On diameter controls and smooth convergence away from singularities

Lakzian, Sajjad

doi:10.1016/j.difgeo.2016.01.003

Cited by 8 publications

(11 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Example 3.2 (Example 5.9 in [Lak16]). In Example 5.9 in [Lak16], Lakzian has shown that there are metrics g j on the sphere S 3 with positive scalar curvature such that the family of rotationally symmetric Riemannian manifolds M j = (S 3 , g j ) has the SWIF limit round sphere S 3 , and the Gromov-Hausdorff limit S 3 ⊔ [0, 1], the round sphere S 3 with an interval of length 1 attached to it. Actually, these M j satisfy all hypotheses in Theorem 1.3.…”

Section: Examplesmentioning

confidence: 99%

See 1 more Smart Citation

A compactness theorem for rotationally symmetric Riemannian manifolds with positive scalar curvature

Park

Tian

Wang

2018

Pure and Applied Mathematics Quarterly

View full text Add to dashboard Cite

Gromov and Sormani conjectured that sequences of compact Riemannian manifolds with nonnegative scalar curvature and area of minimal surfaces bounded below should have subsequences which converge in the intrinsic flat sense to limit spaces which have nonnegative generalized scalar curvature and Euclidean tangent cones almost everywhere. In this paper we prove this conjecture for sequences of rotationally symmetric warped product manifolds. We show that the limit spaces have H 1 warping function that has nonnegative scalar curvature in a weak sense, and have Euclidean tangent cones almost everywhere.

show abstract

Section: Examplesmentioning

confidence: 99%

“…For δ j > 0 with δ j → 0 as j → ∞ and suitable choice of L, M j = (S 3 , g δ j ) give the example. For more details about this example we refer to [Lak16].…”

Section: Examplesmentioning

confidence: 99%

A compactness theorem for rotationally symmetric Riemannian manifolds with positive scalar curvature

Park

Tian

Wang

2018

Pure and Applied Mathematics Quarterly

View full text Add to dashboard Cite

show abstract

“…It has been applied by Lakzian to study Ricci flow through neck pinch singularities in [Lak15a]. It has been applied by Lakzian in [Lak16] to prove F convergence of sequences of manifolds which converge smoothly away from singular sets. In particular, Lakzian proves that if M j = (M, g j ) be a sequence of compact oriented Riemannian manifolds with a set S ⊂ M such that H n−1 (S ) = 0, and a connected precompact exhaustion, W k , of M \ S satisfying…”

Section: Smoothmentioning

confidence: 99%

“…Other theorems about smooth convergence away from singular sets are proven in [Lak16] as well. KAHLER 5.3.…”

Section: Smoothmentioning

confidence: 99%

Scalar Curvature and Intrinsic Flat Convergence

Sormani¹

2017

Measure Theory in Non-Smooth Spaces

View full text Add to dashboard Cite

Herein we present open problems and survey examples and theorems concerning sequences of Riemannian manifolds with uniform lower bounds on scalar curvature and their limit spaces. Examples of Gromov and of Ilmanen which naturally ought to have certain limit spaces do not converge with respect to smooth or Gromov-Hausdorff convergence. Thus we focus here on the notion of Intrinsic Flat convergence, developed jointly with Wenger. This notion has been applied successfully to study sequences that arise in General Relativity. Gromov has suggested it should be applied in other settings as well. We first review intrinsic flat convergence, its properties, and its compactness theorems, before presenting the applications and the open problems.

show abstract

“…The intrisic flat distance has also been studied by Jaramillo et al [13], Lakzian [14], Munn [18], Portegies [23] and the author [21], to mention a few. Gromov proposes to use intrinsic flat distance to solve some of his conjectures [10], [11].…”

Section: Introductionmentioning

confidence: 99%

The Sormani–Wenger intrinsic flat convergence of Alexandrov spaces

Perales²

2018

J. Topol. Anal.

View full text Add to dashboard Cite

We study sequences of integral current spaces (X j , d j , T j ) such that the integral current structure T j has weight 1 and no boundary and, all (X j , d j ) are closed Alexandrov spaces with curvature uniformly bounded from below and diameter uniformly bounded from above. We prove that for such sequences either their limits collapse or the Gromov-Hausdorff and Sormani-Wenger Intrinsic Flat limits agree. The latter is done showing that the lower n dimensional density of the mass measure at any regular point of the Gromov-Hausdorff limit space is positive by passing to a filling volume estimate.In an appendix we show that the filling volume of the standard n dimensional integral current space coming from an n dimensional sphere of radius r > 0 in Euclidean space equals r n times the filling volume of the n dimensional integral current space coming from the n dimensional sphere of radius 1.

show abstract

On diameter controls and smooth convergence away from singularities

Cited by 8 publications

References 18 publications

A compactness theorem for rotationally symmetric Riemannian manifolds with positive scalar curvature

A compactness theorem for rotationally symmetric Riemannian manifolds with positive scalar curvature

Scalar Curvature and Intrinsic Flat Convergence

The Sormani–Wenger intrinsic flat convergence of Alexandrov spaces

Contact Info

Product

Resources

About