arXiv subjects

Ilaria Mondello

Publications and source records attributed to Ilaria Mondello.

12 recordsLinked to original sources

arXiv2026

Strong Kato limit can be branching

We provide an example of a non-collapsed strong Kato limit that is branching, essentially branching, and satisfies neither the $\mathrm{CD}(K,\infty)$ nor the $\mathrm{MCP}(K,N)$ conditions for any $K \in \mathbb{R}$ and $N \in [1,+\infty)$. In particular, this space is not a Ricci limit space. We also construct a compact non-collapsed strong Kato limit that cannot be obtained as Gromov-Hausdorff limit of closed Riemannian surfaces satisfying a uniform small $L^p$ bound à la Petersen--Wei.

math.DG↗

arXiv2025

Kato meets Bakry-Émery

We prove that any complete Riemannian manifold with negative part of the Ricci curvature in a suitable Dynkin class is bi-Lipschitz equivalent to a finite-dimensional $\mathrm{RCD}$ space, by building upon the transformation rule of the Bakry-Émery condition under time change. We apply this result to show that our previous results on the limits of closed Riemannian manifolds satisfying a uniform Kato bound carry over to limits of complete manifolds. We also obtain a weak version of the Bishop-Gromov monotonicity formula for manifolds satisfying a strong Kato bound.

math.DG↗

arXiv2024

Gromov-Hausdorff stability of tori under Ricci and integral scalar curvature bounds

We establish a nonlinear analogue of a splitting map into a Euclidean space, as a harmonic map into a flat torus. We prove that the existence of such a map implies Gromov-Hausdorff closeness to a flat torus in any dimension. Furthermore, Gromov-Hausdorff closeness to a flat torus and an integral bound {on $r_M(x)$, the smallest eigenvalue of the Ricci tensor $\text{ric}_x$ in $x$}, imply the existence of a harmonic splitting map. Combining these results with Stern's inequality, we provide a new Gromov-Hausdorff stability theorem for flat $3$-tori. The main tools we employ include the harmonic map heat flow, Ricci flow, and both Ricci limits and RCD theories.

math.DG↗

arXiv2022

Torus stability under Kato bounds on the Ricci curvature

We show two stability results for a closed Riemannian manifold whose Ricci curvature is small in the Kato sense and whose first Betti number is equal to the dimension. The first one is a geometric stability result stating that such a manifold is Gromov-Hausdorff close to a flat torus. The second one states that, under a stronger assumption, such a manifold is diffeomorphic to a torus: this extends a result by Colding and Cheeger-Colding obtained in the context of a lower bound on the Ricci curvature.

math.DG↗

arXiv2022

Limits of manifolds with a Kato bound on the Ricci curvature. II

We prove that metric measure spaces obtained as limits of closed Riemannian manifolds with Ricci curvature satisfying a uniform Kato bound are rectifiable. In the case of a non-collapsing assumption and a strong Kato bound, we additionally show that for any $α\in (0,1)$ the regular part of the space lies in an open set with the structure of a $\mathcal{C}^α$-manifold.

math.DG↗

arXiv2022

An upper bound on the revised first Betti number and a torus stability result for RCD spaces

We prove an upper bound on the rank of the abelianised revised fundamental group (called "revised first Betti number") of a compact $RCD^{*}(K,N)$ space, in the same spirit of the celebrated Gromov-Gallot upper bound on the first Betti number for a smooth compact Riemannian manifold with Ricci curvature bounded below. When the synthetic lower Ricci bound is close enough to (negative) zero and the aforementioned upper bound on the revised first Betti number is saturated (i.e. equal to the integer part of $N$, denoted by $\lfloor N \rfloor$), then we establish a torus stability result stating that the space is $\lfloor N \rfloor$-rectifiable as a metric measure space, and a finite cover must be mGH-close to an $\lfloor N \rfloor$-dimensional flat torus; moreover, in case $N$ is an integer, we prove that the space itself is bi-Hölder homeomorphic to a flat torus. This second result extends to the class of non-smooth $RCD^{*}(-δ, N)$ spaces a celebrated torus stability theorem by Colding (later refined by Cheeger-Colding).

