The existence of least area surfaces in $3$-manifolds
نویسندگان
چکیده
منابع مشابه
The Existence of Least Area Surfaces in 3-manifolds
This paper presents a new and unified approach to the existence theorems for least area surfaces in 3-manifolds. Introduction. A surface F smoothly embedded or immersed in a Riemannian manifold M is minimal if it has mean curvature zero at all points. It is a least area surface in a class of surfaces if it has finite area which realizes the infimum of all possible areas for surfaces in this cla...
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Let M be a Riemannian manifold and let F be a closed surface. A map f: F---,M is called least area if the area of f is less than the area of any homotopic map from F to M. Note that least area maps are always minimal surfaces, but that in general minimal surfaces are not least area as they represent only local stationary points for the area function. The existence of least area immersions in a ...
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Motivated by problems on apparent horizons in general relativity, we prove the following theorem on minimal surfaces: Let g be a metric on the three-sphere S satisfying Ric(g) ≥ 2g. If the volume of (S, g) is no less than one half of the volume of the standard unit sphere, then there are no closed minimal surfaces in the asymptotically flat manifold (S \ {P}, Gg). Here G is the Green’s function...
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We show that if F is a smooth, closed, orientable surface embedded in a closed, orientable 3-manifold M such that for each Riemannian metric g on M , F is isotopic to a least-area surface F (g), then F is incompressible.
متن کاملIncompressibility of Surfaces in Surgered 3-Manifolds
The problem we consider in this paper was raised in [3]. Suppose T is a torus on the boundary of an orientable 3-manifold X, and S is a surface on ∂X − T which is incompressible in X. A slope γ is the isotopy class of a nontrivial simple closed curve on T . Denote by X(γ) the manifold obtained by attaching a solid torus to X so that γ is the slope of the boundary of a meridian disc. Given two s...
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 1988
ISSN: 0002-9947
DOI: 10.1090/s0002-9947-1988-0965747-6