Related papers: On the flatness of Riemannian cylinders without co…
In this article, we propose the following conjecture: if the Strominger connection of a compact Hermitian manifold has constant non-zero holomorphic sectional curvature, then the Hermitian metric must be K\"ahler. The main result of this…
In this paper, we prove a rigidity theorem for Poincar\'e-Einstein manifolds whose conformal infinity is a flat Euclidean space. The proof relies on analyzing the propagation of curvature tensors over the level sets of an adapted boundary…
We study $n$-dimensional Ricci flows with non-negative Ricci curvature where the curvature is pointwise controlled by the scalar curvature and bounded by $C/t$, starting at metric cones which are Reifenberg outside the tip. We show that any…
Aim of this paper is to discuss convergence of pointed metric measure spaces in absence of any compactness condition. We propose various definitions, show that all of them are equivalent and that for doubling spaces these are also…
This article surveys results for Riemannian manifolds of positive and non-negative sectional curvature with symmetries.
In this paper we take a look at conditions that make a Riemann soliton trivial, compacity being one of them. We also show that the behaviour at infinity of the gradient field of a non-compact gradient Riemann soliton might cause the soliton…
We prove the following result: Let $(M,g_0)$ be a complete noncompact manifold of dimension $n\geq 12$ with isotropic curvature bounded below by a positive constant, with scalar curvature bounded above, and with injectivity radius bounded…
Let $M$ be a complete Riemannian manifold possessing a strictly convex Lipschitz continuous exhaustion function. We show that the isoperimetric profile of $M$ is a continuous and non-decreasing function. Particular cases are Hadamard…
In 1996, Huisken-Yau proved that every three-dimensional Riemannian manifold can be uniquely foliated near infinity by stable closed surfaces of constant mean curvature (CMC) if it is asymptotically equal to the (spatial) Schwarzschild…
We study compact Riemannian manifolds for which the light between any pair of points is blocked by finitely many point shades. Compact flat Riemannian manifolds are known to have this finite blocking property. We conjecture that amongst…
We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature uniformly bounded from below and diameter uniformly bounded above, Gromov-Hausdorff convergence essentially agrees with…
Let (M,J) be a compact complex 2-manifold which which admits a Kaehler metric for which the integral of the scalar curvature is non-negative. Also suppose that M does not admit a Ricci-flat K\"ahler metric. Then if M is blown up at…
In this paper we prove that every Riemannian metric on a locally conformally flat manifold with umbilic boundary can be conformally deformed to a scalar flat metric having constant mean curvature. This result can be seen as a generalization…
We show that after forming a connected sum with a homotopy sphere, all (2j-1)-connected 2j-parallelisable manifolds in dimension 4j+1, j > 0, can be equipped with Riemannian metrics of 2-positive Ricci curvature. The condition of 2-positive…
In this paper, we bend a closed Riemannian manifold in the conformal class, through solving a fully nonlinear equation. As a result, we prove that each metric of quasi-negative Ricci curvature is conformal to a metric with negative Ricci…
Consider a manifold with boundary, and such that the interior is equipped with a pseudo-Riemannian metric. We prove that, under mild asymptotic non-vanishing conditions on the scalar curvature, if the Levi-Civita connection of the interior…
In this paper, we investigate geometric conditions for isometric immersions with positive index of relative nullity to be cylinders. There is an abundance of noncylindrical $n$-dimensional minimal submanifolds with index of relative nullity…
We establish a uniform entropy bound for simply connected Ricci shrinkers with a finite second homotopy group and a uniform curvature bound. Additionally, we extend the non-collapsing result to a broader class of smooth metric measure…
The present work establishes necessary and sufficient conditions for a nonlinear system with two inputs to be described by a specific triangular form. Except for some regularity conditions, such triangular form is flat. This may lead to the…
Motivated by the quasi-local mass problem in general relativity, we apply the asymptotically flat extensions, constructed by Shi and Tam in the proof of the positivity of the Brown--York mass, to study a fill-in problem of realizing…