A compactness theorem for surfaces withL p -bounded second fundamental form

Langer, Joel

doi:10.1007/bf01456183

Cited by 76 publications

(80 citation statements)

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“…In [14] Langer proved a compactness theorem for surfaces with A L q ≤ Λ for q > 2, using that the surfaces are represented as C 1 -bounded graphs over discs of radius r(n, q, Λ) > 0. Clearly, the relevant Sobolev embedding fails for q = 2.…”

Section: Introductionmentioning

confidence: 99%

$W^{2,2}$-conformal immersions of a closed Riemann~surface into $\R^n$

Kuwert

2012

Communications in Analysis and Geometry

124

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We study sequences f k : Σ k → R n of conformally immersed, compact Riemann surfaces with fixed genus and Willmore energy W(f ) ≤ Λ. Assume that Σ k converges to Σ in moduli space, i.e. φ * k (Σ k ) → Σ as complex structures for diffeomorphisms φ k . Then we construct a branched conformal immersion f : Σ → R n and Möbius transformations σ k , such that for a subsequenceloc away from finitely many points. For Λ < 8π the map f is unbranched. If the Σ k diverge in moduli space, then we show lim inf k→∞ W(f k ) ≥ min(8π, ω n p ). Our work generalizes results in [K-S2] to arbitrary codimension.

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Section: Introductionmentioning

confidence: 99%

$W^{2,2}$-conformal immersions of a closed Riemann~surface into $\R^n$

Kuwert

2012

Communications in Analysis and Geometry

124

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“…A strong form of the required compactness theorem has been proved by Langer [6] for the case of compact two-manifolds immersed in Euclidean three-space. A similar result should be true in the general case (with p > dim M), but here we accept the more limited objective of proving 0.2.…”

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confidence: 99%

Immersions with bounded curvature

Corlette

1990

Geom Dedicata

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This article proves that the immersions of compact manifolds into a fixed compact Riernannian manifold, with bounds on the second fundamental forms and either the diameter or volume of the induced metrics, fall into only finitely many topological types. FORMULATION OF THE RESULTCheeger's finiteness theorem [-1] yields control over the diffeomorphism class of a compact Riemannian manifold, given bounds on its geometric invariants. The purpose of this paper is to prove an analogous result for immersions of compact manifolds into a fixed Riemannian manifold.Suppose that N is a Riemannian manifold, and f: M ---, N andf': M' ---, N are immersions of compact manifolds into N. DEFINITION 0.1. The immersions f and f' are said to be equivalent if there is a diffeomorphism ~p: M--, M' and a homotopy F: M x [-0, 1]--, N with Fo = f' o ~p, F1 = f, and F, an immersion for any t ~ [0, 1]. F is called a regular homotopy. Associated with an immersion f: M --, N are: the normal bundle vM of M in N, the second fundamental form l-I: TM ® TM -, vM, and Riemannian invariants of the metric on M induced by the immersion, e.g. the volume and diameter of M. The main result is the following. THEOREM 0.2. Let N be a compact Riemannian manifold (without boundary), B, d, and v three positive constants, and J the set of all immersions f: M --, N satisfying: 1. M is a compact manifold (without boundary) 2. I III <<. B, where IH[ is the pointwise operator norm of the second fundamental form, and 3. either diam (M) <<. d or vol(M) ~< v. J contains only finitely many equivalence classes of immersions.A similar result can be given when N is a homogeneous complete Riemannian manifold, by minor modifications of the argument to be given here.Theorem 0.2 probably reflects a compactness result, in much the same way Geometriae Dedicata 33: [153][154][155][156][157][158][159][160][161] 1990.

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“…Hence, we can extract a subsequence of smooth hypersurfaces ϕ i = ϕ ti and diffeomorphisms σ i : M → M such that, for a fixed metric h on M , the sequence {ϕ i •σ i } converges in the H 2,p weak topology to an immersion ψ : M → R n+1 . With the arguments of the proof of Theorem 8.1 in [13,27] and keeping into account that in our case we have also the estimates (7.6), it is possible to conclude that actually the convergence is in the C ∞ topology and the limit hypersurface is smooth (see also [22], Prop. 3.4).…”

Section: Convergencementioning

confidence: 66%

“…To find limit hypersurfaces, we need the following compactness result of Langer and Delladio [13,27].…”

Section: Convergencementioning

confidence: 99%

Smooth geometric evolutions of hypersurfaces

Mantegazza

2002

Geometric And Functional Analysis

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Abstract. We consider the gradient flow associated to the following functionalsThe functionals are defined on hypersurfaces immersed in R n+1 via a map ϕ : M → R n+1 , where M is a smooth closed and connected n-dimensional manifold without boundary.Here µ and ∇ are respectively the canonical measure and the Levi-Civita connection on the Riemannian manifold (M, g), where the metric g is obtained by pulling back on M the usual metric of R n+1 with the map ϕ. The symbol ∇ m denotes the m-th iterated covariant derivative and ν is a unit normal local vector field to the hypersurface. Our main result is that if the order of derivation m ∈ N is strictly larger than the integer part of n/2 then singularities in finite time cannot occur during the evolution.These geometric functionals are related to similar ones proposed by Ennio De Giorgi, who conjectured for them an analogous regularity result. In the final section we discuss the original conjecture of De Giorgi and some related problems.

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A compactness theorem for surfaces withL p -bounded second fundamental form

Cited by 76 publications

References 5 publications

$W^{2,2}$-conformal immersions of a closed Riemann~surface into $\R^n$

$W^{2,2}$-conformal immersions of a closed Riemann~surface into $\R^n$

Immersions with bounded curvature

Smooth geometric evolutions of hypersurfaces

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