相关论文: Self-Dual Manifolds with Positive Ricci Curvature

In Riemannian geometry the prescribed Ricci curvature problem is as follows: given a smooth manifold $M$ and a symmetric 2-tensor $r$, construct a metric on $M$ whose Ricci tensor equals $r$. In particular, DeTurck and Koiso proved the…

We show that a uniformly Euclidean metric with isolated singularity on $M^n = T^n \# M_0$, where $4\leq n\leq 7$ or $n\geq 4$, $M_0$ spin, and nonnegative scalar curvature on the smooth part is Ricci flat and extends smoothly over the…

The present paper deals with the study of totally real submanifolds and $\textit{C}$-totally real submanifolds of $(LCS)_n$-manifolds with respect to Levi-Civita connection as well as quarter symmetric metric connection. It is proved that…

We show that the positive mass theorem holds for continuous Riemannian metrics that lie in the Sobolev space $W^{2, n/2}_{loc}$ for manifolds of dimension less than or equal to $7$ or spin-manifolds of any dimension. More generally, we give…

In this second part of our overview of the different metric curvatures and their various applications, we concentrate on the Ricci curvature and flow for polyhedral surfaces and higher dimensional manifolds, and we largely review our…

We show that a shrinking Ricci soliton with positive sectional curvature must be compact. This extends a result of Perelman in dimension three and improves a result of Naber in dimension four, respectively.

In this paper, we show the existence of (co-oriented) contact structures on certain classes of $G_2$-manifolds, and that these two structures are compatible in certain ways. Moreover, we prove that any seven-manifold with a spin structure…

We show that each of the topological 4-manifolds $CP^2#k\bar{CP^2}, for $k = 6, 7$ admits a smooth structure which has an Einstein metric of scalar curvature $s > 0$, a smooth structure which has an Einstein metric with $s < 0$ and…

We show that in every dimension $n \geq 8$, there exists a smooth closed manifold $M^n$ which does not admit a smooth positive scalar curvature ("psc") metric, but $M$ admits an $\mathrm{L}^\infty$-metric which is smooth and has psc outside…

In this work, we consider sequences of $C^2$ metrics which converge to a $C^2$ metric in $C^0$ sense. We show that if the scalar curvature of the sequence is almost non-negative in the integral sense, then the limiting metric has scalar…

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…

We prove that various spaces of constrained positive scalar curvature metrics on compact 3-manifolds with boundary, when not empty, are contractible. The constraints we mostly focus on are given in terms of local conditions on the mean…

We study a conformal flow for compact Riemannian manifolds of dimension greater than two with boundary. Convergence to a scalar-flat metric with constant mean curvature on the boundary is established in dimensions up to seven, and in any…

The problem of defining correctly geometric objects such as the curvature is a hard one in discrete geometry. In 2009, Ollivier defined a notion of curvature applicable to a wide category of measured metric spaces, in particular to graphs.…

We formulate natural conformally invariant conditions on a 4-manifold for the existence of a metric whose Schouten tensor satisfies a quadratic inequality. This inequality implies that the eigenvalues of the Ricci tensor are positively…

Let $M_\Sigma$ be an $n$-dimensional Thom-Mather stratified space of depth $1$. We denote by $\beta M$ the singular locus and by $L$ the associated link. In this paper we study the problem of when such a space can be endowed with a wedge…

We give a general procedure for gluing together possibly noncompact manifolds of constant scalar curvature which satisfy an extra nondegeneracy hypothesis. Our aim is to provide a simple paradigm for making `analytic' connected sums. In…

Rigidity results are obtained for Riemannian $d$-manifolds with $\sec \geqslant 1$ and spherical rank at least $d-2>0$. Conjecturally, all such manifolds are locally isometric to a round sphere or complex projective space with the…

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 problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension $n\geq 3$. We prove the existence of such…