Related papers: Three-manifolds with constant vector curvature

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…

Let $(M,g)$ be an asymptotically hyperbolic manifold with a smooth conformal compactification. We establish a general correspondence between semilinear elliptic equations of scalar curvature type on $\del M$ and Weingarten foliations in…

We introduce the notion of conformal trajectories in three-dimensional Riemannian manifolds $M^3$. Given a conformal vector field $V\in\mathfrak{X}(M^3)$, a conformal trajectory of $V$ is a regular curve $\gamma$ in $M^3$ satisfying…

We prove that if an orientable 3-manifold $M$ admits a complete Riemannian metric whose scalar curvature is positive and has a subquadratic decay at infinity, then it decomposes as a (possibly infinite) connected sum of spherical manifolds…

A triharmonic map is a critical point of the tri-energy in the space of smooth maps between two Riemannian manifolds. In this paper, we prove that if $M^n (n\ge 4)$ is a CMC proper triharmonic hypersurface in a space form…

Let M be an almost Hermitian manifold of dimension greater or equal to 6. The following theorems are proved: Theorem 1. If M is of pointwise constant {\theta}-holomorphic sectional curvature for a number {\theta} in (0,{\pi}/2) then M is of…

We propose a notion of scalar curvature lower bounds in a three-dimensional Riemannian manifold endowed with a $C^0$ metric based on the monotonicity of the Hawking mass along the inverse mean curvature flow. We present a stability theorem…

In this paper we prove that any complete locally conformally flat quasi-Einstein manifold of dimension $n\geq 3$ is locally a warped product with $(n-1)$-dimensional fibers of constant curvature. This result includes also the case of…

Let $(M^{n+1},g)$ be a closed Riemannian manifold, $n+1\geq 3$. We will prove that for all $m \in \mathbb{N}$, there exists $c^{*}(m)>0$, which depends on $g$, such that if $0<c<c^{*}(m)$, $(M,g)$ contains at least $m$ many closed $c$-CMC…

A 3-dimensional Riemannian manifold equipped with a tensor structure of type $(1,1)$, whose fourth power is the identity, is considered. This structure acts as an isometry with respect to the metric. A Riemannian almost product manifold…

We show that a compact Riemannian $3$-manifold $M$ with strictly convex simply connected boundary and sectional curvature $K\leq a\leq 0$ is isometric to a convex domain in a complete simply connected space of constant curvature $a$,…

For k at least 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not locally affine homogeneous (and hence not locally homogeneous). The curvature tensor of these manifolds is modeled on…

A submanifold of a pseudo-Riemannian manifold is said to have parallel mean curvature vector if the mean curvature vector field H is parallel as a section of the normal bundle. Submanifolds with parallel mean curvature vector are important…

We establish quantitative topological and singularity properties for (certain) prescribed mean curvature (PMC) hypersurfaces $V^n$ in Riemannian manifolds $(N^{n+1},h)$. Indeed, if $V$ has area at most $A>0$ with PMC given by a…

We apply the Riemannian Penrose inequality and the Riemannian positive mass theorem to derive inequalities on the boundary of a class of compact Riemannian $3$-manifolds with nonnegative scalar curvature. The boundary of such a manifold has…

In this paper, we can prove the existence and uniqueness of solutions to the constant mean curvature (CMC for short) equation with nonzero Neumann boundary data in product manifold $M^{n}\times\mathbb{R}$, where $M^{n}$ is an…

By ECS manifolds one means pseudo-Riemannian manifolds of dimensions $\,n\ge4\,$ which have parallel Weyl tensor, but not for one of the two obvious reasons: conformal flatness or local symmetry. As shown by Roter [10, 2], they exist for…

We prove that every closed, smooth $n$-manifold $X$ admits a Riemannian metric together with a smooth, transversely oriented CMC foliation if and only if its Euler characteristic is zero, where by CMC foliation we mean a codimension-one,…

Given a closed 3-manifold with an initial Riemannian metric of negative sec- tional curvature, we consider the cross curvature flow an evolution equation of metric on M3. We prove long-time existence of a solution to the cross curvature…

In this paper we consider three-manifolds with weakly umbilic boundary (the Second Fundamental form of the boundary is a constant multiple of the metric). We show that if the initial manifold has positive Ricci curvature and the boundary is…