math.DG↗

arXiv2021

Limits of manifolds with a Kato bound on the Ricci curvature

We study the structure of Gromov-Hausdorff limits of sequences of Riemannian manifolds $\{(M_α^n,g_α)\}_{α\in A}$ whose Ricci curvature satisfies a uniform Kato bound. We first obtain Mosco convergence of the Dirichlet energies to the Cheeger energy and show that tangent cones of such limits satisfy the $\mathrm{RCD}(0,n)$ condition. When assuming a non-collapsing assumption, we introduce a new family of monotone quantities, which allows us to prove that tangent cones are also metric cones. We then show the existence of a well-defined stratification in terms of splittings of tangent cones. We finally prove volume convergence to the Hausdorff $n$-measure.

math.DG↗

arXiv2019

Non-existence of Yamabe minimizers on singular spheres

We prove that a minimizer of the Yamabe functional does not exist for a sphere $\mathbb{S}^n$ of dimension $n \geq 3$, endowed with a standard edge-cone spherical metric of cone angle greater than or equal to $4π$, along a great circle of codimension two. When the cone angle along the singularity is smaller than $2π$, the corresponding metric is known to be a Yamabe metric, and we show that all Yamabe metrics in its conformal class are obtained from it by constant multiples and conformal diffeomorphisms preserving the singular set.

math.DG↗

arXiv2019

Sphere theorems for RCD and stratified spaces

We prove topological sphere theorems for RCD(n-1, n) spaces which generalize Colding's results and Petersen's result to the RCD setting. We also get an improved sphere theorem in the case of Einstein stratified spaces.

math.DG↗

arXiv2018

Stratified spaces and synthetic Ricci curvature bounds

We prove that a compact stratied space satises the Riemannian curvature-dimension condition RCD(K, N) if and only if its Ricci tensor is bounded below by K $\in$ R on the regular set, the cone angle along the stratum of codimension two is smaller than or equal to 2$π$ and its dimension is at most equal to N. This gives a new wide class of geometric examples of metric measure spaces satisfying the RCD(K, N) curvature-dimension condition, including for instance spherical suspensions, orbifolds, K{ä}hler-Einstein manifolds with a divisor, Einstein manifolds with conical singularities along a curve. We also obtain new analytic and geometric results on stratied spaces, such as Bishop-Gromov volume inequality, Laplacian comparison, L{é}vy-Gromov isoperimetric inequality.

math.DG↗

arXiv2015

An Obata singular theorem for stratified spaces

Consider a stratified space with a positive Ricci lower bound on the regular set and no cone angle larger than 2$π$. For such stratified space we know that the first non-zero eigenvalue of the Laplacian is larger than or equal to the dimension. We prove here an Obata rigidity result when the equality is attained: the lower bound of the spectrum is attained if and only if the stratified space is isometric to a spherical suspension. Moreover, we show that the diameter is at most equal to $π$, and it is equivalent for the diameter to be equal to $π$ and for the first non-zero eigenvalue of the Laplacian to be equal to the dimension. We finally give a consequence of these results related to the Yamabe problem. Consider an Einstein stratified space without cone angles larger than 2$π$: if there is a metric conformal to the Einstein metric and with constant scalar curvature, then it is an Einstein metric as well. Furthermore, if its conformal factor is not a constant, then the space is isometric to a spherical suspension.

math.DG↗

arXiv2014

The Local Yamabe Constant of Einstein Stratified Spaces

On a compact stratified space (X, g) there exists a metric of constant scalar curvature in the conformal class of g, if the scalar curvature satisfies an integrability condition and if the Yamabe constant of X is strictly smaller than the local Yamabe constant , another conformal invariant introduced in the recent work of K. Akutagawa, G. Carron and R. Mazzeo. Such invariant depends on the local structure of X, in particular on the links, but its explicit value is not known. We are going to show that if the links satisfy a Ricci positive lower bound, then we can compute the local Yamabe constant. In order to achieve this, we prove a lower bound for the spectrum of the Laplacian, by extending a well-known theorem by Lichenrowicz, and a Sobolev inequality, inspired by a result due to D. Bakry. Furthermore, we prove the existence of an Euclidean isoperimetric inequality on particular stratified space, with one stratum of codimension 2 and cone angle bigger than 2$π$.

math.DG↗