arXiv research

In this paper, we give a proof of the quantitative Morse theorem stated by {Y. Yomdin} in \cite{Y1}. The proof is based on the quantitative Sard theorem, the quantitative inverse function theorem and the quantitative Morse lemma.

Proves quantitative Alexandrov theorem for capillary surfaces.

We obtain sharp quantitative Laplacian upper and lower estimates under no assumption on curvatures. As a result, we derive quantitative Laplacian, area and volume comparison theorems for tubes in Riemannian and Kähler manifolds under weak integral curvature assumptions. We also give some applications, such as a general…

Proves theorem for Riemannian manifolds, extending previous work.

The study proves a quantitative functional CLT for neural networks with smooth activation functions.

Proves a quantitative index theorem for positive scalar curvature metrics.

We prove a quantitative version of Obata's Theorem involving the shape of functions with null mean value when compared with the cosine of distance functions from single points. The deficit between the diameters of the manifold and of the corresponding sphere is bounded likewise. These results are obtained in the genera…

Paper analyzes arbitrage in uncertain markets, providing quantitative asset pricing.

The study examines hypersurfaces close to constant mean curvature and their proximity to spheres.

Study geodesic Lie groups' convergence to limits with quantitative estimates.

Quantum neural networks can approximate noisy functions accurately.

Study connects manifold complexity to scalar curvature bounds.

The Alexandrov Soap Bubble Theorem asserts that the distance spheres are the only embedded closed connected hypersurfaces in space forms having constant mean curvature. The theorem can be extended to more general functions of the principal curvatures $f(k_1,\ldots,k_{n-1})$ satisfying suitable conditions. In this paper…

Quantifies the crossing number of knots based on genus and braid index.

The paper proves density and positive mass theorems for incomplete manifolds.

Quantifies Schur's theorem for curves in CAT(k) spaces.

We prove the following quantitative version of the celebrated Soap Bubble Theorem of Alexandrov. Let $S$ be a $C^2$ closed embedded hypersurface of $\mathbb{R}^{n+1}$, $n\geq1$, and denote by $osc(H)$ the oscillation of its mean curvature. We prove that there exists a positive $\varepsilon$, depending on $n$ and upper …

Proves positive mass theorem for hyperbolic manifolds with ends.

We obtain a Bernstein theorem for special Lagrangian graphs in n-dimensional complex space for arbitrary n only assuming bounded slope, but no quantitative restriction.

Quantifies closeness of special Lagrangians under Floer conditions.

We prove a Frobenius-type theorem for singular distributions generated by a family of locally Lipschitz continuous vector fields satisfying almost everywhere a quantitative finite type condition.

We review classical results where the method of the moving planes has been used to prove symmetry properties for overdetermined PDE's boundary value problems (such as Serrin's overdetermined problem) and for rigidity problems in geometric analysis (like Alexandrov soap bubble Theorem), and we give an overview of some r…

Sharp Talenti-type comparison theorem for p-Laplacian on RCD(K,N) spaces.

The paper proves a quantitative positive mass theorem for spin manifolds with distance estimates.

In this article we extend to generic $p$-energy minimizing maps between Riemannian manifolds a regularity result which is known to hold in the case $p=2$. We first show that the set of singular points of such a map can be quantitatively stratified: we classify singular points based on the number of almost-symmetries of…

This paper studies neural network operators and their convergence properties.

Quantitative rigidity theorem for Alexandrov spaces with curvature bounds.

This paper proves a new, more precise version of Cheng's maximal diameter theorem for manifolds with positive Ricci curvature.

The paper sharpens a theorem about surfaces with zero Gaussian curvature.

We study generalizations of Reifenberg's Theorem for measures in $\mathbb R^n$ under assumptions on the Jones' $β$-numbers, which appropriately measure how close the support is to being contained in a subspace. Our main results, which holds for general measures without density assumptions, give effective measure bounds…

Proves a limit on hyperplanes in complex manifolds.

We prove a qualitative and a quantitative stability of the following rigidity theorem: an anisotropic totally umbilical closed hypersurface is the Wulff shape. Consider $n \geq 2$, $p\in (1, \, +\infty)$ and $Σ$ an $n$-dimensional, closed hypersurface in $\mathbb{R}^{n+1}$, boundary of a convex, open set. We show that …

We establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points is bounded from below by a universal positive number, (ii) the solutions are very close to a single stand…

New theorem shows shapes close to balls, flow converges to balls in 2D and 3D.

Study almost rigidity of super Ricci flow with non-negative Muller quantity.

In this paper, we quantitative convergence in $W_2$ for a family of Langevin-like stochastic processes that includes stochastic gradient descent and related gradient-based algorithms. Under certain regularity assumptions, we show that the iterates of these stochastic processes converge to an invariant distribution at a…

The study shows how discrete graphs can resemble hypercube structures under certain curvature conditions.

We give soft, quantitatively optimal extensions of the classical Sphere Theorem, Wilking's connectivity principle and Frankel's Theorem to the context of ${k}$-th Ricci curvature. The hypotheses are soft in the sense that they are satisfied on sets of metrics that are open in the $C^{2}$-topology.

The paper analyzes harmonic maps to flat model spaces to prove geometric stability of the positive mass theorem.

There is a gap in the proof of the main theorem in the article [ShCh13a] on optimal bounds for the Morse lemma in Gromov-hyperbolic spaces. We correct this gap, showing that the main theorem of [ShCh13a] is correct. We also describe a computer certification of this result.

We prove a quantitative openness theorem for $C^1$ submersions under suitable assumptions on the differential. We then apply our result to a class of exponential maps appearing in Carnot-Carathéodory spaces and we improve a classical completeness result by Palais.

We introduce a technique for proving quantitative representation stability theorems for sequences of representations of certain finite linear groups over a field of characteristic zero. In particular, we prove a vanishing result for higher syzygies of VIC- and SI-modules, which can be thought of as a weaker version of …

Quantifies nearly spherical subsets in complex ball geometry.

Proves spacetime positive mass theorem for spin initial data sets with arbitrary ends.

Given a finite collection of $C^1$ complex vector fields on a $C^2$ manifold $M$ such that they and their complex conjugates span the complexified tangent space at every point, the classical Newlander-Nirenberg theorem gives conditions on the vector fields so that there is a complex structure on $M$ with respect to whi…

Improved optimal regularity for harmonic almost complex structures.

This is an appendix to arXiv:1610.04888, "Quantitative null-cobordism", which improves one of the main results of that paper to a near-sharp one. It is not a self-contained paper.

Sharp stability of Möbius group among sphere-valued maps proved in arbitrary dimensions.