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We find constant scalar curvature Type-N and Type-D solutions in all higher curvature gravity theories with actions of the form f(Ricci) that are built on the Ricci tensor, but not on its derivatives. In our construction, these higher…

In this paper, we investigate classifications of $4$-dimensional simply connected complete noncompact nonflat shrinkers satisfying $Ric+\mathrm{Hess}\,f=\tfrac 12g$ with nonnegative Ricci curvature. One one hand, we show that if the…

We explore to what extent one may hope to preserve geometric properties of three dimensional manifolds with lower scalar curvature bounds under Gromov-Hausdorff and Intrinsic Flat limits. We introduce a new construction, called sewing, of…

In this paper, we present finite topological type theorems for open manifolds with non-negative Ricci curvature, under almost maximal local rewinding volume. Unlike previous related research, our theorems remove the constraints of sectional…

In this paper we prove that, under an explicit integral pinching assumption between the $L^2$-norm of the Ricci curvature and the $L^2$-norm of the scalar curvature, a closed 3-manifold with positive scalar curvature admits an Einstein…

We introduce a metric notion of Ricci curvature for $PL$ manifolds and study its convergence properties. We also prove a fitting version of the Bonnet-Myers Theorem, for surfaces as well as for a large class of higher dimensional manifolds.

The Theorem of Bonnet--Myers implies that manifolds with topology $M^{n-1} \times \mathbb{S}^1$ do not admit a metric of positive Ricci curvature, while the resolution of Geroch's conjecture implies that the torus $\mathbb{T}^n$ does not…

In this paper, we prove the three-dimensional $CPE$ conjecture with non-negative Ricci curvature. Moreover, we establish a classification result on three-dimensional vacuum static space with non-negative Ricci curvature. Finally, we show…

We study three-dimensional Alexandrov spaces with a lower curvature bound, focusing on extending three classical results on three-dimensional manifolds: First, we show that a closed three-dimensional Alexandrov space of positive curvature,…

We consider Ricci flow of complete Riemannian manifolds which have bounded non-negative curvature operator, non-zero asymptotic volume ratio and no boundary. We prove scale invariant estimates for these solutions. Using these estimates, we…

In this paper, we study the injectivity radius bound for 3-d Ricci flow. As applications we show the long time existence of the Ricci flow with positive Ricci curvature. We also partially settle a question in page 302 of the book of…

We show $L^1$-bounds of the Riemann curvature tensor on a smooth closed $n$-dimensional Ricci flow. To achieve this we introduce the notion of a neck of maximal symmetry, similar to the one in Cheeger-Jiang-Naber and Jiang-Naber and…

Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, has…

We show that any Riemannian metric conformal to the round metric on $S^n$, for $n\geq 4$, arises as a limit of a sequence of Riemannian metrics of positive scalar curvature on $S^n$ in the sense of uniform convergence of Riemannian…

We show that for a complete Ricci shrinker there exists a sequence of points tending to infinity whose norms of the Ricci tensor grow at most linearly.

For an immortal Ricci flow on an $m$-dimensional $(m\ge 3)$ closed manifold, we show the following convergence results: (1) if the curvature and diameter are uniformly bounded, then any unbounded sequence of time slices sub-converges to a…

Discrete forms of the scalar, sectional and Ricci curvatures are constructed on simplicial piecewise flat triangulations of smooth manifolds, depending directly on the simplicial structure and a choice of dual tessellation. This is done by…

We study the geometry and topology of Riemannian 3-orbifolds which are locally volume collapsed with respect to a curvature scale. We show that a sufficiently collapsed closed 3-orbifold without bad 2-suborbifolds either admits a metric of…

There is a conjecture that a complete Riemannian 3-manifold with bounded sectional curvature, and pointwise pinched nonnegative Ricci curvature, must be flat or compact. We show that this is true when the negative part (if any) of the…

We prove that Ahlfors-regular RCD spaces are uniformly rectifiable. The same is shown for Ahlfors regular boundaries of non-collapsed RCD spaces. As an application we deduce a type of quantitative differentiation for Lipschitz functions on…