Related papers: A Generalization of the Geroch Conjecture with Arb…

We prove that the moduli space of complete Riemannian metrics of bounded geometry and uniformly positive scalar curvature on an orientable 3-manifold is path-connected. This generalizes the main result of the fourth author [Mar12] in the…

In this note we show that a closed Riemannian manifold does not admit a fill-in with nonnegative scalar curvature if the mean curvature is point-wise large. Similar result also holds for fill-ins with a negative scalar curvature lower…

In this paper, we consider {\em mixed curvature} $\mathcal{C}_{a,b}$, which is a convex combination of Ricci curvature and holomorphic sectional curvature introduced by Chu-Lee-Tam. We prove that if a compact complex manifold $M$ admits a…

Positiveness of scalar curvature and Ricci curvature requires vanishing the obstruction $\theta(M)$ which is computed in some KK-theory of C*-algebras index as a pairing of spin Dirac operator and Mishchenko bundle associated to the…

We prove that an admissible manifold (as defined by Apostolov, Calderbank, Gauduchon and T{\o}nnesen-Friedman), arising from a base with a local K\"ahler product of constant scalar curvature metrics, admits Generalized Quasi-Einstein…

Combining the intrinsic and extrinsic geometry, we generalize Einstein manifolds to Integral-Einstein (IE) submanifolds. A Takahashi-type theorem is established to characterize minimal hypersurfaces with constant scalar curvature (CSC) in…

We define enlargeable length-structures on closed topological manifolds and then show that the connected sum of a closed $n$-manifold with an enlargeable Riemannian length-structure with an arbitrary closed smooth manifold carries no…

In 2014, Gromov asked if nonnegative scalar curvature is preserved under intrinsic flat convergence. Here we construct a sequence of closed oriented Riemannian $n$-manifolds, $n\geq 3$, with positive scalar curvature such that their…

In 2014, Gromov conjectured that sequences of manifolds with nonnegative scalar curvature should have subsequences which converge in some geometric sense to limit spaces with some notion of generalized nonnegative scalar curvature. In…

We study the question of existence of a Riemannian metric of positive scalar curvature metric on manifolds with the Sullivan-Baas singularities. The manifolds we consider are Spin and simply connected. We prove an analogue of the…

Suppose $M$ is a closed $n$-dimensional spin$^c$ manifold with spin$^c$ structure $\sigma$ and associated spin$^c$ line bundle $L$. If one fixes a Riemannian metric $g$ on $M$ and a connection $\nabla_L$ on $L$, the generalized scalar…

We establish new mean curvature rigidity theorems for spin fill-ins with non-negative scalar curvature using two different spinorial techniques. Our results address two questions by Miao and Gromov, respectively. The first technique is…

For a compact K\"{a}hler-Einstein manifold $M$ of dimension $n\ge 2$, we explicitly write the expression $-c_1^n(M)+\frac{2(n+1)}{n}c_2(M)c_1^{n-2}(M)$ in the form of certain integral on the holomorphic sectional curvature and its average…

We prove several finiteness results for the class $M_{a,b,G,n}$ of $n$-manifolds that have fundamental groups isomorphic to $G$ and that can be given complete Riemannian metrics of sectional curvatures within $[a,b]$ where $a\le b<0$. In…

The Shapley-Folkman theorem shows that Minkowski averages of uniformly bounded sets tend to be convex when the number of terms in the sum becomes much larger than the ambient dimension. In optimization, Aubin and Ekeland [1976] show that…

In this paper, for any compact Lie group $G$, we show that the space of $G$-invariant Riemannian metrics with positive scalar curvature (PSC) on any closed three-manifold is either empty or contractible. In particular, we prove the…

We show that the existence of an embedded compact, boundaryless hypersurface S of strictly positive mean curvature in a noncompact, connected, complete Riemannian n-manifold N of nonnegative Ricci curvature implies that the homomorphism…

In this paper we use stable capillary surfaces (analogous to the $\mu$-bubble construction) to study manifolds with strictly mean convex boundary and nonnegative scalar curvature. We give an obstruction to filling 2-manifolds by such…

Classical results of Milnor, Wood, Mather, and Thurston produce flat connections in surprising places. The Milnor-Wood inequality is for circle bundles over surfaces, whereas the Mather-Thurston Theorem is about cobording general manifold…

We give examples of asymptotically flat three-manifolds $(M,g)$ which admit arbitrarily large constant mean curvature spheres that are far away from the center of the manifold. This resolves a question raised by G. Huisken and S.-T. Yau in…