• Corpus ID: 229371270

Existence of constant mean curvature 2-spheres in Riemannian 3-spheres

  title={Existence of constant mean curvature 2-spheres in Riemannian 3-spheres},
  author={Da Rong Cheng and Xin Zhou},
We prove the existence of branched immersed constant mean curvature 2spheres in an arbitrary Riemannian 3-sphere for almost every prescribed mean curvature, and moreover for all prescribed mean curvatures when the 3-sphere is positively curved. To achieve this, we develop a min-max scheme for a weighted Dirichlet energy functional. There are three main ingredients in our approach: a bi-harmonic approximation procedure to obtain compactness of the new functional, a derivative estimate of the min… 
Min-max theory for capillary surfaces
We develop a min-max theory for the construction of capillary surfaces in 3manifolds with smooth boundary. In particular, for a generic set of ambient metrics, we prove the existence of nontrivial,
Existence of curves with constant geodesic curvature in a Riemannian 2-sphere
  • D. Cheng, Xin Zhou
  • Mathematics
    Transactions of the American Mathematical Society
  • 2021
We prove the existence of immersed closed curves of constant geodesic curvature in an arbitrary Riemannian 2-sphere for almost every prescribed curvature. To achieve this, we develop a min-max scheme
$L^p$-regularity for fourth order elliptic systems with antisymmetric potentials in higher dimensions
We establish an optimal L-regularity theory for solutions to fourth order elliptic systems with antisymmetric potentials in all supercritical dimensions n ≥ 5: ∆u = ∆(D · ∇u) + div(E · ∇u) + (∆Ω +G)
Morse Index Bound for Minimal Torus
The min-max construction of minimal spheres using harmonic replacement is introduced by Colding and Minicozzi [CM08] and generalized by Zhou [Zho10] to conformal harmonic torus. We prove that the
Motivated by the seminal works of Rivière and Struwe [Comm. Pure. Appl. Math. 2008] and Wang [Comm. Pure. Appl. Math. 2004] and the interesting L regularity theory of Moser [Tran. Amer. Math. Soc.
Prescribed Mean Curvature Min-Max Theory in Some Non-Compact Manifolds
. This paper develops a technique for applying one-parameter prescribed mean curvature min-max theory in certain non-compact manifolds. We give two main applications. First, fix a dimension 3 ≤ n + 1


Existence of hypersurfaces with prescribed mean curvature I – generic min-max
We prove that, for a generic set of smooth prescription functions $h$ on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature $h$. The
Constant mean curvature spheres in homogeneous three-spheres
We give a complete classification of the immersed constant mean curvature spheres in a three-sphere with an arbitrary homogenous metric, by proving that for each $H\in\mathbb{R}$, there exists a
Existence of infinitely many minimal hypersurfaces in positive Ricci curvature
In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infinite number of closed immersed minimal surfaces. We use min–max theory for the area functional to
Min–max theory for constant mean curvature hypersurfaces
In this paper, we develop a min–max theory for the construction of constant mean curvature (CMC) hypersurfaces of prescribed mean curvature in an arbitrary closed manifold. As a corollary, we prove
Degree Theory of Immersed Hypersurfaces
We develop a degree theory for compact immersed hypersurfaces of prescribed $K$-curvature immersed in a compact, orientable Riemannian manifold, where $K$ is any elliptic curvature function. We apply
Constant mean curvature spheres in homogeneous three-manifolds
We prove that two spheres of the same constant mean curvature in an arbitrary homogeneous three-manifold only differ by an ambient isometry, and we determine the values of the mean curvature for
Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes
There has been much interest among differential geometers in finding relationships between curvature and topology of Riemannian manifolds. For the most part, efforts have been directed towards
Constant mean curvature spheres in Riemannian manifolds
We prove the existence of embedded spheres with large constant mean curvature in any compact Riemannian manifold (M, g). This result partially generalizes a result of R. Ye which handles the case
On the existence of embedded minimal 2-spheres in the 3-sphere, endowed with an arbitrary metric
  • F. Smith
  • Mathematics
    Bulletin of the Australian Mathematical Society
  • 1983
First, by modifying the minimax techniques of Pitts [2] , i t is shown that there exists in N a stationary 2-varifold V which can be written as the (varifold) limit of embedded 2-spheres and which
Assume that n ≥ 3 and that we are given a compact (n+ 1)-dimensional Riemannian manifold (M, g) and a compact n-dimensional manifold Λ. We define M(M, g,Λ) to be the set of immersed hypersurfaces in