Related papers: Weyl curvature and the Euler characteristic in dim…
We prove the existence of metrics with prescribed $Q$-curvature under natural assumptions on the sign of the prescribing function and the background metric. In the dimension four case, we also obtain existence results for curvature forms…
In this paper, we will present some characterizations for the upper bound of the Bakry-Emery curvature on a Riemannian manifold by using functional inequalities on path space. Moreover, some characterizations for general lower and upper…
We show that a compact oriented riemannian four-manifold with harmonic and pinched self-dual Weyl curvature is anti-self-dual if the type is nonpositive. The main part is to show that there is an almost-K\"ahler structure outside the zero…
We develop minimal slicing via capillary hypersurfaces to understand positive scalar curvature metric on manifolds with boundary. The method provides rigidity statements once the regularity of minimizers of capillary area functional holds.…
An explicit construction of closed, orientable, smooth, aspherical 4-manifolds with any odd Euler characteristic greater than 12 is presented. The manifolds constructed here are all Haken manifolds in the sense of B. Foozwell and H.…
Following Gromov, a Riemannian manifold is called area-extremal if any modification that increases scalar curvature must decrease the area of some tangent 2-plane. We prove that large classes of compact 4-manifolds, with or without…
We give a sufficient condition to rule out complete Riemannian metrics with nonnegative scalar curvature on the interiors of handlebodies. In higher dimensions, we give examples of ends of manifolds with positive scalar curvature metrics.
This paper investigates the question of which smooth compact 4-manifolds admit Riemannian metrics that minimize the L2-norm of the curvature tensor. Metrics with this property are called OPTIMAL; Einstein metrics and scalar-flat…
Inspired by Goette-Semmelmann \cite{GSSU2002}, we derive an estimate for the scalar curvature without a nonnegativity assumption on curvature operator. As an application, we show that, on an even dimensional closed manifold with nonzero…
In this paper we describe the topology of 4-dimensional closed orientable Riemannian manifolds with a uniform lower bound of sectional curvature and with a uniform upper bound of diameter which collapse to metric spaces of lower dimensions.…
We classify, up to homeomorphisms, the closed simply-connected 4-manifolds that admit a Riemannian metric for which averages of pairs of sectional curvatures of orthogonal planes are positive.
Seiberg-Witten theory leads to a delicate interplay between Riemannian geometry and smooth topology in dimension four. In particular, the scalar curvature of any metric must satisfy certain non-trivial estimates if the manifold in question…
In the article we introduce new conformal and smooth invariants on compact, oriented four-manifolds with boundary. In the first part, we show that "positivity" conditions on these invariants will impose topological restrictions on…
We examine relations between geometry and the associated curvature decompositions in Weyl geometry.
In this work we prove that for a compact odd-dimensional orbifold its Euler characteristic is half of the Euler characteristic of its boundary.
Comparisons on $L^{n\over 2}$-norms of scalar curvatures between Riemannian metrics and standard metrics are obtained. The metrics are restricted to conformal classes or under certain curvature conditions.
Let (M,J) be a minimal compact complex surface of Kaehler type. It is shown that the smooth 4-manifold M admits a Riemannian metric of positive scalar curvature iff (M,J) admits a KAEHLER metric of positive scalar curvature. This extends…
On Riemannian manifolds of dimension 4, for prescribed scalar curvature equation, under lipschitzian condition on the prescribed curvature, we have an uniform estimate for the solutions of the equation if we control their minimas.
We show any Weyl curvature model can be geometrically realized by a Weyl manifold
We continue the study of the question of when a pseudo-Riemannain manifold can be locally characterised by its scalar polynomial curvature invariants (constructed from the Riemann tensor and its covariant derivatives). We make further use…