with small signature. We construct various sequences with bounded or unbounded Betti numbers and Euler characteristic.…

For the Bach-flat closed manifold with positive scalar curvature, we prove a rigidity result under a given inequality involving the Weyl curvature and the traceless Ricci curvature. Moveover, under an inequality involving…