相关论文: Complete surfaces with negative extrinsic curvatur…

Let $n\ge 2$ and $k\ge 1$ be two integers. Let $M$ be an isometrically immersed closed $n$-submanifold of co-dimension $k$ that is homotopic to a point in a complete manifold $N$, where the sectional curvature of $N$ is no more than…

We prove that every complete, minimally immersed submanifold $f\: M^n \to \mathbb{S}^{n+p}$ whose second fundamental form satisfies $|A|^2 \le np/(2p-1)$, is either totally geodesic, or (a covering of) a Clifford torus or a Veronese surface…

We prove that every open Riemann surface $M$ is the complex structure of a complete surface of constant mean curvature 1 (CMC-1) in the 3-dimensional hyperbolic space $\mathbb{H}^3$. We go further and establish a jet interpolation theorem…

We show that the image of a nonconstant conformal harmonic map $\mathbb C\to \mathbb R^3$, not necessarily proper and possibly with branch points, intersects every properly embedded nonflat minimal surface of bounded curvature in $\mathbb…

We construct a weakly complete flat surface in hyperbolic 3-space having a pair of hyperbolic Gauss maps both of whose images are contained in an arbitrarily given open disc in the ideal boundary of H^3. This construction is accomplished as…

In this note, we prove that for a complete noncompact three dimensional Riemannian manifold with bounded sectional curvature, if it has uniformly positive scalar curvature, then there is a uniform lower bound on the injectivity radius.

Using harmonic mean curvature flow, we establish a sharp Minkowski type lower bound for total mean curvature of convex surfaces with a given area in Cartan-Hadamard 3-manifolds. This inequality also improves the known estimates for total…

In this paper, we first study isometric immersions $f: M^n\rightarrow M^{n+k}(c), n\geq 3,$ into space forms with flat normal bundle and constant scalar curvature $R.$ Under a suitable multiplicity condition on the second fundamental form…

We study the old problem of isometrically embedding a 2-dimensional Riemannian manifold into Euclidean 3-space. It is shown that if the Gaussian curvature vanishes to finite order and its zero set consists of two Lipschitz curves…

The theory of complete surfaces of (nonzero) constant mean curvature in $\RR^3$ has progressed markedly in the last decade. This paper surveys a number of these developments in the setting of Alexandrov embedded surfaces; the focus is on…

This paper concerns the global theory of properly embedded spacelike surfaces in three-dimensional Minkowski space in relation to their Gaussian curvature. We prove that every regular domain which is not a wedge is uniquely foliated by…

We prove the nonexistence of stable immersed minimal surfaces uniformly conformally equivalent to the complex plane in any complete orientable four-dimensional Riemannian manifold with uniformly positive isotropic curvature. We also…

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…

We consider a non-negative biminimal properly immersed submanifold $M$ (that is, a biminimal properly immersed submanifold with $\lambda\geq0$) in a complete Riemannian manifold $N$ with non-positive sectional curvature. Assume that the…

We will prove that \emph{there are no stable complete hypersurfaces of $\mathbb{R}^4$ with zero scalar curvature, polynomial volume growth and such that $\dfrac{(-K)}{H^3}\geq c>0$ everywhere, for some constant $c>0$}, where $K$ denotes the…

We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold M. Under the assumption that the sectional curvature of M is strictly positive, we prove the existence of a smoothly immersed sphere…

We use pinched smooth hyperbolization to show that every closed, nonpositively curved $n$-dimensional manifold $M$ can be embedded as a totally geodesic submanifold of a closed, nonpositively curved $(n+1)$-dimensional manifold $\hat{M}$ of…

Given a compact 3-manifold N without boundary, we prove that for a bumpy metric of positive scalar curvature the space of minimal surfaces having a uniform upper bound on the Morse index is always finite unless the manifold itself contains…

Among other things, we prove the following two topologcal statements about closed hyperbolic 3-manifolds. First, every rational second homology class of a closed hyperbolic 3-manifold has a positve integral multiple represented by an…

We prove that for any given compact Riemannian manifold $N$ of dimension $n+1 \geq 3$ and any non-negative Lipschitz function $g$ on $N$, there exists a quasi-embedded, boundaryless hypersurface $M \subset N,$ of class $C^{2, \alpha}$ for…