arXiv research

Mathematical formulas for elliptic curve integrals solve anomaly equations.

Global regularity proved for 4D Ricci flow with scalar curvature integral bound.

A new method integrates forms on Riemann surfaces, leading to modular forms.

The study shows how certain ODEs and integrals are regular under Borel summation.

New framework for logarithmically divergent integrals on manifolds with corners.

Extends Campanato theory to multi-valued functions for geometric variational problems.

Study the Lax equation in infinite-dimensional Lie algebras and Lie groups.

Generalizes Frobenius theorem to quasiconformal deformations.

We extend the Newlander-Nirenberg theorem to manifolds with almost complex structures that have somewhat less than Lipschitz regularity. We also discuss the regularity of local holomorphic coordinates in the integrable case, with particular attention to Lipschitz almost complex structures.

We discuss a recurrent geometrical method, due to Élie Cartan and von Weber ([1],[11]) enabling us to determine, step by step, the maximal integral manifolds of a not necessarily integrable nor regular Pfaffian system. The dimensions of such integral manifolds can, of course, vary from point to point but more so can va…

The paper examines partial regularity of Lipschitz solutions to minimal surface system.

We prove a regularity result for unit volume conformal metrics with integral scalar curvature bounds for $p>n/2$ and first eigenvalue of $Δ$ bounded from below by a constant $B > Λ_1(S^n,[g_{st.}]).$

This work is concerned with Mishchenko-Fomenko subalgebras and their restrictions to the adjoint orbits in a finite-dimensional complex semisimple Lie algebra. In this setting, it is known that each Mishchenko-Fomenko subalgebra restricts to a completely integrable system on every orbit in general position. We improve …

We show a higher order integrability theorem for distributions generated by a family of vector fields under a horizontal regularity assumption on their coefficients. We use as chart a class of almost exponential maps which we discuss in details

For a $2$-dimensional non-flat spray we associate a Berwald frame and a $3$-dimensional distribution that we call the Berwald distribution. The Frobenius integrability of the Berwald distribution characterises the Finsler metrizability of the given spray. In the integrable case, the sought after Finsler function is pro…

Regular Lie groups are infinite dimensional Lie groups with the property that smooth curves in the Lie algebra integrate to smooth curves in the group in a smooth way (an `evolution operator' exists). Up to now all known smooth Lie groups are regular. We show in this paper that regular Lie groups allow to push surprisi…

Integrates rough geometric forms on manifolds.

Study calculates first $p$-widths of unit disk.

The paper proves existence and partial regularity for Legendrian area-minimizing currents.

Regularizes persistent homology gradients for neural network integration.

The paper studies the smoothness of critical points of variational integrals on Hessian spaces.

Given a closed submanifold, or a compact regular domain, in euclidean space, we consider the Riesz energy defined as the double integral of some power of the distance between pairs of points. When this integral diverges, we compare two different regularization techniques (Hadamard's finite part and analytic continuatio…

Researchers find second-order estimates for $p$-Laplacian in RCD spaces.

SinSim improves self-supervised learning by integrating optimal transport into contrastive learning.

We show that the integral foliated simplicial volume of a connected compact oriented smooth manifold with a regular foliation by circles vanishes.

We provide a draft of a theory of geometric integration of rough differential forms which are generalizations of classical (smooth) differential forms to similar objects with very low regularity, for instance, involving Hölder continuous functions that may be nowhere differentiable. Borrowing ideas from the theory of r…

We study conditions for the integrability of the distribution defined on a regular Poisson manifold as the orthogonal complement (with respect to some (pseudo)-Riemannian metric) to the tangent spaces of the leaves of a symplectic foliation. Examples of integrability and non-integrability of this distribution are provi…

Study on smoothness of solutions to nonlinear equations on Riemannian manifolds.

The paper provides gradient estimates for solutions on manifolds with integral Ricci bounds.

A symplectic integration of a Poisson manifold $(M,Λ)$ is a symplectic groupoid $(Γ,η)$ which realizes the given Poisson manifold, i.e. such that the space of units $Γ_0$ with the induced Poisson structure $Λ_0$ is isomorphic to $(M,Λ)$. This notion was introduced by A. Weinstein in order to quantize Poisson manifolds …

We provide new bounds on a flux integral over the portion of the boundary of one regular domain contained inside a second regular domain, based on properties of the second domain rather than the first one. This bound is amenable to numerical computation of a flux through the boundary of a domain, for example, when ther…

Proposes a multivariate regression model for better analysis of multiple datasets.

We study the Hadamard finite part of divergent integrals of differential forms with singularities on submanifolds. We give formulae for the dependence of the finite part on the choice of regularization and express them in terms of a suitable local residue map. The cases where the submanifold is a complex hypersurface i…

We survey - by means of 20 examples - the concept of varifold, as generalised submanifold, with emphasis on regularity of integral varifolds with mean curvature, while keeping prerequisites to a minimum. Integral varifolds are the natural language for studying the variational theory of the area integrand if one conside…

Sharp criteria for 2-varifolds to be induced by smooth immersions.

Proves interior singular set dimension for area-minimizing currents in smooth submanifolds.

Extends method for solving certain hydrodynamic systems.

Paper introduces a differentiable regularizer for condition number to improve neural network stability.

LSTD is a popular algorithm for value function approximation. Whenever the number of features is larger than the number of samples, it must be paired with some form of regularization. In particular, L1-regularization methods tend to perform feature selection by promoting sparsity, and thus, are well-suited for high-dim…

The paper quantifies the regularity of attention operations.

We give two structural conditions on a codimension $1$ integral $n$-varifold with first variation locally summable to an exponent $p>n$ that imply the following: whenever each orientable portion of the $C^{1}$-embedded part of the varifold (which is non-empty by the Allard regularity theory) is stationarity and the $C^…

Integrates transitive Lie algebroids to Lie groupoids, explaining obstructions.

The paper proves regularity for varifolds with bounded anisotropic mean curvature.

Hopf's Umlaufsatz relates the total curvature of a closed immersed plane curve to its rotation number. While the curvature of a curve changes under local deformations, its integral over a closed curve is invariant under regular homotopies. A natural question is whether one can find some non-trivial densities on a curve…

Unique solutions found for Plateau problems in smooth and continuous calibrations.

Special Legendrian Integral Cycles in $S^5$ are the links of the tangent cones to Special Lagrangian integer multiplicity rectifiable currents in Calabi-Yau 3-folds. We show that such Special Legendrian Cycles are smooth except possibly at isolated points.

The behavior of geodesic curves on even seemingly simple surfaces can be surprisingly complex. In this paper we use the Hamiltonian formulation of the geodesic equations to analyze their integrability properties. In particular, we examine the behavior of geodesics on surfaces defined by the spherical harmonics. Using t…

Proposes a method for multi-view clustering that integrates consistent and complementary graph regularizers.