arXiv research

The article explores the fundamental gap in Bakry-Emery geometry.

Study shows fundamental gap of horoconvex domains in hyperbolic space has no positive lower bound.

New convex domains in hyperbolic space can have lower fundamental gap than constant potentials.

Fundamental gap vanishes for convex domains in hyperbolic space.

Study calculates eigenvalues and eigenfunctions for spherical triangles and finds fundamental gap behavior.

Study simplicial volume for fixed fundamental groups, finding gaps.

The study improves fundamental gap estimates for surfaces with non-constant positive curvature.

The article proves a conjecture about the fundamental gap for horoconvex domains in hyperbolic space.

Proves a fundamental gap lower bound for horoconvex domains in hyperbolic space.

In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap (the difference between the first two eigenvalues) conjecture for convex domains in the Euclidean space and conjectured similar results holds for spaces with constant sectional curvature. We prove the conjecture for the sphere. Namely wh…

Probabilistic method proves gap estimates on sphere.

Log-concavity of eigenfunctions on curved surfaces is proven, leading to fundamental gap estimates.

Negative curvature restricts the gap between the first and second eigenvalues of convex domains.

Formula found for surfaces in Sol_3, leading to gap results.

We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spectral gap) of a Schrödinger operator with convex potential and Dirichlet boundary data on a convex domain is bounded below by the spectral gap on an interval of the same diameter with zero poten…

Paper proves Simon's third gap conjecture for minimal surfaces in spheres.

Motivated by an example of Shih, we compute the fundamental gap of a family of convex domains in the hyperbolic plane $\mathbb H^2$, showing that for some of them $λ_2 - λ_1 < \frac{3π^2}{D^2}$, where $D$ is the diameter of the domain and $λ_1$, $λ_2$ are the first and second Dirichlet eigenvalues of the Laplace operat…

The study provides energy estimates for Willmore surfaces and derives a gap statement.

Study concavity of solutions to elliptic equations under conformal deformations.

Flat minimal tori counterexamples refute Lu's second-gap conjecture.

New study on neural network calibration, linking it to generalization gap.

Study bounds on self-shrinkers with bounded HA for applications.

For an immersed Lagrangian submanifold, let $\check{A}$ be the Lagrangian trace-free second fundamental form. In this note we consider the equation $\nabla^*T=0$ on Lagrangian surfaces immersed in $\mathbb{C}^2$, where $T=-2\nabla^*(\check{A}\lrcornerω)$, and we prove a gap theorem for the Whitney sphere as a solution …

Estimates the mass gap for domains with integral Ricci curvature bounds.

The paper proves gap results for self-shrinkers in $r$-mean curvature flow.

New analysis shows a gap between Gaussian RKHS and neural networks on unbounded domains.

We consider critical points of the functionals $Π$ and $Ψ$ defined as the global $L^2$-norm of the second fundamental form and mean curvature vector of isometric immersions of compact Riemannian manifolds into a background Riemannian manifold, respectively, as functionals over the space of deformations of the immersion…

Crowdsourcing platforms provide marketplaces where task requesters can pay to get labels on their data. Such markets have emerged recently as popular venues for collecting annotations that are crucial in training machine learning models in various applications. However, as jobs are tedious and payments are low, errors …

In this note, we give a new and simple proof of a result in {\cite{DX1}} which states that any smooth complete self-shrinker in $\mathbb{R}^3$ with second fundamental form of constant length must be a generalized cylinder $\mathbb{S}^k \times \mathbb{R}^{2-k}$ for some $k\leq2$. Moreover, we prove a gap theorem for smo…

In [SWW], S. Seto, L. Wang and G. Wei proved that the gap between the first two Dirichlet eigenvalues of a convex domain in the unit sphere is at least as large as that for an associated operator on an interval with the same diameter, provided that the domain has the diameter at most $π/2$. In this paper, we extend Set…

Study shows cooperation can reduce investment risk and price gaps.

In this paper we establish a gap phenomenon for immersed surfaces with arbitrary codimension, topology and boundaries that satisfy one of a family of systems of fourth-order anisotropic geometric partial differential equations. Examples include Willmore surfaces, stationary solitons for the surface diffusion flow, and …

Simplified proofs for splitting homotopy idempotents.

We show that, among free boundary minimal surfaces in the unit ball in the three-dimensional Euclidean space, the flat equatorial disk and the critical catenoid are characterised by a pinching condition on the length of their second fundamental form.

Paper uses 3-circle theorem to study Willmore surfaces and prove decay estimates.

Stable commutator length scl_G(g) of an element g in a group G is an invariant for group elements sensitive to the geometry and dynamics of G. For any group G acting on a tree, we prove a sharp bound scl_G(g)>=1/2 for any g acting without fixed points, provided that the stabilizer of each edge is relatively torsion-fre…

We extend to higher codimension earlier characterization of the equatorial disk and the critical catenoid by a pinching condition on the length of their second fundamental form among free boundary minimal surfaces in the three dimensional Euclidean ball due to L. Ambrozio and I. Nunes.

Study establishes monodromy equivalence for Lamé-type equations and constructs cone spherical metrics.

These are expanded notes of a course given in Grenoble in june 2004. After a brief description of the harmonic map proof of Margulis' superrigidity and arithmeticity theorems, it is shown how the method might generalize to fundamental groups of simplicial complexes whose links have large enough nonlinear spectral gaps,…

In part I it was shown that for each k>0 the generalized Sato-Levine invariant detects a gap between k-quasi-isotopy of link and peripheral structure preserving isomorphism of the finest quotient G_k of its fundamental group, `functorially' invariant under k-quasi-isotopy. Here we show that Cochran's derived invariant …

Deep learning models outperform classical methods in forecasting company fundamentals.

We study $λ$-hypersurfaces that are critical points of a Gaussian weighted area functional $\int_Σ e^{-\frac{|x|^2}{4}}dA$ for compact variations that preserve weighted volume. First, we prove various gap and rigidity theorems for complete $λ$-hypersurfaces in terms of the norm of the second fundamental form $|A|$. Sec…

Proves a pinching theorem for self-shrinkers of mean curvature flow.

Machine learning's predictive power is limited by sample size, as shown by the Limits-to-Learning Gap.

The study classifies complete self-shrinkers in Euclidean space.

Paper addresses eigenvector perturbation in small eigen-gap scenarios.

In this paper, we prove that the mean curvature blows up at the same rate as the second fundamental form at the first singular time $T$ of any compact, Type I mean curvature flow. For the mean curvature flow of surfaces, we obtain similar result provided that the Gaussian density is less than two. Our proofs are based …

Open manifolds with nonnegative Ricci curvature have virtually abelian fundamental groups if they escape from bounded balls at a small rate.