arXiv research

Study of Ricci flow convergence on surfaces with boundary.

We prove under suitable hypotheses that convergence of integral varifolds implies convergence of associated mod 2 flat chains and subsequential convergence of associated integer-multiplicity rectifiable currents. The convergence results imply restrictions on the kinds of singularities that can occur in mean curvature f…

We investigate compactness phenomena involving free boundary minimal hypersurfaces in Riemannian manifolds of dimension less than eight. We provide natural geometric conditions that ensure strong one-sheeted graphical subsequential convergence, discuss the limit behaviour when multi-sheeted convergence happens and deri…

We study the behaviour of the Kähler-Ricci flow on projective bundles. We show that if the initial metric is in a suitable Kähler class, then the fibers collapse in finite time and the metrics converge subsequentially in the Gromov-Hausdorff sense to a metric on the base.

Given a sequence of properly embedded minimal surfaces in a $3$-manifold with local bounds on area and genus, we prove subsequential convergence, smooth away from a discrete set, to a smooth embedded limit surface, possibly with multiplicity, and we analyze what happens when one blows up the surfaces near a point where…

Given a Hermitian line bundle $L\to M$ over a closed, oriented Riemannian manifold $M$, we study the asymptotic behavior, as $ε\to 0$, of couples $(u_ε,\nabla_ε)$ critical for the rescalings \begin{align*} &E_ε(u,\nabla)=\int_M\Big(|\nabla u|^2+ε^2|F_\nabla|^2+\frac{1}{4ε^2}(1-|u|^2)^2\Big) \end{align*} of the self-dua…

We establish the short-time existence of the Ricci flow on surfaces with a finite number of conic points, all with cone angle between 0 and $2π$, where the cone angles remain fixed or change in some smooth prescribed way. For the angle-preserving flow we prove long-time existence and convergence. When the Troyanov angl…

Compactness theory for super Ricci flows provides convergence results.

We consider the continuous $W^{2,d}$ immersions of $d$-dimensional hypersurfaces in $\mathbb{R}^{d+1}$ with second fundamental forms uniformly bounded in $L^d$. Two results are obtained: first, a family of such immersions is constructed, whose limit fails to be an immersion of a manifold. This addresses the endpoint ca…

We propose inertial versions of block coordinate descent methods for solving non-convex non-smooth composite optimization problems. Our methods possess three main advantages compared to current state-of-the-art accelerated first-order methods: (1) they allow using two different extrapolation points to evaluate the grad…

Compactness theorem for Fueter sections yields non-zero harmonic 1-forms.

Proposes BMME for optimizing nonsmooth nonconvex problems with block structure.

Study shows translators can have non-removable singularities at infinity but eventually converge to unique planes.

Uniform estimates lead to Gromov-Hausdorff limits for Hermitian minimal models.

This paper studies the convergence of penalized energy to harmonic maps in Riemannian manifolds.

Study examines Yang-Mills-Higgs energy convergence to codimension-three area functional.

This paper improves MI-based BCIs by applying transfer learning across all components.

The study finds critical points of Yang-Mills-Higgs energy on 3-manifolds.

For Ginzburg-Landau vortices, energy quantization holds only when density is less than 2.

Proves weak convergence equals mean convergence in GGC.

This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We begin with a description of three extrinsic notions which have been applied to study sequences of submanifolds in Euclidean…

The abstract discusses convergence properties of Lipschitz functions and sets defined by equations.

Study shows intrinsic timed Hausdorff convergence leads to Gromov-Hausdorff and big bang convergence.

Studied SGD convergence under weak conditions.

Study on convergence rate of $Q$-curvature flow in 6 dimensions.

The objective of this paper is to introduce the notion of generalized almost statistical (briefly, GAS) convergence of bounded real sequences, which generalizes the notion of almost convergence as well as statistical convergence of bounded real sequences. As a special kind of Banach limit functional, we also introduce …

Establishes geometric convergence of iterative optimization algorithms.

Uniform counting formulas for orthogeodesics in Kleinian groups converge.

The paper explores null distance convergence for warped product spacetimes.

New quasi-Newton method guarantees global superlinear convergence.

We investigate finite-time decoupled convergence in nonlinear two-time-scale stochastic approximation.

The article introduces a new convergence concept for Lorentzian spaces and applies it to generalized cones.

AdaBoost's classifier and margins converge to a known value.

Study shows gap between uniform convergence and test error in random feature models.

The Sinkhorn-Knopp derivatives converge with linear rate.

The paper examines convergence of distances in Lipschitz structures on manifolds.

We introduce a natural definition of $L^p$-convergence of maps, $p \ge 1$, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the $L^p$-convergence, we establish a theory of …

We prove a criterion of convergence in the augmented Teichmueller space that can be phrased in terms of convergence of the hyperbolic metrics or of quasiconformal convergence away from the nodes.

We develop a new approach to geometric quantization using the theory of convergence of metric measure spaces. Given a family of Kähler polarizations converging to a non-singular real polarization on a prequantized symplectic manifold, we show the spectral convergence result of $\bar{\partial}$-Laplacians, as well as th…

Sharp convergence theorem for sphere submanifolds proved.

Proves curvature tensor convergence for smoothable spaces.

DCDC calculates convergence rates for Markov chains using neural networks.

Develops clustering methods based on likelihood and convergence proved.

New technique explains convergence in ML models with data modifications.

Characterizes problems solvable via linear convergence algorithms.

Equivalence shown between two mathematical concepts for hyperbolic surfaces.

The paper provides convergence guarantees for multicalibration gradient boosting.

Study Gromov-Hausdorff convergence of metric pairs and tuples.