arXiv subjects

Raquel Perales

Publications and source records attributed to Raquel Perales.

25 records · Page 2Linked to original sources

arXiv2020

A generalized tetrahedral property

We present examples of metric spaces that are not Riemannian manifolds nor dimensionally homogeneous that satisfy the Tetrahedral Property. In spite of that, Euclidean cones over metric spaces with small diameter do not satisfy this property. We extend Sormani's Tetrahedral Property to a less restrictive property and prove that this generalized definition retains all the results of the original Tetrahedral Property proven by Portegies-Sormani: it provides a lower bound on the sliced filling volume and a lower bound on the volumes of balls. Thus sequences with uniform bounds on this Generalized Tetrahedral Property also have subsequences which converge in both the Gromov-Hausdorff and Sormani-Wenger Intrinsic Flat sense to the same non-collapsed and countably rectifiable limit space.

math.DG↗

arXiv2020

Stability of graphical tori with almost nonnegative scalar curvature

By works of Schoen-Yau and Gromov-Lawson any Riemannian manifold with nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov conjectured subconvergence of tori with respect to a weak Sobolev type metric when the scalar curvature goes to $0$. We prove flat and intrinsic flat subconvergence to a flat torus for noncollapsing sequences of $3$-dimensional tori $M_j$ that can be realized as graphs of certain functions defined over flat tori satisfying a uniform upper diameter bound and scalar curvature bounds of the form $R_{g_{M_j}} \geq -1/j$. We also show that the volume of the manifolds of the convergent subsequence converges to the volume of the limit space. We do so adapting results of Huang-Lee, Huang-Lee-Sormani and Allen-Perales-Sormani. Furthermore, our results also hold when the condition on the scalar curvature of a torus $(M, g_M)$ is replaced by a bound on the quantity $-\int_T \min\{R_{g_M},0\} d{\mbox{vol}_{g_T}}$, where $M=\mbox{graph}(f)$, $f: T \to \mathbb R$ and $(T,g_T)$ is a flat torus. Using arguments developed by Alaee, McCormick and the first named author after this work was completed, our results hold for dimensions $n \geq 4$ as well.

math.DG↗

arXiv2018

On the Sormani-Wenger Intrinsic Flat Convergence of Alexandrov Spaces

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 $1$.

math.MG↗

arXiv2017

Alexandrov Spaces with Integral Current Structure

We endow each closed, orientable Alexandrov space $(X, d)$ with an integral current $T$ of weight equal to 1, $\partial T = 0 and \set(T) = X$, in other words, we prove that $(X, d, T)$ is an integral current space with no boundary. Combining this result with a result of Li and Perales, we show that non-collapsing sequences of these spaces with uniform lower curvature and diameter bounds admit subsequences whose Gromov-Hausdorff and intrinsic flat limits agree.

math.DG↗

arXiv2014

Volumes and Limits of Manifolds with Ricci Curvature and Mean Curvature Bounds

We prove a laplacian comparison theorem in the barrier sense for the function distance to the boundary of Riemannian manifolds with nonnegative Ricci curvature, area and mean curvature of the boundary bounded above. As an application we get volume estimates, different to the ones obtained by Heintz-Karcher, and area estimates. With those estimates we prove Sormani-Wenger Intrinsic Flat (SWIF) compactness theorems for sequences of such manifolds with uniform bounds.

math.MG↗

arXiv2013

A survey on the Convergence of Manifolds with Boundary

This survey reviews precompactness theorems for classes of Riemannian manifolds with boundary. We begin with the works of Kodani, Anderson-Katsuda-Kurylev-Lassas-Taylor and Wong. We then present new results of Knox and the author with Sormani.

math.DG↗

arXiv2013

Sequences of Open Riemannian Manifolds with Boundary

We consider sequences of open Riemannian manifolds with boundary that have no regularity conditions on the boundary. To define a reasonable notion of a limit of such a sequence, we examine "$δ$ inner regions" which avoid the boundary by a distance $δ$. We prove Gromov-Hausdorff compactness theorems for sequences of these "$δ$ inner regions". We then build "glued limit spaces" out of the Gromov-Hausdorff limits of these $δ$ interior regions and study the properties of these glued limit spaces. Our main applications assume the sequence is noncollapsing and has nonnegative Ricci curvature. We include open questions.

math.MG↗