Related papers: Extremality and rigidity for scalar curvature in d…

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.

We give lower bounds, in terms of the Euler characteristic, for the $L^2$-norm of the Weyl curvature of closed Riemannian 4-manifolds. The same bounds were obtained by Gursky, in the case of positive scalar curvature metrics.

We study critical Riemannian 4-manifolds with a lower bound on Ricci curvature, but no a priori analytic constraints such as on Sobolev constants. We derive elliptic-type estimates for the local curvature radius, which itself controls…

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.…

We prove that any smooth Riemannian manifold of non-negative scalar curvature and with a strictly mean convex and compact boundary component can be (C^2) extended beyond the component to have non-negative scalar curvature and to enjoy…

We prove that every locally conformally flat metric on a closed, oriented hyperbolic 4-manifold with scalar curvature bounded below by -12 satisfies Schoen's conjecture. We also classify all closed Riemannian 4-manifolds of positive scalar…

Sectional curvature bounds are of central importance in the study of Riemannian manifolds, both in smooth differential geometry and in the generalized synthetic setting of Alexandrov spaces. Riemannian metrics along with metric spaces of…

We classify positively curved Alexandrov spaces of dimension 4 with an isometric circle action up to equivariant homeomorphism, subject to a certain additional condition on the infinitesimal geometry near fixed points which we conjecture is…

In this paper, we study closed four-dimensional manifolds. In particular, we show that under various new pinching curvature conditions (for example, the sectional curvature is no more than 5/6 of the smallest Ricci eigenvalue) then the…

We prove a generalization of Gromov's conjecture on scalar curvature rigidity of convex polytopes to arbitrary convex Riemannian polytope type domains via harmonic spinors on convex domians with boundary condition constructed by Brendle. In…

We consider a 4-dimensional Riemannian manifold M endowed with a right skew-circulant tensor structure S, which is an isometry with respect to the metric g and the fourth power of S is minus identity. We determine a class of manifolds (M,…

We prove that a locally compact space with an upper curvature bound is a topological manifold if and only if all of its spaces of directions are homotopy equivalent and not contractible. We discuss applications to homology manifolds, limits…

We prove a splitting theorem for Riemannian n-manifolds with scalar curvature bounded below by a negative constant and containing certain area-minimising hypersurfaces (Theorem 3). Thus we generalise [25,Theorem 3] by Nunes. This splitting…

We prove explicit upper and lower bounds for the torsional rigidity of extrinsic domains of submanifolds P^m with controlled radial mean curvature in ambient Riemannian manifolds N^n with a pole p and with sectional curvatures bounded from…

In this article, we extend the example constructed in the paper by Sormani-Tian-Wang to build new examples that satisfy the assumptions of the conjecture by Gromov. Each of these new examples of sequence converges to a limit space with…

Let $\big(M,g^{TM}\big)$ be a noncompact complete spin Riemannian manifold of even dimension $n$, with $k^{TM}$ denote the associated scalar curvature. Let $f\colon M\rightarrow S^{n}(1)$ be a smooth area decreasing map, which is locally…

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…

We investigate the geometry and topology of extremal domains in a manifold with negative sectional curvature. An extremal domain is a domain that supports a positive solution to an overdetermined elliptic problem (OEP for short). We…

In this paper we introduce two new notions of sectional curvature for Riemannian manifolds with density. Under both notions of curvature we classify the constant curvature manifolds. We also prove generalizations of the theorems of…

By making use of the classification of real simple Lie algebra, we get the maximum of the squared length of restricted roots case by case, thus we get the upper bounds of sectional curvature for irreducible Riemannian symmetric spaces of…