Search arXiv⌕ Search

arXiv · 0704.2102

A Remark on Compact Minimal Surfaces in $S^5$ with Non-Negative Gaussian Curvature

Abstract

In this paper we classify compact minimal surfaces in $S^5$ with non-negative Gaussian curvature using the notion of a contact angle.

Explore related subjects

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Rodrigo Ristow Montes. 2007-04-17. A Remark on Compact Minimal Surfaces in $S^5$ with Non-Negative Gaussian Curvature. https://arxiv.org/abs/0704.2102

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

Spiral Harmonic Products

Motivated by the systematic study of spiral minimal products, we consider harmonic maps obtained by coupling two spherical eigenmaps through a profile curve in $\mathbb S^3$, with a doubly warped product metric on the source. The source warps are independent of the profile magnitudes. This gives two complementary problems: reconstructing a compatible source metric from prescribed profile or source data, and finding closed profiles when the source metric is fixed. For arbitrary prescribed warps, homogenization of the reduced Lagrangian gives a strongly convex conic Finsler metric whose oriented graph geodesics are exactly the harmonic profiles. In the unit-speed inverse problem, the phase momenta reduce reconstruction to a scalar magnitude equation. We obtain global reconstructions from a prescribed magnitude, one source warp, or a monotone volume factor, together with periodic families in which profiles closing on finite covers are dense. In the fixed-metric problem, a change of time gives an autonomous Neumann system. One fixed source metric then carries continuous families of closed profiles and infinitely many closed graph geodesics. For fixed even $p,q\geq 2$, every fixed metric in this Neumann class on $\mathbb S^1\times\mathbb S^p\times\mathbb S^q$ supports harmonic maps into $\mathbb S^{p+q+1}$ of degree $4m$ for every $m\geq 1$. In the matched constant-warp case, these maps reduce to explicit eigenmaps.

math.DG↗

Gravitational instantons and hyperbolic space

We construct a degeneration of Hitchin's non-standard minitwistor spaces associated with toric gravitational instantons of type A$_{\rm odd}$ to the minitwistor space of a simple hyperbolic orbifold. We show that the corresponding families of minitwistor lines also converge under this degeneration. Consequently, the associated non-standard Einstein--Weyl spaces converge to the hyperbolic orbifold.

math.DG↗

Equivariant isometric immersions of surfaces in hyperbolic space

Let $S$ be a closed, oriented surface of genus at least $2$, let $h$ be a smooth Riemannian metric on $S$ with curvature $K\in (-1,0]$, and let $c\in \cT_S$ be a conformal structure on $S$. There exists a unique equivariant isometric immersion of $(S,h)$ in $\HH^3$ such that the pull-back by the hyperbolic Gauss map of the conformal structure at infinity is $c$. Dually, if $h^*$ is a smooth metric on $S$ with curvature $K^*\in (-\infty, 0)$ and if $c\in \cT_S$, there exists a unique equivariant immersion of $S$ into $\HH^3$ with third fundamental form $h^*$ and such that the pull-back of the conformal class at infinity by the Gauss map is $c$. Equivalently, given $h$ and $c\in \cT_S$, there is a unique pair $(E,u)$ where $E$ is a hyperbolic end with conformal structure at infinity $c$ and $u$ is an isometric embedding of $(S,h)$ in $E$. Given $h^*$ and $c$, there exists a unique pair $(E,u^*)$, where $E$ is a hyperbolic end with conformal structure at infinity $c$, and $u^*:S\to E$ is an embedding inducing the third fundamental form $h^*$. Those statements can be considered as a smooth counterpart of known or conjectural statements on the grafting map and on circle patterns.

math.DG↗