Related papers: Some Uniformization Problems for a Fourth order Co…

We consider a four dimensional Riemannian manifold M with a metric g and affinor structure q. The local coordinates of these tensors are circulant matrices. Their first orders are (A, B, C, B), A, B, C\in FM and (0, 1, 0, 0), respectively.…

A Riemannian manifold is said to be almost positively curved if the sets of points for which all $2$-planes have positive sectional curvature is open and dense. We show that the Grassmannian of oriented $2$-planes in $\mathbb{R}^7$ admits a…

We obtain existence results for the $Q$-curvature equation of order $2k$ on a closed Riemannian manifold of dimension $n\ge 2k+1$, where $k\ge1$ is an integer. We obtain these results under the assumptions that the Yamabe invariant of order…

It is well known in Riemannian geometry that the metric components have the best regularity in harmonic coordinates. These can be used to characterize the most regular element in the isometry class of a rough Riemannian metric. In this…

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 describe a construction of Riemannian metrics of nonnegative sectional curvature on a closed smooth nonorientable 4-manifold with fundamental group of order two that realizes a homotopy class that was not previously known to contain…

Our goal is to identify curvature conditions that distinguish Euclidean space in the case of open, contractible manifolds and the disk in the case of compact, contractible manifolds with boundary. First, we show that an open manifold that…

We use PDE methods as developed for the Liouville equation to study the existence of conformal metrics with prescribed singularities on surfaces with boundary, the boundary condition being constant geodesic curvature. Our first result shows…

We study the Riemann curvature tensor of (\kappa,\mu,\nu)-contact metric manifolds, which we prove to be completely determined in dimension 3, and we observe how it is affected by D_a-homothetic deformations. This prompts the definition and…

Let $(M,g)$ be a compact Riemannian manifold with non-empty boundary. Provided $f$ an isoparametric function of $(M,g)$ we prove existence results for positive solutions of the Yamabe equation that are constant along the level sets of $f$.…

The Yamabe invariant Y(M) of a smooth compact manifold is roughly the supremum of the scalar curvatures of unit-volume constant-scalar curvature Riemannian metrics g on M. (To be absolutely precise, one only considers…

Consider a compact manifold $M$ with smooth boundary $\partial M$. Suppose that $g$ and $\tilde{g}$ are two Riemannian metrics on $M$. We construct a family of metrics on $M$ which agrees with $g$ outside a neighborhood of $\partial M$ and…

It is proved that every locally conformal flat Riemannian manifold all of whose Jacobi operators have constant eigenvalues along every geodesic is with constant principal Ricci curvatures. A local classification (up to an isometry) of…

In this paper, I shall demonstrate that sufficiently high-dimensional closed positively-curved Riemannian manifolds are either diffeomorphic to a spherical space form, or isometric to a locally compact rank one symmetric space. This…

Let (M,g) be a four or six dimensional compact Riemannian manifold which is locally conformally flat and assume that its boundary is totally umbilical. In this note, we prove that if the Euler characteristic of M is equal to 1 and if its…

Curvature properties of a metric connection with totally skew-symmetric torsion are investigated. It is shown that if either the 3-form $T$ is harmonic, $dT=\delta T=0$ or the curvature of the torsion connection $R\in S^2\Lambda^2$ then the…

In this note, we derive explicit formulae for the curvature of a convex sum of Riemannian metrics, \(g_t = (1-t)g_0 + t g_1\). We study whether such a deformation can increase the \emph{average} of the Riemann curvature component…

In this paper we develop a bubble tree structure for a degenerating class of Riemannian metrics satisfying some global conformal bounds on compact manifolds of dimension 4. Applying the bubble tree structure, we establish a gap theorem, a…

A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold…

Let $(M, \bar g)$ be an $n$-dimensional complete Riemannian manifold. In this paper, we considers the following conformal scalar curvature rigidity problem: Given a compact smooth domain $\Omega$ with $\partial \Omega$, can one find a…