Related papers: Ricci surfaces

Biconservative surfaces of Riemannian 3-space forms $N^3(\rho)$, are either constant mean curvature (CMC) surfaces or rotational linear Weingarten surfaces verifying the relation $3\kappa_1+\kappa_2=0$ between their principal curvatures…

We classify all surfaces with constant Gaussian curvature $K$ in Euclidean $3$-space that can be expressed as an implicit equation of type $f(x)+g(y)+h(z)=0$, where $f$, $g$ and $h$ are real functions of one variable. If $K=0$, we prove…

In this paper, we prove that any compact 2-sided smooth stable minimal hypersurface in gradient Ricci soliton $(M^{n},g,f)$ with scalar curvature $R\geq(n-1)\lambda$ must have vanished second fundamental form and vanished normal Ricci…

Let $M$ be a closed hyperbolic 3-manifold that admits no infinitesimal conformally-flat deformations. Examples of such manifolds were constructed by Kapovich. Then if $g$ is a Riemannian metric on $M$ with scalar curvature greater than or…

Let (M,g) be a compact Riemannian manifold of dimension 3, and let \mathscr{F} denote the collection of all embedded surfaces homeomorphic to \mathbb{RP}^2. We study the infimum of the areas of all surfaces in \mathscr{F}. This quantity is…

We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature. We show that if there is an integer $k$ such that any tangent cone at infinity of the Riemannian universal cover of $M$ is a metric cone, whose…

We show that any closed biquotient with finite fundamental group admits metrics of positive Ricci curvature. Also, let M be a closed manifold on which a compact Lie group G acts with cohomogeneity one, and let L be a closed subgroup of G…

Let (M, g) be a compact Einstein Riemannian manifold with boundary. We show that under certain conditions, the map that associates to a metric on M its Ricci curvature, its induced conformal class on the boundary, and its mean curvature on…

This paper is concerned with properties of maximal solutions of the Ricci and cross curvature flows on locally homogeneous three-manifolds of type SL(2,R). We prove that, generically, a maximal solution originates at a sub-Riemannian…

In this paper we consider an inverse problem of determining a minimal surface embedded in a Riemannian manifold. We show under a topological condition that if $\Sigma$ is a $2$-dimensional embedded minimal surface, then the knowledge of the…

Ricci curvature was proposed by Ollivier in a general framework of metric measure spaces, and it has been studied extensively in the context of graphs in recent years. In this paper we prove upper bounds for Ollivier's Ricci curvature for…

Any closed, connected Riemannian manifold $M$ can be smoothly embedded by its Laplacian eigenfunction maps into $\mathbb{R}^m$ for some $m$. We call the smallest such $m$ the maximal embedding dimension of $M$. We show that the maximal…

In the study manifolds of Ricci curvature bounded below, a stumbling obstruction is the lack of links between large-scale geometry and small-scale geometry at a fixed reference point. There have been few links (volume, dimension) when the…

We show that a complete Ricci flow of bounded curvature which begins from a manifold with a Ricci lower bound, local entropy bound, and small local scale-invariant integral curvature control will have global point-wise curvature control at…

We show that, in toric Kaehler geometry, the sign of the Ricci curvature corresponds exactly to convexity properties of the volume functional. We also discuss analogous relationships in the more general context of quasi-homogeneous…

Let $(M^3,g_0)$ be a complete noncompact Riemannian 3-manifold with nonnegative Ricci curvature and with injectivity radius bounded away from zero. Suppose that the scalar curvature $R(x)\to 0$ as $x\to \infty$. Then the Ricci flow with…

Pro and the third author showed that there are Riemannian submersions $\pi: M \to B$ with $M$ a compact manifold with positive Ricci curvature, whose base $B$, has Ricci curvatures with both signs. Thus, Riemannian submersions need not…

We investigate the topology of the compact hypersurfaces in round spheres whose Ricci curvature satisfies an appropriate bound that only depends on the mean curvature of the submanifold. In this paper, the use of the Bochner technique…

We prove nonexistence of nonconstant local minimizers for a class of functionals, which typically appears in the scalar two-phase field model, over a smooth N-dimensional Riemannian manifold without boundary with non-negative Ricci…

In this paper, an n-dimensional complete open manifold with nonnegative Ricci curvature and collapsing volume has been investigated. If its radial sectional curvature bounded from below, it shows that such a manifold is of finite…