Yamabe flow on manifolds with edges

Bahuaud, Eric; Vertman, Boris

doi:10.1002/mana.201200210

Cited by 33 publications

(73 citation statements)

References 33 publications

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“…To estimate (16) we set a := K 0 κ ∈ [0, 1] and s = κt and observe that the expression Ξ = aψ(s) 1] as long as a ≤ 1 and therefore bounded from above by aψ(0) + ψ ′ (0) = 1 and from below by a 3 ≥ 0. Substituting the term 0 ≤ Ξ ≤ 1 in (16), we conclude…”

Section: Local Estimatesmentioning

confidence: 99%

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Instantaneously complete Yamabe flow on hyperbolic space

Schulz

2019

Calc. Var.

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We prove global existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension m ≥ 3. The initial metric is assumed to be conformally hyperbolic with conformal factor and scalar curvature bounded from above. We do not require initial completeness or bounds on the Ricci curvature. If the initial data are rotationally symmetric, the solution is proven to be unique in the class of instantaneously complete, rotationally symmetric Yamabe flows.

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Section: Local Estimatesmentioning

confidence: 99%

“…In the interval [1,2], the function ϕ can be chosen explicitly as a polynomial and it can be arranged that |ϕ normal coordinates with origin p 0 on (H, g H ). We introduce a parameter A > 1 to define the functions w(r, ϑ) is non-decreasing, as w it is assumed to be rotationally symmetric.…”

Section: Upper and Lower Boundsmentioning

confidence: 99%

Instantaneously complete Yamabe flow on hyperbolic space

Schulz

2019

Calc. Var.

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“…converges in C 0,α (M 2 × [0, T ]) to some solution V 2 of equation (1) in M 2 × [0, T ]. Iterating this procedure leads to a diagonal subsequence of {U k } 4<k which converges to a limit U satisfying the Yamabe flow equation (1) in M ×[0, T ]. Moreover, the uniform bounds from Lemmata 1.1 and 2.1 are preserved in the limit and by construction, the initial condition is satisfied.…”

Section: Scalar Curvature Estimatesmentioning

confidence: 99%

Yamabe Flow on Non-compact Manifolds with Unbounded Initial Curvature

Schulz

2019

J Geom Anal

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We prove global existence of Yamabe flows on non-compact manifolds M of dimension m ≥ 3 under the assumption that the initial metric g 0 = u 0 g M is conformally equivalent to a complete background metric g M of bounded, non-positive scalar curvature and positive Yamabe invariant with conformal factor u 0 bounded from above and below. We do not require initial curvature bounds. In particular, the scalar curvature of (M, g 0 ) can be unbounded from above and below without growth condition.Richard Hamilton's [10] Yamabe flow describes a family of Riemannian metrics g(t) subject to the evolution equation ∂ ∂t g = −R g g, where R g denotes the scalar curvature corresponding to the metric g. This equation tends to conformally deform a given initial metric towards a metric of vanishing scalar curvature. Hamilton proved existence of Yamabe flows on compact manifolds without boundary. Their asymptotic behaviour was subsequently analysed by Chow [6], Ye [19], Schwetlick and Struwe [15] and Brendle [3,4]. The theory of Yamabe flows on non-compact manifolds is not as developed as in the compact case. Daskalopoulos and Sesum [7] analysed the profiles of self-similar solutions (Yamabe solitons). The question of existence in general was addressed by Ma and An who obtained the following results on complete, non-compact Riemannian manifolds (M, g 0 ) satisfying certain curvature assumptions:• If (M, g 0 ) has Ricci curvature bounded from below and uniformly bounded, nonpositive scalar curvature, then there exists a global Yamabe flow on M with g 0 as initial metric [13].

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“…We mention the work of Bahuaud, Dryden and Vertman [3,4] as initial steps. However, this problem has turned out to present some formidable technical obstacles and further progress has been difficult.…”

Section: Extensionsmentioning

confidence: 99%

Sharp parabolic regularity and geometric flows on singular spaces

Mazzeo¹

2016

Journées Équations Aux Dérivées Partielles

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This is a brief survey about regularity expansions for solutions of elliptic and parabolic problems on spaces with conic singularities. The results themselves are closely related to classical results about elliptic boundary problems, and analogues of these are expected to hold on quite general stratified spaces with incomplete iterated edge metrics. The emphasis here is on the interpretation and application of these expansions to geometric problems.

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Yamabe flow on manifolds with edges

Cited by 33 publications

References 33 publications

Instantaneously complete Yamabe flow on hyperbolic space

Instantaneously complete Yamabe flow on hyperbolic space

Yamabe Flow on Non-compact Manifolds with Unbounded Initial Curvature

Sharp parabolic regularity and geometric flows on singular spaces

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