arXiv research

We present a compensated compactness theorem in Banach spaces established recently, whose formulation is originally motivated by the weak rigidity problem for isometric immersions of manifolds with lower regularity. As a corollary, a geometrically intrinsic div-curl lemma for tensor fields on Riemannian manifolds is ob…

We are concerned with the global weak rigidity of the Gauss-Codazzi-Ricci (GCR) equations on Riemannian manifolds and the corresponding isometric immersions of Riemannian manifolds into the Euclidean spaces. We develop a unified intrinsic approach to establish the global weak rigidity of both the GCR equations and isom…

Paper extends Weyl's lemma to RCD(K,N) spaces.

This paper is devoted to give a complete unified study of several weak forms of $\ddb-$Lemma on compact complex manifolds.

Dense neural networks can't approximate all functions.

The study proves a theorem on Riemannian manifolds for wedge products of weakly convergent differential forms.

Extends Margulis Lemma to RCD(K,N) spaces.

We prove the Focal Index Lemma and the Rauch and Berger comparison theorems on a weak Riemannian Hilbert manifold with a smooth Levi-Civita connection and we apply these results to the free loop spaces of a compact manifold with the L^2 metrics

We prove a Schwarz-type lemma for noncompact manifolds with possibly noncompact boundary. The result is a consequence of a suitable form of the weak maximum principle of independent interest. The paper is enriched with applications to conformal deformations of noncompact manifolds with boundary, among them a generaliza…

In 1982, Uhlenbeck \cite {U2} established the well-known gauge fixing theorem, which has played a fundamental role for Yang-Mills theory. In this paper, we apply the idea of Uhlenbeck to establish a parabolic type of gauge fixing theorems for the Yang-Mills flow and prove existence of a weak solution of the Yang-Mills …

Stein's method (Stein, 1973; 1981) is a powerful tool for statistical applications and has significantly impacted machine learning. Stein's lemma plays an essential role in Stein's method. Previous applications of Stein's lemma either required strong technical assumptions or were limited to Gaussian distributions with …

Paper proves optimal decomposition for matrix fields, reducing convex integration steps.

We give several applications of a lemma on completeness used by Osserman to show the meromorphicity of Weierstrass data for complete minimal surfaces with finite total curvature. Completeness and weak completeness are defined for several classes of surfaces which admit singular points. The completeness lemma is a usefu…

In this paper we establish two boundary versions of the Schwarz lemma. The first is for general holomorphic self maps of bounded convex domains with $C^2$ boundary. This appears to be the first boundary Schwarz lemma for general holomorphic self maps that requires no strong pseudoconvexity or finite type assumptions. T…

Develops higher arity VC theory and characterizes PAC learning in product spaces.

The Hopf Lemma for second order elliptic operators is proved to hold in domains with $C^{1,α}$, and even less regular, boundaries. It need not hold for $C^1$ boundaries. Corresponding results are proved for second order parabolic operators.

We prove a sharp version of the Hopf boundary point lemma for Black-Scholes type equations. We also investigate the existence and the regularity of the spatial derivative of the solutions at the spatial boundary.

We prove weak and strong maximum principles, including a Hopf lemma, for smooth subsolutions to equations defined by linear, second-order, partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the smooth subsolutions along this boundary…

Paper proves regularity and existence of Riemannian splines.

Defines weak geodesics on specific subsets of manifolds.

Quantum computing techniques improve graph analysis and community detection.

We consider a general path-dependent version of the hedging problem with price impact of Bouchard et al. (2019), in which a dual formulation for the super-hedging price is obtained by means of PDE arguments, in a Markovian setting and under strong regularity conditions. Using only probabilistic arguments, we prove, in …

Extends optimal regularity and Uhlenbeck compactness to non-Riemannian manifolds.

We prove a Morse Lemma for coarsely regular quasigeodesics in nonpositively curved symmetric spaces and euclidean buildings X. The main application is a simpler coarse geometric characterization of Morse subgroups of the isometry groups Isom(X) as undistorted subgroups which are coarsely uniformly regular. We show furt…

The paper proves properties of curves in Riemannian manifolds.

Smooth Yang-Mills fields proved in supercritical dimensions.

Study proves existence of weak mean curvature flow with contact angle.

Study on fourth order Lamm-Riviere system for biharmonic mappings in 4D.

In this note we discuss the fundamental groups and diameters of positively Ricci curved $n$-manifolds. We use a method combining the results about equivarient Hausdorff convergence developed by Fukaya and Yamaguchi with the Ricci version of splitting theorem by Cheeger and Colding to give new information on the topolog…

Study canonical deformations of complex forms and their cohomology properties.

New regularization method corrects over-shrinkage in small data regression.

Sharp regularity for Pfaff system leads to isometric immersions in arbitrary dimensions.

Study optimal transport on globally hyperbolic spacetimes, focusing on weak Kantorovich potentials' regularity.

We review recent work on the Einstein equations of general relativity when the curvature is defined in a weak sense. Weakly regular spacetimes are constructed, in which impulsive gravitational waves, as well as shock waves, propagate.

New method for efficient graph learning on large graphs.

We consider the Kähler-Ricci flow $\frac{\partial}{\partial t}g_{i\bar{j}} = g_{i\bar{j}} - R_{i\bar{j}}$ on a compact Kähler manifold $M$ with $c_1(M) > 0$, of complex dimension $k$. We prove the $ε$-regularity lemma for the Kähler-Ricci flow, based on Moser's iteration. Assume that the Ricci curvature and $\int_M |\r…

Proves higher regularity for anisotropic inverse mean curvature flow.

Study proves uniqueness of Yang-Mills field tangent cones in arbitrary dimensions.

New boundary condition for weak inverse mean curvature flow in bounded domains.

We prove existence and regularity of minimisers for the Canham-Helfrich energy in the class of weak (possibly branched and bubbled) immersions of the $2$-sphere. This solves (the spherical case) of the minimisation problem proposed by Helfrich in 1973, modelling lipid bilayer membranes. On the way to prove the main res…

Researchers derived formulas for joint moments of elliptical distributions.

Method learns graphons from graphs via Gromov-Wasserstein barycenters.

In this paper we study constant scalar curvature equation (CSCK), a nonlinear fourth order elliptic equation, and its weak solutions on Kähler manifolds. We first define a notion of weak solution of CSCK for an $L^\infty$ Kähler metric. The main result is to show that such a weak solution (with uniform $L^\infty$ bound…

This work is a short, self-contained introduction to subriemannian geometry with special emphasis on Chow's Theorem. As an application, a regularity result for the Poincaré Lemma is presented. At the beginning, the definitions of a subriemannian geometry, horizontal vector fields and horizontal curves are given. Then t…

Paper explores weak solutions' regularity in critical dimensions without conservation law.

We prove existence and regularity of minimizers for Hölder densities over general surfaces of arbitrary dimension and codimension in \(\R^n \), satisfying a cohomological boundary condition, providing a natural dual to Reifenberg's Plateau problem. We generalize and extend methods of Reifenberg, Besicovitch, and Adams,…

Proves and tests methods for learning time-series with breaks.

We consider the local analytic behavior for a family of holomorphic differentials on a family of degenerating annuli. Three results and discussion are presented. The first is the normal families Lemma 1. The second is an isomorphism of sheaves, formula (3), giving a direct description of families of regular $k$-differe…